eml.berkeley.educshannon/debreu/taub.pdf · the dynamics of strategic information flows/p. seiler,...

The Dynamics of Strategic Information Flows/P. Seiler, B. Taub dsif˙0805:9-27-05

THE DYNAMICS OF STRATEGIC INFORMATION FLOWS IN STOCK MARKETSP. Seiler and B. Taub∗

1. INTRODUCTION

In stock markets today, thousands of traders aggressively and continually trade in the same room. Theytrade hundreds of different companies’ stocks, as well as derivatives. The prices and quantities transactedare simultaneously displayed to all traders.

The advent of computer technology makes possible the dispersion of these trades, but it has, if anything,increased the numbers of traders who interact in the same markets, increased the number of stocks traded,increased the speed and volume of trade, and increased the amount of information that is available to traders.

Why is it desirable to have so many people proximately trading? Why are so many distinct stockstraded in the same markets? Our thesis is that stock markets are a means of aggregating information. Buthow does information about a stock relate to the price and trading in another, and how does informationdisperse across stocks as time passes?

We answer these questions. We show how the dynamics of information flows in multi-asset stock marketsare different from the dynamics of information flows when trade is restricted to a single stock. We showthat speculators obtain useful information from observing the prices of all stocks, not merely stocks aboutwhich they have private information. And we show how information about one stock dynamically diffusesinto information about all stocks.

Our starting point is to assume that individual speculators have private information about the truevalues of the assets they trade. These speculators are strategic: they structure their trades taking accountof their heterogeneously informed rivals, and this affects the speed with which their private information isembodied in asset prices.

The speculators’ abilities to observe prices means that information embodied in the price of one assetcan influence their trading strategies in another asset. Concretely, speculators use information from theprice of Glaxo to adjust their trading on Merck. In choosing order flow for Merck, speculators must usetheir private information about the valuation process on Merck, private information about Glaxo’s valuationprocess that is correlated with it, information from the price of Merck, and information from the price ofGlaxo. On top of this complicated set of inputs, the speculators must be cognizant of the effects of theirown trades on the prices not only of Merck but of Glaxo indirectly, as they also affect their profits.

Speculators use price information dynamically: the information in the prices of both Merck and Glaxocontinues to be useful for trading Merck stock after its initial realization, and strategic effects of trades onprices also last. The way speculators weight the history of Glaxo prices for trades in Merck is potentiallydifferent from the weighting of the history of Glaxo prices for trading directly in Glaxo: speculators mightwant to discount the effect of older prices differently for the two trades.

The methods we employ here allow us to show that this cross-asset information in fact decays extraor-dinarily rapidly. Our main numerical finding is that multiple assets increase trading intensity and speed upthe incorporation of information through prices. This happens because speculators and especially marketmakers have additional channels through which to gain information, and correspondingly, to release thatinformation.

The faster release of information in a multi-asset environment goes hand in hand with reduced profits forinformed speculators: they compete more heavily on their private information and this lowers their aggregateprofits. This reduced profit is a boon to the liquidity speculators, whose losses must in equilibrium equalthe profits of informed speculators, and we can loosely interpret this outcome as increased liquidity.

Previous models in this literature, such as Kyle (1985), Back, Cao and Willard (2000), Foster andViswanathan (1996), and Admati and Pfleiderer (1998) have presumed a single asset. Moreover, the literature

∗ Honeywell Corporation and University of Illinois, respectively. Taub thanks the National Science Foundation for supportof this research under grant # SES-0317700. This paper is part of a larger project in which Dan Bernhardt is a co-investigator;Bernhardt made essential contributions to the proof of Proposition 3.1. That larger collaboration, and this paper, havebenefited from conversations with Kerry Back, Roger Germundsson, Lars Hansen, Burton Hollifield, Ken Kasa, Pete Kyle,Orlando Merino, Wayne Shafer, Conrad Wolfram, and participants in the 2005 Canadian Economic Theory Conference. Weespecially thank Joseph Ball for extensive help with terminology and for pointing out an error in an earlier version of the paper.

1


has grappled with the information embodied in price in two ways in order to maintain tractability: eitherprice is observable with a lag, or the information embodied in price is fully revealed with a lag. In this paperwe eschew these assumptions and allow speculators to observe prices contemporaneously with their trades.Moreover the private information that speculators have is never publicly revealed. 1

The papers of Caballe and Krishnan (1994) and Admati (1985) examine multi-asset markets in staticsettings. Our model differs from that of Caballe and Krishnan in that speculators can see price and useinformation in price immediately, as well as using lagged information; our model is therefore like the Kyle(1989) model of demand submission and the model of Back, Cao and Willard (2000) in which price iscontinuously observable, or equivalently, in which speculators know the price at which each of their orderswill be executed when submitting their orders. In this sense our model resembles Admati’s (1985) modelof multi-asset trade. However, we are able to characterize the finite-agent, multi-asset equilibrium, whenspeculators interact strategically. 2

Two closely connected papers, Bernhardt, Seiler and Taub (2004) and Bernhardt and Taub (2005b),separately examine dynamic effects in a market for a single asset, and cross-asset effects but in a purelystatic environment, respectively. In the real world, both effects matter, and this paper combines them: thereare multiple stocks, multiple speculators, and a dynamic setting. We show that the technical results ofthose papers—the mathematical demonstration of the existence of equilibrium and its characteristics, andthe numerical implementation of the model, can be generalized to this broader and more realistic setting,but it poses significant technical challenges.

Bernhardt, Seiler and Taub (2004) analyzes dynamics in a single-asset setting. The persistence of assetvalues causes speculators to spread their trades out over time: trading intensity is lower on trades on newinformation than it would be in a purely static setting, but over time cumulative trading intensity is higher.That paper also established that speculators accelerate the release of information due to competition withother speculators. We verify and extend those results to the more general setting here.

Bernhardt and Taub (2005b) demonstrates how a multi-asset environment in which underlying assetvalues are correlated would result in lower profits for informed speculators: here we establish that the resultsextend to the dynamic setting. We are also interested in how information in the prices of assets is usedby speculators and market makers to trade and price assets indirectly as well as directly. Bernhardt andTaub (2005b) investigate this question in the static setting, but are not able to show how this cross-assetinformation evolves dynamically.

As in Bernhardt, Seiler and Taub (2004) our model also differs from the literature in that rather thanhaving a single piece of information revealed to each speculator in the initial period, with no further arrivalof information, we allow new information to arrive continually and from multiple sources. 3 Thus, informedspeculators must weigh the use of current versus old information, and must also weigh strategic interactionswith other informed speculators. Given that real speculators have a constant inflow of information, thisassumption is more realistic.

The strategic interaction between the informed speculators complicates their strategies. Speculators whochoose their orders without regard to strategic interaction will weigh past information too heavily, revealingtoo much of it to other speculators. Speculators who account for strategic interactions use the informationrevealed in prices, and bid more aggressively on new information relative to old information as a result. Inturn, the other speculators respond similarly, leapfrogging the second-round response with strategies thateven more aggressively weight recent information relative to old information. Because strategies are bestresponses to the strategies of the other speculators, at the conclusion of this best response iteration, nostrategy that is finite in character survives. The equilibrium therefore requires strategies that are infinitely

1 In this sense our model also incorporates elements of the Kyle (1989) model, which has a demand submission construction.We expand on this point in Bernhardt and Taub, 2005a.

2 The intractability of the finite-agent multi-asset setting leads Admati to explore the more tractable setting in which thereis a continuum of speculators of each type. With the continuum assumption the speculators do not interact strategically.

3 Back and Pederson (1998) is an exception: they allow the continual arrival of information in a single-informed speculatormodel, and therefore do not analyze strategic interactions between informed speculators. The model of Bernhardt and Miao(2004) is also an exception in that they allow for sequential information arrival, but their model is limited to three periods withlimited characterizations.

2


complicated in a sense that we detail later.The infinite complexity feature of the model expresses “forecasting the forecasts of others” (as in

Townsend, 1985) when there is strategic interaction, and our solution method therefore provides a set oftools for solving other models in which this issue arises.

These tools allow a reinterpretation of the trading strategies of the informed speculators and the pricingstrategy of the market maker. The market-maker can be viewed as pricing the asset in order to minimizethe forecast error conditional on information about current and past order flow, and price is therefore theprojection of the equity-value process on the history of order flow. Informed speculators can profit onlyfrom unforecastable information, and therefore their net trades are forecast errors of their private signalsconditional on information about current and past total order flow. The functional analysis methods weuse allow us to retain these interpretations and to solve for the projections despite the complicated dynamicinteractions in the model.

The centerpiece of our technical approach is to transform the model in such a way that the solutions arefunctions, which we find in a direct way with a variational method. These methods are known as frequency-domain methods. The methods allow us to confront the forecasting-the-forecasts-of-others issue head on,because it handles the infinite dimensions of the information histories and also of the strategies themselveswith relative ease. We use the methods in the proof of existence using a contraction mapping argument on aspace of functions. Although some bound restrictions are imposed, this proof is more general than the proofsin Bernhardt, Seiler and Taub (2004) and Bernhardt and Taub (2005b), and can no doubt be generalized tomany similar settings.

Because the equilibrium involves functions, we must use so-called state-space methods for our numericalanalysis. Our numerical analysis is buttressed by the proof: the numerical iterations converge because ofthe contraction property.

We compute and extensively analyze a numerical example with two assets and two speculators. Theexample differs from the examples in Foster and Viswanathan (1996) and in Back, Cao and Willard (2000),in that there is zero direct correlation between the information that the informed speculators have abouteach asset. In those models, and also in Bernhardt, Seiler and Taub (2004), correlation of signals increasestrading intensity. All the direct correlation in our model is in information across assets. The correlation ofinformation across assets induces two secondary types of correlation. The first is correlation of fundamentalinformation for each asset, because private-information signals about Glaxo’s fundamentals are also a noisysignal about Merck that supplement the direct private signal about Merck. The effects of this correlatedsignal are like those explored in Back, Cao and Willard (2000), and which were also extensively analyzedin Bernhardt, Seiler and Taub (2004). The other secondary correlation appears endogenously in prices, andthis is our main focus here. The information contained in prices is used by speculators to condition theirtrades in all assets, and speeds up the release of the speculators’ private information. The speed-up, asestablished in Bernhardt and Taub (2005a), is because speculators increase their trading intensity due tothe order-reduction effect: a speculator who increases his order for a stock will cause other speculators’projections of their private information on price to increase in response, and this reduces their net order.This knowledge causes speculators to trade with greater intensity and thereby reveal their information faster.This result emerges here as well but in part with the effect occurring across assets.

The technical elements of the model are set out in the next sections. Section 2 sets out the structureof the model. Section 3 demonstrates that the complicated conditional expectations in the model can besimplified, and section 4 exploits this result to convert the problem to frequency-domain form. In sections5-7 the model is solved. We present our numerical results in section 8. There are four appendices. Our firstappendix outlines the frequency-domain methods that we use. The second appendix contains derivations offirst-order conditions and related formulas in the text. The third appendix develops the existence proof. Thefourth appendix has supplementary material on spectral factorization, a key step of the numerical analysis.

3


2. THE MODEL

The model has a number of technical assumptions, most of which stem directly from the market-microstructureliterature. As in Bernhardt, Seiler and Taub (2004), we model asset values as first-order autoregressive pro-cesses. Asset values therefore change dynamically: they are serially correlated and they can be correlatedwith each other. There are N informed traders, and there are also noise traders. The stochastic processesgoverning order flow from noise traders are not motivated by optimization within the model.

Each informed trader has partial information about the realized values of the fundamental processesthat drive the asset value processes. The traders are heterogeneously informed however, and therefore theyreveal information to each other by engaging in asset trade. The information is reflected in asset prices.

Public information does not fully reveal private information. Because the asset value processes arecorrelated with each other, traders use public information produced by all assets, not just assets aboutwhich they have private information, to determine their trades.

Markets are cleared by market makers who act competitively. They use information to price assets notonly from the order flow from that asset alone, but from the order flow from all assets: one can think of themarket makers as specialists who look at activity in the market as a whole as an element of their pricingdecisions.

The time horizon is infinite. The infinite horizon framework allows us to consider stationary equilibria,unlike the more standard finite horizon constructions of the literature, as we explored in our precursor paper(Bernhardt, Seiler and Taub, 2004).

There are a number similarities between the model and the model presented in Back, Cao and Willard(2000). Because the Back, Cao and Willard model is set in continuous time, it effectively has the samecontemporaneously observable price feature as the model here. Like most of the market microstructureliterature, they model the information as single signals, revealed to the informed traders in the initial instantof time. The noise trades are represented as a Brownian process, which is highly serially correlated, whilehere the noise trades are completely serially uncorrelated. Aside from these differences in the structure andtiming of information, the Back, Cao and Willard model must take account of the fact that informationcommences in an initial period, which leads the informed traders to adopt a Kalman-Bucy filter whenextracting information from price. This filter is intrinsically nonstationary. By contrast, the model herepresumes an infinite horizon and stationarity and that at each time there is an infinite history of price fromwhich to extract information. As a result, the informed traders adopt Wiener filters to extract information,which are simply the long-run converged versions of Kalman filters.

We now turn to the technical elements of the model. There is a vector vt of M asset value processes. Eachvalue process expresses the evolution of the true value of an equity. This value is driven by the fundamentalvalue of the firm whose stock it represents. The fundamental value is comprised of the liquidation oracquisition value of the firm at each future date, discounted by the probability of liquidation or acquisitionoccurring. In Bernhardt, Seiler and Taub (2004) it is detailed how to re-express this form of the fundamentalvalue into a form such that traders buy and sell dated equivalents of value. These dated equivalents are thevalue process vt that we consider.

The value processes are linear functions of K fundamental processes which form a vector et. The vectorof serially uncorrelated innovations et can be contemporaneously correlated and is stationary:

E[ete′t] = Σe

The value processes can therefore be represented by

vt = FΦ(L)et

where Φ is a K × K matrix function of the lag operator and F is an M × K matrix. The Φ matrix givesthe vector of innovations dynamic structure, and the F matrix agglomerates the resulting dynamic processesinto the value processes of the assets.

In the interest of keeping the model tractable, the matrix Φ will be assumed to have first-order autore-gressive form. In practical terms this means Φ can be expressed as

Φ = (I − RL)−1

4


where R is a K × K matrix with no eigenvalues outside the disk of radius β1/2. We achieve even greatersimplification by assuming that R is diagonal with identical elements. But in principle the structure is farmore general.

The order flow from the noise traders consists of a vector process ut which will be assumed to be i.i.d.here. The noise trade process will be independent of the value process innovations, and hence of the privateinformation. The noise trade order flow is also independent but with contemporaneous correlation possible:

E[utu′t] = Σu

The structure of information. Each trader i observes a private signal ωit, and public information Ωt, attime t, and both sources of new information are incorporated into histories. Based on those histories, thetraders determine their order flow, xit.

Public information consists of the vector of sums of the order flows for each asset. Private informationconsists of a filtered version of the value-process innovations, Aj(L)et, where Aj is an Li × K noninvertiblefilter specific to the jth trader and where

A =

A1...

AN

Note that Li 6= M in general, although Li = M is natural—that is, the number of information sources isthe same as the number of assets. Also note that the information filter can have a lag structure, that is,Aj can be a function of the lag operator L; this encompasses the possibility that private information mightbe lagged. However, in this paper, the Ai will be assumed to be scalar—that is, traders see informationcontemporaneously with its realization.

The noninvertibility of Aj guarantees that no informed trader has complete information about theinnovations. Thus each trader has only partial information about each asset.

In each period t the ith trader has two sources of information, namely the history of their private signalsof the innovations (eit, ei,t−1, . . .), which will be denoted ωt

i , and the history of public information as embodied

in total order flow(

∑Nj=1 xjt + ut,

∑Nj=1 xj,t−1 + ut−1, . . .

)

, and equivalently in prices, (pt, pt−1, . . .), which

will be denoted Ωt.The ith trader maximizes the difference between value and price times the position, which is the same

as order flow:

maxxit

E

[

∞∑

s=t

βs−t(vs − ps)′xis|ω

t,Ωt

]

(2.1)

The market-maker’s objective is to minimize the discounted forecast error variance of the differencebetween value and price. He thus solves

minpt

E

[

∞∑

s=t

βs−t(vs − ps)2|Ωt

]

(2.2)

The market-maker conditions only on public information—the history of total order flow—and thus hisconditioning differs from that of the informed traders.

The conditioning of the expectations in the two objectives (2.1) and (2.2) presents significant difficulty.Not only must the traders account for their effect on price, but also on the conditioning information of theother informed traders, and this accounting must be done with respect to the entire history.

In order to eliminate this complexity we first establish that linear strategies and a linear pricing rulehold in equilibrium. We demonstrate this by positing that when all but the ith trader follow linear tradingrules, and the market maker uses a linear pricing rule, the ith trader finds it optimal to also follow a lineartrading rule. With that result in hand, the model can be translated into a more tractable form in which(i) all strategies are expressed in terms of the fundamental processes ejt; (ii) the objective functions areunconditional; (iii) the optimal policies are functions. We can then proceed to re-transforming the objectivesinto variational form.

5


3. TRANSFORMING TO STATIONARY FORM

The first step in the transformation of the model is to posit the linear structure. Traders other than trader iwill be assumed to choose order flow as a linearly weighted sum of the histories of their private informationand of public information. The market maker is posited to choose order flow based on the history of orderflow alone. These processes are ultimately functions of the fundamentals et

j and ut.

Denote total order flow by Xt + ut, where Xt ≡∑N

j=1 xjt. We first posit a linear pricing function λ, sothat pt = λ(L)(Xt + ut). Keep in mind that because there are M assets, Xt + ut is an M × 1 vector and λis an M × M matrix function of the lag operator, L. We further conjecture that the pricing function λ, isinvertible, so that observations of the infinite history of price are equivalent to observing the infinite historyof total order flow, Xt + ut. We can thus denote the history of public information at time t by (Xt + ut)

t.We next conjecture that the informed traders, other than trader i, adopt trading strategies that are

stationary linear functions of both the history of their private signals and the history of public informationnet order flows. Specifically, we conjecture that trader j 6= i’s order in period t is given by:

xjt = bje(L)Ajet + bjΩ(L)(Xt + ut). (3.1)

That is, bje(L) is trader i’s weight on the history of his private signal, and bjΩ is his weight on the history ofnet order flow. We call bie the private-information trading intensity filter and call biΩ the public-information

trading intensity filter.We can now re-state trader i’s problem using this new structure: informed trader i chooses his order

xit at time t to maximize expected discounted trading profits:

maxxit

E

[

∞∑

s=t

βs−t (vs − λ(L) (Xs + us))′xi,s

∣

∣

∣

∣

∣

Aiett, (Xt + ut)

t

]

.

The net order flow history can be written as follows:

Xt + ut = xit +∑

j 6=i

(bje(L)Ajet + bjΩ(L)(Xt + ut)) + ut

Assuming that I −∑

j 6=i bjΩ(L) is invertible, define

qi(L) ≡ (I −∑

j 6=i

bjΩ(L))−1,

This equation can be solved and total order flow can be expressed as follows:

Xt + ut = qi(L)

xit +∑

j 6=i

bje(L)Ajet + ut

The first-order condition for trader i at time t with respect to xit is then given by

0 = E

[

(vt − λ(L) (Xt + ut))′−

∞∑

t=0

βss∑

τ=0

λτqi,s−τxi,t+s

∣

∣

∣

∣

∣

(Aiet)t, (Xt + ut)

t

]

,

where qi,s denotes the s-th lag of qi(L). The last term corresponds to the gradient derivative of

βs(Xt + ut)′λ(L)′ = βs

xi,t+s +∑

j 6=i

bje(L)Ajej,t+s + ut+s

λ(L)′qi(L)′.

Using

qi(βL−1)λ(βL−1)xit =∞∑

t=0

βss∑

τ=0

λτqi,s−τxi,t+s,

6


we can re-write the first-order condition as

0 = E

vt − λ(L)

N∑

j=1

xj,t + ut

′

− x′itλ(βL−1)′qi(βL−1)′

∣

∣

∣

∣

∣

(Aiet)t, (Xt + ut)

t

(3.2)

We are now ready to demonstrate that the conditioning can be simplified.PROPOSITION 3.1: Let the market maker have a stationary linear pricing function, and let speculators

other than speculator i have linear stationary trading rules, such that the pricing function has the followingform:

pt = ξt + ξ0xit + ξ1xi,t−1 + ξ2xi,t−2 + . . .

with ξ0 positive-definite, and let the kth eigenvalue δk of ξj be a positive decreasing sequence: δk(ξ0) >δk(ξ1) > . . . > 0. Then speculator i’s order xit is a linear function of the market maker’s forecast error(conditional on the net order flow history, (Xt + ut)

t) of his trade on his private signals, Aietit.

The proof is in Appendix B.The practical consequence of this proposition is that the trader’s net order corresponds to the market

maker’s forecast error of his trade on private information, and substituting this structure, we can dropconditioning on (Xt + ut)

t from the first-order condition. The first-order condition then simplifies to

0 = E

[

(

pt + E[

vt

∣

∣Aiet − E[

Aiet

∣

∣(Xt + ut)t]]

− pt

)


∣

∣

∣

∣

Aiet

]

= E

[

E[

vt

∣

∣Aiet − E[

Aiet

∣

∣(Xt + ut)t]]


∣

∣

∣

∣

Aiet

]

.

It follows immediately that along a stationary equilibrium path, xit is a stationary linear function of Aiet −E[Aiet|(Xt + ut)

t], i.e., biΩ(L) is a matrix of projection coefficients of current and lagged trading intensitieson private information onto the net order flow history. The upshot is that we can express the first-orderconditions in terms of the fundamental processes et.

We now return to the first-order condition (3.2) in order to express it in terms of the fundamentalinnovations, Ajetj and ut. The first step is to add orders across traders to get net order flow,

Xt + ut =

N∑

j=1

bje(L)Ajet +

N∑

j=1

bjΩ(L)(Xt + ut) + ut,

which we then solve to obtain the net order flow in terms of trading intensities on primitives,

Xt + ut = (I −

N∑

j=1

bjΩ(L))−1(

N∑

j=1

bje(L)Ajet + ut)

Define the adjusted net order flow filter

γi(L) ≡ (I −∑

j 6=i

bjΩ(L) − biΩ(L))−1biΩ(L)

and defineΓ ≡ I +

∑

j

γj .

We separate out∑

j 6=i bjΩ(L) from biΩ(L) in order to emphasize that agent i takes as given the tradingstrategies of the other agents,

∑

j 6=i bjΩ(L). In particular, γi(L) can be mapped one-to-one with biΩ(L).Then trader i’s order flow can be written as

xit = bie(L)Aiet + γi(L)

N∑

j=1

bje(L)Ajet + ut

(3.3)

7


and total order flow can be written as

Xt + ut = Γ(L)

N∑

j=1

bje(L)Ajet + ut

+ ut. (3.4)

Next, substitute for these order flows into the first-order condition (3.2), recognizing that we no longer needto condition on net order flow so that ut drops out,

0 = E

φ(L)∑

j

ejt − λ(L)Γ(L)

N∑

j=1

bje(L)Ajet − qi(βL−1)λ(βL−1)

bie(L)Aieit + γi(L)

N∑

j=1

bje(L)Ajet

∣

∣

∣

∣

∣

Aiet

.

Now multiply the first-order condition by I + γi(βL−1), and noting that

I + γi(βL−1) = qi(βL−1)−1Γ(βL−1)

we write the first-order condition in the following convenient form

0 = E

[

(I + γi(βL−1))φ(L)

N∑

j=1

ejt−(I + γi(βL−1))λ(L)Γ(L)

N∑

j=1

bje(L)Ajet

−λ(βL−1)Γ(βL−1)

bie(L)Aiet + γi(L)N∑

j=1

bje(L)Ajet

∣

∣

∣

∣

∣

(Aieit)t

]

,

(3.5)where the third term follows because (I + γi(βL−1))qi(βL−1) = Γ(βL−1). We note that multiplying bynon-positive powers of the lag (by multiplying by γi(βL−1) ) is legitimate because it pushes future values ofejt further into the future, and their expectation is zero.

In order to satisfy equation (3.5), the functions bie must be structured so that the terms inside theexpectation are zero for all realizations of current and past information. In addition, the projections γi

must be calculated, and these projections are coupled with the solutions of the trading intensity filters bie.The solution is to find functions such that all the coefficients generated on the innovations es are for futureinnovations, that is s > t, so that their expectation is zero.

Equation (3.5) corresponds to its frequency-domain counterpart, which provides a method to find thissolution. We now turn to the derivation of the frequency-domain model that is the focus of the remainderof the paper.

4. A FREQUENCY-DOMAIN REFORMULATION

In this section we reformulate the model in the frequency-domain. We first re-state the optimization problemof the informed trader, using the expressions for individual order flow and total order flow in equations (3.3-4).

maxbie,γi

E

[

∞∑

0

βt

Φ(L)et − λ(L)

Γ(L)

N∑

j=1

(bje(L)Aj(L)et + ut)

+ ut

′

×(

bie(L)Ai(L)et + γi(L)(N∑

j=1

bje(L)Aj(L)et + ut))

]

This can be expressed with the matrices and vectors a bit more explicit:

= maxbie,γi

E

[

∞∑

0

βt

(

(

Φ(L) − λ(L)Γ(L)∑N

j=1(bje(L)Aj(L) −λ(L)Γ(L))

(

et

ut

)

)′

×

(

(

bie(L)Ai(L) + γi(L)∑N

j=1 bje(L)Aj(L) γi(L))

(

et

ut

))

](4.1)

8


It is key that the initial expression of the objective, (3.1), has a conditional expectation in it, but (4.1) doesnot. Moreover, the choice variable in (3.1) is the current level of order flow xit, while the controls in (4.1)are the functions bje and bj,Ω.

With this formulation, trader j’s action appears twice in his order flow: first in the direct sense offiltering his private information ωjt = Aj(L)et, and second in the indirect sense due to his contribution topublic information in the γkbjAj terms.

Because the products γkbj appear as elements of a quadratic objective, the possibility exists that thequadratic structure has been lost. However, because γi is a projection—something that was already estab-lished in the time-domain approach of the previous section and that will re-emerge in the next section, thecross-products are orthogonal and the objective retains its quadratic character.

Because the objective in (4.1) is explicitly defined in terms of the fundamental processes eit and ut, andbecause the expectation is unconditional, the variance-covariance matrix can be calculated directly. Thisin turn facilitates the further transformation of the problem. Using methods that we describe in general inAppendix A, the objective can then be restated in frequency-domain form:

The objective can then be restated as

maxbje,γj

1

2πi

∮

tr

(

Φ′ −∑N

k=1

(

bkeAk + γk

∑N`=1 b`eA`

)′

λ′

−(∑N

k=1 γk + IM )′λ′

)

(

b∗jeA∗j + γ∗

j

∑Nk=1 b∗keA

∗k γ∗

j

)

(

Σe 00 Σu

)

dz

z

(4.2)The key feature of this re-expression of the objective is that the optimal functions can be found using avariational method. The first-order condition can be calculated for this objective.

In the next section we develop the solution.

5. SOLUTION OF THE FREQUENCY-DOMAIN MODEL

The variational method described in Whiteman (1985, pp. 235-7) will be applied to find the first orderconditions. Consider a function bje that is the optimum, and a variation αη(z), where η is analytic on theβ−1/2 disk and α is a real scalar. Substitute the sum bje+αη(z) into the objective (4.2), equate the derivativewith respect to α zero, then set α to zero. Using derivations in Appendix B, the resulting expression is

A∗jΣe

N∑

k=1

(

bkAk + γk

N∑

`=1

b`A`

)′

λ′(

I + γ∗j

)

+A∗jΣe

(

bjAj + γj

N∑

k=1

bkAk

)′

λ∗

(

I +

N∑

k=1

γ∗k

)

= A∗jΣeΦ

′(I + γ∗j ) +

−1∑

−∞

(5.1)

Defining

Γ ≡ I +

N∑

k

γk

and

bA ≡ ( b1 b2 . . . bN )

A1

A2...

AN

≡ bA

the equation can be written

A∗jΣeA

′b′Γ′λ′(

I + γ∗j

)

+A∗jΣe (bjAj + γjbA)

′λ∗Γ∗ = A∗

jΣeΦ′(I + γ∗

j ) +

−1∑

−∞

(5.2)

9


or, defining µ ≡ λΓ,A∗

jΣeA′b′µ′

(

I + γ∗j

)

+A∗jΣe (bjAj + γjbA)

′µ∗ = A∗

jΣeΦ′(I + γ∗

j ) +

−1∑

−∞

(5.3)

Equation (5.3) replicates the time-domain first-order condition, (3.5). The equality with the principal part∑−1

−∞ in (5.3) mirrors the requirement that only future values of the innovations et survive in (3.5).This equation displays a lack of multiplicative symmetry because of the cross-corrrelation of information.

Because of the structure, this equation alone cannot be solved directly; the array of first-order conditionsacross traders must be constructed. We carry this out in the next section.

First-order condition for γi. The first-order condition for the γi is found in a similar way. Define

Γ ≡ I +

N∑

k

γk

and

bA ≡ ( b1 b2 · · · bN )

A1

A2...

AN

≡ bA

Let γi = αη. Using results in the appendix, taking the derivative of the objective and setting it to zero yields

( b∗A∗ I )

(

Σe 00 Σu

)(

A′b′

I

)

(

Γ′λ′ + γ′jλ

∗)

=( b∗A∗ I )

(

Σe 00 Σu

)((

Φ′

0

)

−

(

A′jb

′j

0

))

+

−1∑

−∞

(5.4)

We can simplify this using the market maker’s condition, which will be derived next.The market-maker’s problem. In the univariate setting of Bernhardt, Seiler and Taub (2004), the

market-maker’s problem is to minimize the difference between price and value conditional on aggregateorder flow. In the multiple-stock framework, this translates into minimizing the sum of these differences.In this linear-quadratic setting, the minimization is more formally the minimization of the forecast errorbetween price and value, so the price process is the projection of the value process on the public informationprocess. The objective of the market maker is

maxλ

1

2πi

∮

tr

(

Φ′ −∑N

k=1

(

bkAk + γk

∑N`=1 b`A`

)′

λ′

−(∑N

k=1 γk + IM )′λ′

)

×(

Φ∗ −∑N

k=1

(

b∗kA∗k + γ∗

k

∑N`=1 b∗`A

∗`

)

λ∗ −(∑N

k=1 γ∗k + IM )λ∗

)

(

Σe 00 Σu

)

dz

z

Using derivations in the appendix, the market-maker’s first-order condition is

Γ∗ ( b∗A∗ I )

(

Σe 00 Σu

)(

A′b′

I

)

Γ′λ′ = Γ∗ ( b∗A∗ I )

(

Σe 00 Σu

)(

Φ′

0

)

+

−1∑

−∞

(5.5)

In addition to its direct use in expressing the equilibrium, this condition can be used to simplify the expressionfor γi.

Define the function J as follows:

J∗′J ≡ ( b∗A∗ I )

(

Σe 00 Σu

)(

A′b′

I

)

10


We choose J to be both analytic and invertible. The function J captures the dynamic structure of the publicinformation process. There is a fundamental process, ωt, implicitly defined along with J as well.

Also define µ as follows:µ ≡ λΓ

Cancelling Γ∗ from the market-maker’s first-order condition, we have the solution for µ

µ′ = J−1

[

J∗′−1( b∗A∗ I )

(

Σe 00 Σu

)(

Φ′

0

)]

+

(5.6)

This expression is the projection of value on public information—total net order flow. This simply reflectsthe market maker’s use of the public information to estimate the asset value.

Using the market-maker’s condition to simplify the γi condition. Substituting the market-maker’s con-dition (5.5) into the condition for γj (5.4) yields

( b∗A∗ I )

(

Σe 00 Σu

)(

A′jb

′j + A′b′γ′

j

γ′j

)

λ∗ =

−1∑

−∞

This can be written

( b∗A∗ I )

(

Σe 00 Σu

)(

A′b′

I

)

γ′jλ

∗ = − ( b∗A∗ I )

(

Σe 00 Σu

)(

A′jb

′j

0

)

λ∗ +

−1∑

−∞

This can be written in factored form as

J∗′Jγ′jλ

∗ = − ( b∗A∗ I )

(

Σe 00 Σu

)(

A′jb

′j

0

)

λ∗ +

−1∑

−∞

which can be solved if λ is invertible. Assuming such invertibility, the solution is

γ′i = −J−1

[

J∗′−1( b∗A∗ I )

(

Σe 00 Σu

)(

A′ib

′i

0

)]

+

(5.7)

This expression is the generalized projection coefficient of trader i’s order flow on his private informationon public information. The negative sign emphasizes that this projection is subtracted from trader i’s orderflow, because no trading profit is possible on that component of his information. His net order flow is thenthe market maker’s forecast error of his private signal.

6. EQUILIBRIUM

In this section we explore solution algorithms. We begin with a definition of equilibrium.DEFINITION 6.1: A stationary linear equilibrium is a set of functions bi

Ni=1, γi

Ni=0, µ such that (i)

the trading intensity filters bi and public-information filters γi solve the optimization problem of each traderi; (ii) The market-maker’s filter on order flow µ solves the market-maker’s optimal forecasting problem; and(iii) Public information is identical to the total order flow generated by the informed traders and the noisetraders.

The central difficulty in solving for the equilibrium is that the first-order conditions for the tradingintensity filters on private information, bi, are nonstandard in structure. As a practical matter iteration isrequired for solving the model because of the strategic interactions across the traders.

As a first step in the solution, the first-order conditions for the bi, equation (5.3), can be stacked acrosstraders. Stacking the first-order conditions yields

A∗1ΣeA

′b′(µ′(I + γ∗1 ) + (I + γ′

1)µ∗) − A∗

1Σe

∑

j 6=1 A′jb

′jµ

∗

...A∗

NΣeA′b′(µ′(I + γ∗

N ) + (I + γ′N )µ∗) − A∗

NΣe

∑

j 6=N A′jb

′jµ

∗

=

A∗1ΣeΦ

′(I + γ∗1 ) +

∑−1−∞

...A∗

NΣeΦ′(I + γ∗

N ) +∑−1

−∞

(6.1)

11


The resulting matrix expression can then be re-expressed using the Kronecker product, resulting in a non-standard Wiener-Hopf equation, which we do not display here. Instead, we carry out an indirect solutionmethod.

An iterative solution approach. By rearranging the stacked equation (6.1), an iterative solution can bedeveloped. Begin by re-writing (6.1) as follows:

A∗1ΣeA

′b′(µ′(I + γ∗1 ) + (I + γ′

1)µ∗)

...A∗

NΣeA′b′(µ′(I + γ∗

N ) + (I + γ′N )µ∗)

=

A∗1ΣeΦ

′(I + γ∗1 )) − A∗

1Σe

∑

j 6=1 A′jb

′jµ

∗ +∑−1

−∞

...A∗

NΣeΦ′(I + γ∗

N )) − A∗NΣe

∑

j 6=N A′jb

′jµ

∗ +∑−1

−∞

The left hand side now has the Hermitian factors µ′(I + γ∗i ) + (I + γ′

i)µ∗, and it can be factored. Defining

the factorizations

GiG∗i′ ≡ µ′(I + γ∗

i ) + (I + γ′i)µ

∗

The stacked first-order conditions then are

A∗1ΣeA

′b′G1

...A∗

NΣeA′b′GN

=

(A∗1Σe

∑

j 6=1 A′jb

′jµ

∗ + A∗1ΣeΦ

′(I + γ∗1 ))G∗

1′−1

+∑−1

−∞

...(A∗

NΣe

∑

j 6=N A′jb

′jµ

∗ + A∗NΣeΦ

′(I + γ∗N ))G∗

N′−1

+∑−1

−∞

The solution will now be developed when the information matrices, Ai, are scalar. In that case, the annihila-tor can be applied to the right hand side of equation separately, and then each equation can be subsequentlymultiplied by G−1

i .The equations can now be stacked after annihilation:

AΣeA′b′ =

[

(A∗1Σe

∑

j 6=1 A′jb

′jµ

∗ + A∗1ΣeΦ

′(I + γ∗1 ))G∗

1′−1]

+G−1

1

...[

(A∗NΣe

∑

j 6=N A′jb

′jµ

∗ + A∗NΣeΦ

′(I + γ∗N ))G∗

N′−1]

+G−1

N

(6.2)

with solution

b′ = (AΣeA′)−1

[

(A∗1Σe

∑

j 6=1 A′jb

′jµ

∗ + A∗1ΣeΦ

′(I + γ∗1 ))G∗

1′−1]

+G−1

1

...[

(A∗NΣe

∑

j 6=N A′jb

′jµ

∗ + A∗NΣeΦ

′(I + γ∗N ))G∗

N′−1]

+G−1

N

(6.3)

Because b-terms appear on the right hand side, this is not an algebraic solution, it is simply an equilibriumcondition. In numerical implementations the previous iteration of the trading intensity filter b is recursivelysubstituted on the right hand side in order to find the approximate solution of b. The equation also is thefoundation for an existence proof.

7. EXISTENCE

We prove existence in symmetric settings, with a constrained set of parameters. Our proof makes use of acontraction-mapping argument to generate a fixed point for the mapping implicit in the formulas for bi, γi,and µ, but the contraction argument is non-standard in that it does not spring directly from an optimizationproblem such as consumer-surplus maximization. Instead, it simply relies on iteration of best responses, asexpressed in (6.3). We relegate the details to the Appendix C.

12


8. A SYMMETRIC ANTI-CORRELATED EXAMPLE

In this section an example with two equities and two traders is discussed. Our numerical routines use methodsfrom control theory to compute the example. The model is translated to a so-called state-space formulation,which represents the model in terms of matrices rather than directly in terms of rational functions. Allstandard operations such as addition, multiplication, matrix inversion, and so on, can be performed inthe state-space setting. In addition, the state-space approach effects the crucial operation of pole-zerocancellation, which intrinsically requires approximation, using so-called minimal-realization algorithms. Theoperations of annihilation and spectral factorization can also be performed in the state-space setting.

There are two sets of factorizations that must be performed: the factorization of JJ ∗ to recover J andthe factorizations of G∗

i′Gi in order to recover the Gi. Two methods of factorization were used in separate

simulations in order to ensure numerical accuracy. For the first method, for both J and Gi, the structure ofthe model allows a so-called Popov form representation, and this in turn simplifies the spectral factorizationproblem, reducing it essentially to the solution of a discrete algebraic Riccati equation (see Kailath, Sayed,and Hassibi, 2000, p. 277). A more technical description of these ideas, especially of the generation of thePopov form, is presented in Appendix D.

The second method translates an algorithm recently developed by Iakoubovsky and Merino (1999) to astate-space setting. Unlike the Popov form approach, this method is less dependent on the intrinsic structureof the functions being factored. It uses a Newton optimization approach, but in function space, substitutinga Frechet derivative for the derivative that would appear in a conventional Newton optimization problem.

In order to limit the proliferation of the polynomial order, balanced reduction is performed on b(z) aftereach iteration.

A zero-correlation benchmark. We begin with a benchmark example that is similar in structure to themodel of Bernhardt, Seiler and Taub (2004). There is only one asset. The traders are fully symmetric. Inthis example, there is no correlation of information across traders of any kind. Each trader views a singleinformation process about the asset, which is orthogonal to the other trader’s information process.

The correlation matrix Σe is

Σe =

(

.25 00 .25

)

for the fundamental processes e1t and e2t. The noise trade process has variance 2.0:

Σu = 2

The value process is simply the autoregressive sum of the two information processes:

vt =1

1 − .5L(e1t + e2t)

The informed traders each view one of the processes privately. The first trader sees e1t and the second tradersees e2t. The A-matrices are thus

A1 = ( 1 0 ) A2 = ( 0 1 )

Numerically iterating the model with this structure yields the following outcomes. The key structures of themodel are the b-filters. These are

b =.67

1 − .48z+

.70

1 − .36z+

.39

1 − .21z

The γ filters iterate to approximately

γ = −.01

1 − .43z−

.06

1 − .28z−

.15

1 − .16z

The autoregressive coefficients of the trading intensity filter b shrink below the first value of .48, whichapproximates .5, reflecting the increased trading intensity on newer information relative to old information.The γ function reflects this to an even greater degree: most of the weight is on the term with autoregressive

13


parameter .16, and thus the traders subtract the effect of their influence on public information most intenselyon new information.

For these and the other functions of interest—λ, profit π, and the market-maker’s forecast error varianceσ2FE—can be decomposed into impulse-response or moving average coefficients:

Lag 0 1 2 3 4 5 6 7 8 9 10b(L) 1.753 .653 .262 0.110 0.048 0.021 0.010 0.004 0.002 0.001 0.000γ(L) -.227 -.046 -.011 0.003 0.001 0.000 0.000 0.000 0.000 0.000 0.000λ(L) 0.488 .244 .122 0.061 0.030 0.015 0.008 0.004 0.002 0.001 0.000π(L) 0.533 .176 .067 0.028 0.012 0.006 0.003 0.001 0.000 0.000 0.000σ2

FE(L) 0.284 .185 .100 0.052 0.026 0.013 0.007 0.003 0.003 0.002 0.001

The key properties are as follows. First, the trading-intensity coefficients decay significantly faster than theintrinsic rate of decay of the value process—that is, .653/1.753 = .373, significantly less than the decay rateof the value process, 0.5. Similarly, the decay rate of the γ impulse-response coefficients on the first lag is.046/.227 = .203, an even more extreme rate of decay. Thus, traders trade on new information intensely,revealing it in price at an earlier stage, and traders thus subtract the price impact of their trades via γ muchmore on new information than on old.

Another perspective on this high decay rate of trading intensity is the negative correlation of traders’information that develops as a result of their release of information. This negative correlation, which wasalso noted by Foster and Viswanathan and by Back, Cao and Willard as the “waiting game,” is the result ofthe market-maker’s acquiring information and generating a forecast of value that is an average of the traders’information. As a result, the traders’ net information—that is, the information that is orthogonal to themarket-maker’s information (i.e. public information) necessarily brackets the market-maker’s information.

The decay rate of the λ coefficients reflects the decay rate of the underlying process. The initialcoefficient, .488, is the impact of total order flow on contemporaneous price. The smaller this number is, thehigher the liquidity of the market (see Kyle, 1985).

The total profit of the informed traders is .827. There is only one asset traded here, and we will latercompare with the two-asset case, and for comparison to that case the profit needs to be doubled to 1.654.The impulse-response coefficients, like the other coefficients, decay at a faster rate than the underlying valueprocess. The first pair of coefficients decays at a rate of .176/.533 = .330. This also stems from the increasedtrading intensity on new versus old information: the increased trading intensity on new information entanglesthe information and results in negative correlation and the “waiting game” after the initial rounds of trading.

The forecast error variance of the market maker is a measure of the quality of his information. The totalforecast error variance is .674, but this quantity decays more slowly than the value process: the first pair ofcoefficients show a decay rate of .185/.284 = .651, much larger than .5. This also reflect the early release ofinformation: most of it is released immediately. The ensuing negative correlation slows down trading andretards the reduction of the market-maker’s forecast error variance.

A single-asset with correlated information benchmark. We now embellish our benchmark to allow cor-relation in the traders’ information about a single asset. When we examine the multi-asset example wewill preserve the degree of correlation of the traders’ information but we will change the structure of thatcorrelation.

We can represent a single asset with correlated trader information by the following covariance matrix:

Σe =

(

σ2 θθ σ2

)

In our numerical example this is

Σe =

(

.185185 .0648148.0648148 .185185

)

and again the noise trade isΣu = 2.0

14


The simulation results for this case are as follows. After numerical convergence, the approximate tradingintensity functions are

b =.63

1 − .49z+

1.09

1 − .37z+

.49

1 − .20z

γ = −.03

1 − .28z−

.02

1 − .25z−

.23

1 − .13z

Comparing the numerator weights to the uncorrelated case, we see that the weight on the 11−.49z term has

shrunk, while the weights on the other two terms have increased. This reflects an overall increase in tradingintensity, but also a relative increase in the intensity of trading on recent innovations eit versus older ones.

We can see this effect more clearly by examining the impulse response coefficients on the eit. Theimpulse response coefficients are as follows.

Lag 0 1 2 3 4 5 6 7 8 9 10b(L) 2.242 .815 .321 0.133 0.058 0.026 0.012 0.005 0.003 0.001 0.000γ(L) -.281 -.042 .007 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000λ(L) .597 .290 .140 0.067 0.032 0.015 0.007 0.003 0.002 0.001 0.000π(L) .522 .152 .056 0.023 0.010 0.005 0.002 0.001 0.000 0.000 0.000σ2

FE(L) .222 .128 .065 0.033 0.016 0.008 0.004 0.002 0.001 0.000 0.000

Not only is the level of the trading intensity high, the decay rate for the trading intensity impulse responsecoefficients is fast, .242; this is faster than the rate of decay of the value process, and also slightly faster thanthe decay rate for the trading intensity in the single-asset uncorrelated case.

The decay rate of the γ coefficients is extreme here: correlation of information for the same asset inducesthe greatest degree of competition and causes information to be released at the fastest rate.

Total per-period informed-trader profit is 0.772. Again, to facilitate later comparison with the two-assetcase we must double this to 1.544. The first observation is that this profit is lower than the profit in theuncorrelated case, an outcome driven by the intensified competition on common information. The effect ofthe increased trading intensity at short lags due to strategic best responses shows up in the impulse responsepattern of profits: profits also decrease at a significantly faster rate than the value process: .34 instead of .5.Thus, the major fraction of profit (66%) is obtained by trading immediately on new information.

The market-maker’s forecast-error variance is .480. Except for the first lag, the forecast error variancesdecay at roughly the intrinsic rate of decay of the value process.

Two correlated assets. The trading on two assets might simply replicate the trading that would occur onsome equivalent single asset, that is, a single asset that has a correlation structure reflecting the correlationstructure of the two assets. Does the presence of two assets make a difference? Intuitively two assets shouldmake a difference in the equilibrium: with two prices reflecting order flow information separately, informationshould get out faster. This is reflected in the nonzero off-diagonal γi term.

The information structure is such that the Σe matrix has an X pattern:

Σe =

σ2 0 0 θ0 σ2 θ 00 θ σ2 0θ 0 0 σ2

The variance-covariance matrix of the e-processes is constructed so that all the entries add to one. Thenumerical values are

Σe =

.185185 0 0 .06481480 .185185 .0648148 00 .0648148 .185185 0

.0648148 0 0 .185185

The processes are all symmetrically correlated with correlation 0.35, so information about one stock hasvalue in predicting the value of the other stock, but there is no direct correlation between the informationprocesses driving the value of a particular stock.

15


The X-shape of the Σe-matrix is key: an informed trader cannot obtain information about Merckdirectly just by looking at the price of Merck. In order to obtain information about Merck stock beyond theinformation derived from his private signals, the trader must look not only at the price of Merck, he mustalso look at the price of Glaxo.

There are two noise trade processes as well. The noise processes are uncorrelated with each other, andalso are not correlated with the fundamental value processes:

Σu =

(

2 00 2

)

The value processes are chosen to be symmetric and serially correlated:

v1t =1

1 − .5L(e1t + e2t) v2t =

1

1 − .5L(e3t + e4t)

that is,

FΦ =

( 11−.5z

11−.5z 0 0

0 0 11−.5z

11−.5z

)

The informed traders each view two of the processes privately. The first trader sees e1t and e3t and the secondtrader sees e2t and e4t. Thus trader 1 sees partial information about each stock and the second trader alsosees partial information about each stock, but his information is different from trader 1’s information. TheA-matrices are thus

A1 =

(

1 0 0 00 0 1 0

)

A2 =

(

0 1 0 00 0 0 1

)

An initial naive trading intensity filter is posited in which each trader trades intensely on the processthat drives the appropriate equity and less intensely on the other process, which provides only indirectinformation. That is, trader 1 has the following order flow for equity 1:

1.5

1 − .5Le1t +

.1

1 − .5Le3t

while his order flow for equity 2 is:

+.1

1 − .5Le1t +

1.5

1 − .5Le3t

Trader 2 has a similar pattern for his information.After several rounds of iteration the coefficients converge to the following approximate values.

(

b11(z) b12(z)b21(z) b22(z)

)

=

( .901−.48z + .88

1−.43z + .061−.22z + .03

1−.16z + .17 .931−.48z − .77

1−.43z − .061−.22z + .03

1−.16z + .04.99

1−.48z − .851−.43z − .06

1−.22z + .041−.16z + .03 1.02

1−.48z + .771−.43z + .06

1−.22z + .041−.16z + .16

)

The slight asymmetry is a consequence of asymmetric initial conditions and the characteristics of the nu-merical algorithm.

The γ matrix is

(

γ11(z) γ12(z)γ21(z) γ22(z)

)

=

(

− .071−.30z − .11

1−.23z − .04 .071−.30z − .11

1−.23z − .02.07

1−.30z − .111−.23z − .02 − .07

1−.30z − .111−.23z − .04

)

Again, note that the coefficients of the diagonal terms of the trading intensity filter are larger on the termswith less persistence than in the single uncorrelated asset case. Also, the γ coefficients show an evenmore pronounced emphasis on the low-persistence terms, underscoring how the information in prices decaysextraordinarily fast. These effects are clearer if we examine the impulse response coefficients.

16


The impulse-response coefficients (rounded to three decimal places) are

Lag 0 1 2 3 4 5 6 7 8 9 10b11(L)e1t 2.031 .821 .368 0.167 0.076 0.035 0.016 0.007 0.003 0.002 0.001b12(L)e3t .146 .093 .062 0.037 0.021 0.011 0.006 0.003 0.002 0.001 0.000

γ11(L) -.220 -.047 - .012 - 0.003 - 0.001 - 0.000 - 0.000 - 0.000 0.000 0.000 0.000γ12(L) -.060 -.005 .000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

λ11(L) .429 .215 .107 0.054 0.027 0.013 0.007 0.003 0.002 0.001 0.000λ12(L) .171 .086 .043 0.021 0.011 0.005 0.003 0.001 0.001 0.000 0.000

π(L) .880 .267 .091 0.034 0.014 0.006 0.003 0.001 0.000 0.000 0.000

σ2FE(L) .380 .243 .122 0.059 0.029 0.014 0.007 0.003 0.002 0.001 0.000

The rate of drop-off of the direct trading intensity terms on the main diagonal for b is faster than the drop-offof the value process: it is .404 (.821/2.031) between the initial two periods instead of .5: the best responsesof the informed traders to each other tend to accelerate trading—or from another perspective, flatten thespectrum of the trading intensity filter.

The indirect trading intensity is positive—the positive correlation of e3t with e2t makes the signal usefulin trading asset 1—but it is significantly smaller than the on-diagonal trading intensity. The fall-off in theoff-diagonal trading intensity after just one round is slower than that for the direct trading intensity terms:.637 instead of .5. Despite their small magnitudes, the indirect trading intensity terms are significant: whenadded to the direct trading, the overall trading intensity on each asset increases over the correlated single-asset case. In the correlated single-asset case the first three trading intensity terms are 2.242, .815, and .321,summing to more than 3.378; the summed diagonal and off-diagonal terms in the two-asset case are 2.177,.914, and .430, summing to more than 3.521—the summed trading intensities are higher in the two-assetcase.

As in the static multi-asset model of Bernhardt and Taub (2005b), the dynamic structure causes spec-ulators to spread out their trades over time. In a similarly parameterized but purely static version of thecurrent example, the static trading intensities are b11 = 2.586 and b12 = 0.243.

The direct effect of γ11(L) subtracts from trading intensity: recall that it is in fact a projection of thetrader’s trading intensity on public information, so that the net order flows of the informed traders are theforecast error of this projection. The drop-off of the direct term after the first lag is extreme: .22, much lessthan .5. The value of public information in forecasting traders’ information thus shrinks drastically after thefirst round.

The indirect γ terms differ from zero. These also serve to reduce trading intensity. Although theseeffects are small, they are the most interesting. These terms reflect the use of the price of the second assetto forecast the value of the first asset: recalling that the γ term in the single-asset case reflects the trader’ssubtracting the projection of his private information on price from his trade, the indirect term here representsthe subtraction of the projection of his asset-1 information on the price of the second asset from his trade onasset 1. Again, the fall-off of the coefficients is very steep after just one lag, .082, far lower than .5. Thus,the indirect impact of information on prices fades very fast. But the coefficients are not zero, and this meansthat traders find it useful to have a stock market: they gain useful information from simultaneously andrepeatedly trading multiple assets.

Total per-period informed-trader profit is 1.30. The effect of the increased trading intensity at shortlags due to strategic best responses shows up in the impulse response pattern of profits: profits also decreaseat a significantly faster rate than the value process: .32 instead of .5. Thus, the major fraction of profit(70%) is obtained by trading immediately on new information.

Profit is clearly lower than the adjusted profit in the single-uncorrelated asset case, and is also lower thanthe single asset correlated case. Thus, the additional channel of information available through the prices ofother assets intensifies trading beyond simple one-asset correlation, driving down profit and correspondinglyincreasing liquidity. It should be emphasized that the degree of correlation of information across informed

17


traders is the same in both the single asset and two-asset cases. The essential difference between them isthat the market maker has two sources of information rather than one in the two-asset case. The marketmaker uses the information he has from both markets to determine pricing, and his improved informationdrives down the profits of the informed traders.

This same reasoning illustrates why the magnitude of the indirect γ filters is so small and dynamicallyshrinks so precipitously. Because the market makers get information from both assets, the usable informationabout Merck from the Glaxo price has almost all been incorporated in the Merck price already by the marketmaker. Moreover, any such residual information is quickly detected and used, leading to the extremely rapiddecay of the indirect γ’s.

The market-maker’s forecast-error variance is .880. Except for the first lag, the forecast errors decay atroughly the intrinsic rate of decay of the value process.

9. CONCLUSION

We have developed a model of strategic informed trade in a dynamic environment, with multiple correlatedassets and multiple informed traders whose information is correlated.

Our main conclusion is that correlated information is channeled through prices and this intensifiestrading intensity and the speed of information release. With the equivalent degree of correlation of privateinformation in a one-asset benchmark, the presence of multiple assets speeds information release by providingmore channels for market makers to acquire information and incorporate that information in prices. Sec-ond, the additional information generated by the correlation of private information, whether it is expressedthrough the direct correlation of private information or through prices, ultimately increases trading intensityat short lags. This increased trading intensity reduces informed-trader profits and correspondingly reducesthe losses of the liquidity traders. This can be viewed tentatively as a welfare-improving increase in liquidity,enhanced by simultaneous trade in multiple assets. We have thus normatively motivated the role of stockmarkets.

18


REFERENCES

Admati, A (1985), “A noisy rational expectations equilibrium for multi-asset securities markets,” Economet-

rica 53 , 629-658.Admati, A., and P. Pfleiderer (1998), “A theory of intraday patterns: volume and price variability,” Review

of Financial Studies 1, 3-40.Back, K., H. Cao and G. Willard (2000), “Imperfect competition among informed traders,” Journal of

Finance, 55 2117-2155.Back, K. and D. Pederson (1998), “Continuous arrival of information in stock markets,” Journal of Quanti-

tative Finance.Baruch, S. and G. Saar (2004), “Asset returns and the listing choice of firms,” working paper, University of

Utah.Bernhardt, D. and Jianjun Miao (2003), “Informed trading when information becomes stale,” forthcoming,

Journal of Finance.Bernhardt, D. and B. Taub (2005a), “Kyle versus Kyle (’85 versus ’89),” forthcoming, Annals of Finance.Bernhardt, D. and B. Taub (2005b), “Strategic information flows in stock markets,” working paper, Univer-

sity of Illinois.Bernhardt, D., P. Seiler, and B. Taub (2004), “Speculative dynamics,” working paper, University of Illinois.Conway, John B. (1985), A Course in Functional Analysis. New York: Springer-Verlag.Davenport, Wilbur B., Jr., and William L. Root (1958), An Introduction to the Theory of Random Signals

and Noise. New York: McGra w-Hill.Dhrymes, Phoebus J. (1978), Mathematics for Econometrics. New York: Springer.Foster, Douglas, and S. Viswanathan (1996), “Strategic trading when agents forecast the forecasts of others,”

Journal of Finance LI (4), 1437-1478.Futia, C. (1982), “Rational expectations in stationary linear models,” Econometrica 49(1), 171-192.Hansen, L., and T. Sargent (1980), “Formulating and estimating dynamic linear rational expectations mod-

els,” Journal of Economic Dynamics and Control 2.Hansen, L., and T. Sargent (1981), “Linear rational expectations models for dynamically interrelated vari-

ables,” in Rational Expectations and Econometric Practice v. 1, ed. by R.E. Lucas, Jr., and T. Sargent.Minneapolis: University of Minnesota Press.

He, H. and J. Wang (1995), “Differential information and dynamic behavior of stock trading volume”, Review

of Financial Studies, 920-972.Holden, C. W., and A. Subrahmanyam, 1992, “Long-lived private information and imperfect competition”,

Journal of Finance 47, 247-270.Hutson, V., and J. S. Pym (1980), Applications of functional analysis and operator theory. London: Academic

Press.Iakoubovski, M., and O. Merino (1999), “Calculating spectral factors of low and full rank matrix valued

functions on the unit circle,” Proceedings, 38th Conference on Decision and Control.Kasa, K. (2000), “Forecasting the forecasts of others in the frequency domain”, Review of Economic Dy-

namics 3(4).Kailath, T., A. Sayed, and B. Hassibi (2000), Linear Estimation. Upper Saddle River: Prentice Hall.Kyle, A., (1985), “Continuous auctions and insider trading”, Econometrica 53, 1315-1335.Kyle, A., (1989), “Informed speculation and monopolistic competition”, Review of Economic Studies 56,

317-355.Iakoubovski, M., and Orlando Merino (1999), “Calculating spectral factors of low and full rank matrix-

valued functions on the unit circle”, Proceedings of the 38th IEEE Conference on Decision and Control,505-506.

Pearlman, J. and T. Sargent (2002), “Knowing the forecasts of others,” working paper, Stanford University.Rozanov, Yu A. (1967), Stationary Random Processes. San Francisco: Holden-Day.Rudin, W. (1974), Real and Complex Analysis. New York: McGraw-Hill.Sanchez-Pena, Ricardo, and Mario Sznaier (1998), Robust Systems: Theory and Applications. New York:

John Wiley.Taub, Bart (1986), “Optimal policy in a model of endogenous fluctuations and assets,” Journal of Economic

Dynamics and Control 21, 1669-1697 (1997).

19


Townsend, R. (1985), “Forecasting the forecasts of others,” Journal of Political Economy 91, 546-588.Wang, J. (1994), “A model of competitive stock trading volume,” Journal of Political Economy 102(1),

127-168.Whiteman, C. (1985), “Spectral utility, Wiener-Hopf techniques, and rational expectations,” Journal of

Economic Dynamics and Control 9, 225-240.

20


APPENDIX A: z-Transform Methods∗

Consider a serially-correlated stochastic process at that can be expressed as a weighted sum of i.i.d. innova-tions:

at =∞∑

k=0

Aket−k.

While the innovations change through time, the weights Ak remain fixed. The stochastic process can thereforebe written succinctly as a function of the lag operator, L: at = A(L)et. The list of weights Ak can beviewed as a sequence, and by a fundamental theorem of analysis (Riesz-Fischer theorem, see Rudin (1974),pp. 86-90), are equivalent to functions of a complex variable z. The function of the lag operator A(L) is thenmathematically equivalent to a function A(z) of a complex variable z. The function A(z) can be analyzedwith the rules of complex analysis, and this, in turn, fully characterizes the stochastic process at.

An important feature of complex analysis is that the properties of a function are characterized by thedomain over which they are specified—the unit disk, or sets that are topologically equivalent to the unitdisk, are often the domains of interest. If a complex function on the disk can be expressed as a Taylorexpansion—an infinite series where the powers of the independent variable, z, range from zero to infinity—then the function is said to be analytic on the disk. However, some functions, termed meromorphic functions,when expressed as a generalized Taylor expansion—a Laurent expansion—have both positive and negativepowers of z, defined in an annular region containing the unit circle. This implies that they correspond tofunctions containing negative powers of the lag operator, which means that they operate on future values ofa variable. If a variable is stochastic, this is not permissible, as it would mean that the future is predictable,contradicting its stochastic aspect. In particular, solutions to an agent’s optimization problem cannot beforward-looking.

The negative powers of z in meromorphic functions arise from poles. The sum of the negative powers isthe principal part. 4

To eliminate negative powers of z in a posited solution to an agent’s optimization problem, we usethe annihilator operator, [·]+. The annihilator operator sets the coefficients of negative powers of z in theLaurent expansion to zero, while preserving all coefficients on the non-negative powers of z. This leaves apermissible, backward-looking solution to an agent’s optimization problem. A function with both backward-and forward-looking parts is converted to one with only backward-looking parts by the application of theannihilator. 5

A second property of a function concerns its invertibility. 6 If a serially-correlated stochastic processcan be represented by an invertible operator, the innovations of the process can be completely and exactlyrecovered by observing the history of the process. That is, the inverse of the operator applied to thevector of realizations of the process yields the vector of innovations, exactly as it would if a finite vector ofinnovations were converted into a finite vector of realizations by an invertible matrix. A function is invertibleon its domain if it does not take on a value of zero at any point inside the domain, and its inverse is thenanalytic. If, instead, an analytic function takes on a value of zero at a point inside the domain, then itis noninvertible. The inverse of a noninvertible function is not analytic. Hence, one cannot recover thevector of innovations by observing the vector of realizations, because inverting a function with a zero resultsin a function with negative powers of z. Recovery of the innovations would then depend on knowledge offuture realizations. The factorization theorem of Rozanov (1967) ensures that any process described by az-transform with either negative powers of z or zeroes can be converted into an observationally-equivalent

∗ This appendix duplicates the similar appendix in Bernhardt, Seiler and Taub (2005), which in turn builds on a muchearlier exposition in Whiteman (1985).

4 More precisely, a pole is a singularity located inside a region in the complex plane. Poles are only one possible type ofsingularity: there are also so-called essential singularities. Moreover, singularities need not be isolated points. In this paperthe discussion focuses on rational functions, which are characterized by poles alone. Engineering terminology also refers to afunction that is analytic as “causal”, and the presence of poles makes it non-causal.

5 For domain D it would be more appropriate to refer to [·]+

as the projection operator from L2(D) to H2(D), but the termis in widespread use.

6 In engineering parlance a function that is analytic and invertible is called minimum phase.

21


process that is characterized by an operator that is invertible and has only non-negative powers of z, so thatit is backward-looking.

To illustrate the variational method that we employ in the frequency domain, we present a simpleconsumer optimization problem. Consider an individual whose earnings evolve stochastically according toyt = A(L)et, where et is an i.i.d., zero mean, “white noise” period innovation to earnings. The consumer’sproblem is to adjust bond holdings bt

∞t=0 to maximize quadratic utility,

maxbt

−E∞∑

t=0

βt(yt + rbt−1 − bt)2, (A.1)

where r is the gross interest rate satisfying βr > 1. 7 The decision problem is to choose not just the initialvalue of bt, but the entire sequence bt

∞t=0. This problem implicitly requires the choice of functions that

react to current and possibly past states. Stationarity results in the same function applying each period.The stochastic component of a quadratic utility function is essentially a conditional variance. If innova-

tions are i.i.d., then the expectation of cross-products of random variables yields the sum of variances. Forwhite-noise innovations, for k > s, k > r,

Et−k

[

et−ret−s

]

=

0, r 6= sσ2

e , r = s,(A.2)

because of the independence of the innovations. Expressed in lag operator notation, this is

Et−k

[

(Lret)(Lset)

]

=

0, r 6= sσ2

e , r = s.

Notice that the “action” is in the exponents of the lag operators. From Cauchy’s theorem (Conway, 1978),it is equivalent to write

σ2e

1

2πi

∮

zrz−s dz

z=

0, r 6= sσ2

e , r = s,

where the integration is counterclockwise around the unit circle. In Cauchy’s theorem, z, which is a complexnumber with unit radius (it is on the boundary of the disk), is represented in polar form: z = e−iθ. Nowa more conventional integral can be undertaken, integrating over θ ∈ [0, 2π]. Using Euler’s theorem, whichrepresents complex numbers in trigonometric form, e−iθ = cos θ − i sin θ, gives θ the interpretation of afrequency, so that z and functions of z are in the frequency domain.

Whiteman (1985) demonstrated that a discounted conditional covariance involving complicated lags canbe succinctly expressed as a convolution. Consider two serially-correlated processes, at and bt, where

at =

∞∑

k=0

Aket−k and bt =

∞∑

k=0

Bket−k.

The discounted conditional covariance as of time t, setting realized innovations to zero, is

Et

[ ∞∑

s=1

βsat+sbt+s

]

= Et

[ ∞∑

s=1

βs

(

∞∑

k=0

Aket+s−k

)(

∞∑

k=0

Bket+s−k

)

]

. (A.3)

Because cross-product terms drop out, coefficients of like lags of et can be grouped:

β[A0B0 + βA1B1 + β2A2B2 + . . .]Et[e2t+1] + β2[A0B0 + βA1B1 + β2A2B2 + . . .]Et[e

2t+2] + . . .

= β[A0B0 + βA1B1 + β2A2B2 + . . .]σ2e + β2[A0B0 + βA1B1 + β2A2B2 + . . .]σ2

e + . . .

=βσ2

e

1 − β

∞∑

s=0

βkAkBk =βσ2

e

1 − β

1

2πi

∮

A(z)B(βz−1)dz

z.

(A.4)

7 To make this problem well-defined a (small) adjustment cost must also be included, but we suppress it here because thenet effect of the adjustment cost is just to make the solution stationary. Alternatively, one could simply impose the requirementthat any solution be stationary.

22


This is a useful transformation because the integrand is a product. Because the optimal policy for anoptimization problem in which the objective is an expected value like that in (A.3), the representation in(A.4) permits a direct variational approach. Equation (A.4) is an instance of Parseval’s formula, which statesthat the inner product of analytic functions is the sum of the products of the coefficients of their power seriesexpansions.

Optimization in the frequency domain. We apply these insights to the consumer’s optimization prob-lem. Hansen and Sargent (1978, 1979) showed that the first-order conditions of linear-quadratic stochasticoptimization problems could be expressed in lag-operator notation, z-transformed, and solved. Whitemannoticed that the z-transformation could be performed on the objective function itself, skipping the step offinding the time-domain version of the Euler condition. 8 The objective is then a functional, i.e., a mappingof functions into the real line. One can then use the calculus of variations to find the optimal policy function.

The first step is to conjecture that the solution to the agent’s optimization problem must be an analyticfunction of the fundamental process et:

bt = B(L)et.

The agent’s objective can then be restated in terms of the functions A and B, and the innovations:

maxB(·)

−E

[ ∞∑

t=0

βt(

(A(L) − (1 − rL)B(L))et

)2]

.

Expressing the objective in frequency-domain form, using the equivalence established in (A.4), the agent’sobjective can be written as

maxB(·)

−βσ2

e

1 − β

1

2πi

∮

(A(z) − (1 − rz)B(z))(A(βz−1) − (1 − rβz−1)B(βz−1))dz

z.

The variational method. Let ζ(z) be an arbitrary analytic function on the domain z : |z| ≤ β12 , and

let a be a real number. Let B(z) be the agent’s optimal choice. His objective can be restated as

J(a) = maxa

−βσ2

e

1 − β

1

2πi

∮

(A(z)− (1− rz)(B(z) + aζ(z)))(A(βz−1)− (1− rβz−1)B(βz−1) + aζ(βz−1)))dz

z.

This is a conventional problem. Differentiating with respect to a and setting a = 0 yields the first-ordercondition describing the agent’s optimal choice of B(·):

J ′(0) = 0 = −βσ2

e

1 − β

1

2πi

∮

ζ(z)(1 − rz)(A(βz−1) − (1 − rβz−1)B(βz−1))dz

z

−βσ2

e

1 − β

1

2πi

∮

ζ(βz−1)(1 − rβz−1)(A(z) − (1 − rz)B(z))dz

z.

Observe the symmetry between the two integrals—everywhere βz−1 appears in the first integral, z appearsin the second, and conversely. Whiteman establishes that the two integrals are in fact equal; we refer to thisproperty as “β-symmetry”. Therefore, the first-order condition simplifies to

0 = −1

2πi

∮

(A(z) − (1 − rz)B(z))(1 − rβz−1)ζ(βz−1)dz

z, (A.5)

where we have dropped the constantβσ2

e

1−β .

The integral in first-order condition (A.5) must be zero for arbitrary analytic functions ζ. By Cauchy’sintegral theorem, a contour integral around a meromorphic function with all its singularities inside the

8 A similar variational approach in continuous time can be found in Davenport and Root, p. 223.

23


domain—a function of z that has no component that can be represented as a convergent power seriesexpansion within the domain—is zero. Thus, all that is needed to make the integral in (A.5) zero is to makethe integrand singular inside the disk, and to have no singularities outside the disk. 9

Recall that a solution to the agent’s optimization problem must be an analytic function. The next step inthe solution is to separate the forward-looking components in (A.5) from the backward-looking components,so that we can then eliminate the non-analytic portion from our solution. Examining equation (A.5), notethat by construction ζ is analytic, so that it can be represented by a power series,

ζ(z) =

∞∑

j=0

ζjzj .

This means that ζ(βz−1) has an expansion of the form

ζ(βz−1) =

∞∑

j=0

ζjβjz−j ,

which has only nonpositive powers of z. The negative powers of z—all but the first term—define singularitiesat z = 0, which is an element of the unit disk. However, the rest of the integrand in (A.5), (1−rβz−1)(A(z)−(1−rz)B(z)), can have both positive and negative powers of z in its power series expansion. If it were possibleto guarantee that only negative powers of z appeared in (1−rβz−1)(A(z)− (1−rz)B(z)), then its expansionwould take the form

(1 − rβz−1)(A(z) − (1 − rz)B(z)) =∞∑

j=1

fjβjz−j ,

for some fj, and the product of this with ζ(βz−1) would take the form

ζ(βz−1)(1 − rβz−1)(A(z) − (1 − rz)B(z)) =∞∑

j=1

gjβjz−j .

for some gj. Every term in the sum is a singularity, and the integral of the sum is therefore zero.The first-order condition (A.5) can now be broken out of the integral and stated as follows:

(1 − rβz−1)(A(z) − (1 − rz)B(z)) =

−1∑

−∞

, (A.6)

where∑−1

−∞ is shorthand for an arbitrary function that has only negative powers of z, and hence cannot bepart of the solution to the agent’s optimization problem. This type of equation is known as a Wiener-Hopf

equation.

Factorization. To solve the Wiener-Hopf equation of a stochastic linear-quadratic optimization problem,we must factor the equation to separate the nonanalytic parts from the analytic parts. The factorizationproblem is a generalization of the problem of solving a quadratic equation, but there is no general formulafor the solution. However, if a candidate factorization can be found, then even if it is not analytic andinvertible, there is a general formula for converting that solution into an analytic and invertible factorization(Ball, Gohberg, Rodman (1990)).

The Wiener-Hopf equation (A.6) can be restated as:

(1 − rβz−1)(1 − rz)B(z) = (1 − rβz−1)A(z) +

−1∑

−∞

.

9 The assertion is an indirect way of stating that the contour of integration is treating the outside of the circle (including∞) as the domain over which the meromorphic function has no poles so that it is analytic there: Cauchy’s theorem assertsthat the integral in this sense is zero.

24


At this point it should be emphasized that the solution will be a Wiener filter, as opposed to a Kalman filter.A Kalman filter recursively reacts to information from the previous period and converges as the history ofinformation evolves after its initiation. A Wiener filter explicitly treats history as infinite and therefore astarting date in the infinite past; the stationarity of the model dictates the use of the Wiener approach.

It is tempting to solve for B(z) by dividing the left-hand side by the coefficient of B(z), (1− rβz−1)(1−

rz). However, this would multiply the∑−1

−∞ term by positive powers of z, making it impossible to establishthe coefficients of the positive powers of z in the solution.

The correct procedure is first to factor the coefficient of B(z) into the product of analytic and non-analytic functions:

(1 − rβz−1)(1 − rz) = βr2(1 − (βr)−1βz−1)(1 − (βr)−1z).

Because by assumption 1r < β1/2, the first factor on the right-hand side, (1 − (βr)−1βz−1), when inverted

has a convergent power series (on the disk defined by z∣

∣|z| ≤ β1/2)) in negative powers of z. Hence, wecan divide through by this factor to rewrite the Wiener-Hopf equation as

βr2(1 − (βr)−1z)B(z) =(1 − rβz−1)

1 − (βr)−1βz−1A(z) +

−1∑

−∞

, (A.7)

where we use the fact that1

(1 − (βr)−1βz−1

−1∑

−∞

has only negative powers of z. Because the left-hand side of (A.7) is the product of analytic functions,applying the annihilator to (A.7) yields

βr2(1 − (βr)−1z)B(z) =

[

(1 − rβz−1)

(1 − (βr)−1βz−1)A(z)

]

+

.

Because (β1/2r)−1 < 1, it follows that the inverse of (1− (βr)−1z) is also analytic, so that we can divide by(1 − (βr)−1z) to solve for the optimal B(z),

B(z) =

[

(1 − (βr)−1βz−1)−1(1 − rβz−1)A(z)]

+

[(βr2)(1 − (βr)−1z)].

A more explicit solution for B(z) obtains if the endowment process is AR(1), so that

A(z) =1

1 − ρz.

Proposition A.1 establishes a key result that is used repeatedly: the annihilate when there is an AR(1)construct can be simply calculated—if A(z) is an AR(1), then

[

f(βz−1)A(z)]

+= f(βρ)A(z).

PROPOSITION A.1: If f is analytic on β−1/2 and ρ < β−1/2, then[

f∗(1 − ρz)−1]

+= f(βρ)(1− ρz)−1.

PROOF: Direct computation (see, e.g., Taub (1986)).

Proposition A.2 shows that the proposition about annihilates of first-order AR functions must be usedwith caution. If there is a zero in the annihiland, the proposition changes.

PROPOSITION A.2: Let a < β−1/2. Then[

f∗ 1− 1a z−1

1−az

]

+= 0.

PROOF:[

f∗ 1 − 1az−1

1 − az

]

+

=1

a

[

z−1f∗ az − 1

1 − az

]

+

=1

a

[

−f∗z−1]

+= 0.

Using Proposition A.1, it follows that

B(z) =(1 − rβ)A(z)

[(βr2)(1 − (βr)−1βρ)(1 − (βr)−1z)].

25


This formula has a simple “permanent income” interpretation: the agent applies the filter

1 − rβ

[(βr2)(1 − (βr)−1βρ)(1 − (βr)−1L)]

to the endowment process A(L)et in order to smooth consumption.

Vector formulation. The ideas presented above apply with little change to our multi-agent model ofstrategic informed stock trading. The primary difference is that our economy has multiple informed traders,so there is a vector of fundamental processes. Because of this, a vector formulation of the translation to thefrequency domain and manipulations within the frequency domain must be used.

Consider a vector model with objective

maxB(·)

−E

[

∞∑

t=0

βt(

(A(L) − B(L)F (L))et

)2]

,

where et is now a vector of fundamental processes with covariance matrix S. Using the trace operator, theobjective can be written as

maxB(·)

−E

[

∞∑

t=0

βttr(

(A(L) − B(L)F (L))et

)2]

.

Commuting under the trace, taking the expectation, and transforming to the frequency-domain yields theobjective

maxB(z)

1

2πi

∮

tr[(A(βz−1) − B(βz−1)F (βz−1))S(A(z) − B(z)F (z))′]dz

z, (A.8)

where F (z) is a function analogous to the net bond trade ρ − rz in a vector setting and ′ denotes thetranspose.

As in the scalar case, a variational procedure is used to solve (A.8). The variation is B(z) + aζ(z).When the derivative is taken inside the integral and trace, the first-order condition can be stated:

0 = −1

2πi

∮

tr[(A(βz−1) − B(βz−1)F (βz−1))SF (z)′ζ(z)′]dz

z−

1

2πi

∮

tr[ζ(βz−1)F (βz−1)S(A(z) − B(z)F (z))′]dz

z

Exploiting β-symmetry (the integrals are equal), the Wiener-Hopf equation simplifies to

(A(z) − B(z)F (z))SF (βz−1)′ =

−1∑

−∞

,

where∑−1

−∞ is a vector of functions of strictly negative powers of z. Taking transposes, the equation can berewritten as

B(z)F (z)SF (βz−1)′ = A(z)SF (βz−1)′ −

−1∑

−∞

.

The factorization theorem applies here as well: F (z)SF (βz−1) can be factored into the product of twomatrixes, H(z) and H(βz−1),

H(z)H(βz−1) = F (z)SF (βz−1)′,

where every entry in the matrix H is an analytic function and its determinant has no zeroes. The solutionis then

B(z) =[

A(z)SF (βz−1)′H(βz−1)−1]

+H(z)−1.

The factorization problem can be significantly more difficult in a matrix setting than in a scalar setting; themethods set out in Ball and Taub (1991) typically must be used. Solving the annihilate remains possible ifthe functional forms are tractable; if A(z) has an AR structure then explicit solutions again obtain, just asin the scalar case.

Because matrix formulations are notationally complicated, it is standard to abbreviate functions asfollows: instead of f(z), one simply writes f , and instead of f(βz−1), one writes f∗. In the mathematicalliterature f∗ is understood to mean the transpose as well: f ∗ = f(βz−1)′. However, in this paper, thetranspose is kept as an explicit separate notation.

26


APPENDIX B: PROOFS AND DERIVATIONS

This appendix has two main parts. The first part is the proof of Proposition 3.1. The second partprovides the derivations for the frequency-domain objectives in section 5.

We begin with the proof of Proposition 3.1. Elements of this proof were developed jointly with DanBernhardt. The proof uses two lemmas that are also presented here. Lemma B.1 establishes that matriceswith a particular structure arising from geometric series are positive definite, and this is used to establishthat a second-order condition is satisfied. The second lemma, Lemma B.2, establishes that the geometricseries structure in Lemma B.1 holds in equilibrium.

PROOF: The first-order condition (3.2) has three elements. The middle element, λ(L)(Xt+ut), containsonly current and past xit and requires no additional analysis. We analyze the other two elements separately.

We begin with the term E[vt

∣

∣(Aiet)t, (Xt + ut)

t]. Using the law of recursive projections,

E[

vt

∣

∣(Aiet)t, (Xt + ut)

t]

= E[

vt

∣

∣(Xt + ut)t]

+ E

[

vt − E[

vt

∣

∣(Xt + ut)t]

∣

∣

∣

∣

Aiet − E[

Aiet

∣

∣(Xt + ut)t]

]

= pt + E

[

vt − E[

vt

∣

∣(Xt + ut)t]

∣

∣

∣

∣

Aiet − E[

et

∣

∣(Xt + ut)t]

]

,

becauseE[

vt

∣

∣(Xt + ut)t]

= pt.

By construction, (Xt + ut)t is orthogonal to the forecast error Aiet − E

[

Aiet

∣

∣(Xt + ut)t]

so that

E

[

E[

vt

∣

∣(Xt + ut)t]

∣

∣

∣

∣

Aiet − E[

Aiet

∣

∣(Xt + ut)t]

]

= 0

This establishes the result for the vt term.We next turn to the analysis of the final term in the first order condition, q(βL−1)λ(βL−1)xit. We

must rule out dependence of xi,t+s on public information (Xt + ut)t+s. Suppose the contrary: that xi,t+s is

a function of public information (Xt + ut)t+s. The discounted expected payoff for the speculator i can be

broken into two components:

E

[

s∑

τ=0

βτ (vt+τ − pt+τ )xi,t+τ

∣

∣

∣ei,t, (Xt + ut)t

]

+ E

[

∞∑

τ=s


∣

∣

∣ei,t, (Xt + ut)t

]

(B.1)

We proceed in two steps. First, we demonstrate that expected period profits are bounded, and therefore thesecond term shrinks to zero as s approaches infinity. Second, we demonstrate that the first term, and theoptimal strategy that optimizes the first term, are independent of (Xt + ut)

t and converge to the infinitehorizon case.

Both steps rest on the assumption that the pricing function and also the strategies of the other spec-ulators are linear and stationary. As a result, the response of price to speculator i’s trades is a fixed linearresponse. We can therefore represent the price process pt as a linear function of current and lagged xit, withthe response to trader i’s order by the other traders already been incorporated into the price function in astationary fashion:

pt = ξt + ξ0xit + ξ1xi,t−1 + ξ2xi,t−2 + . . .

where the term ξt captures the conditional expectation (because of certainty equivalence) of price terms thatdo not interact directly with speculator i’s trade.

(i) Boundedness. Because speculator i’s objective is quadratic, any stationary linear strategy will resultin bounded expected period profits. Therefore, in order for speculator i to obtain unbounded profits he mustuse a nonstationary or unbounded strategy. Suppose there is such a strategy. The strategy would exploitthe dependence of price on speculator i’s past as well as current order flow xit. For example, speculator imight manipulate price by shorting the stock at time t so that price becomes negative in period t + 1, andthen exploit the negative price at time t + 1 with a large long trade. The loss he incurs by going short attime t would be made up at time t + 1 by the long trade.

27


In the argument that follows we show that the equilibrium pricing function puts more weight on currentorder flow than on lagged order flow, and this guarantees that the costs incurred in driving the price negativeone period ahead can never be recouped. As a result the arbitrage strategy fails. The optimal strategy isthen characterized by first order conditions, and the objective, conditional on time-t information, is bounded.

If there is an arbitrage strategy, then at time t the speculator could set it up as a deterministic responseto the realized outcomes at time t, and ignore the (stationary) subsequent outcomes in expectation. At timet the speculator attempting to execute this arbitrage solves this deterministic problem:

maxxi,t+s∞

0

∞∑

s=0

βs(vt+s − pt+s)xi,t+s

By our assumption that the price function can be represented as linear stationary function of current andpast xit, the objective then becomes

maxxi,t+s∞

0

∞∑

s=0

βs(vt+s − (ξt,t+s + ξ0xi,t+s + ξ1xi,t+s−1 + ξ2xi,t+s−2 + . . .))xi,t+s

where the v terms are the conditional expected values of the vt+s terms. They evolve from the current andpast realizations of the ejt and ut, but the future realizations of the e’s and u’s are zero in expectation. Therealized parts of the e’s and u’s effectively create deterministic terms that could be the foundation of anarbitrage strategy. Because the problem is deterministic, the speculator can choose the entire sequence offuture orders xi,t+s as well as the current order xit.

It is helpful to first consider the finite-horizon version of this deterministic problem. We can representthe finite horizon problem as optimizing a quadratic form with a linear part and a quadratic part in whichthe optimization is over the vector of current and future orders (xi,t+s)

T0 :

( xi,0 xi,1 . . . xt . . . xi,t+T )

`0β`1...

βt`t...

βT `T

− ( xi,0 xi,1 . . . xt . . . xi,t+T )

ξ0 0 0 0 . . . 0βξ1 βξ0 0 0 . . . 0...

...βtξt βtξt−1 . . . βtξ0 . . . 0

......

βt+T ξt+T βt+T ξt+T−1 βt+T ξt+T−2 . . . . . . βt+T ξ0

xi,0

xi,1

. . .xit

. . .xi,t+T

where the vector ` abstractly represents the linear coefficients of the objective.We can write the problem more compactly as

maxxT

x′T `T − x′

T MT xT

where the time subscripts emphasize that we are considering the finite-horizon problem.We can calculate the second-order condition of the finite-horizon problem explicitly. Since our interest

is in the second order condition, we need only determine whether the central matrix MT is negative definite.Removing the negative sign, this is equivalent to establishing that the internal matrix is positive definite.We can establish negative definiteness by considering the symmetrized version of the problem:

maxxT

x′T `T −

1

2x′

T (MT + M ′T )xT = max

xT

x′T `T − x′

T AT xT

28


Because AT is symmetric, AT is positive definite if its eigenvalues are positive. In Lemma B.2 we establishthat ξ0 is positive definite, and denoting the kth eigenvalue of ξj by δk(ξj), that the eigenvalues form apositive decreasing sequence δk(ξ0) > δk(ξ1) > . . . > 0. This in turn means that AT satisfies the matrixstructure in the hypothesis of Lemma B.1, and therefore it is positive definite.

In the limit, the final row of the matrix MT , which is multiplied by βt+T , converges to zero, and thestrict negative definiteness of −A∞ then might fail. Because this term converges to zero, the quadraticpenalty also converges to zero and the infinite-horizon objective effectively becomes linear. A nonstationaryarbitrage strategy might then exist which manipulates price and then unwinds the position at infinity.

An arbitrage strategy that attempted to exploit this asymptotic linearity would fail for the followingreason. Consider a strategy xit that exploits this. The growth of xit must be bounded below 1/βt so thatthe shrinkage of βt will dominate it, driving the quadratic penalty to zero. But the gain from this strategyoccurs from the linear term in the objective, and this is also discounted at rate βt. Therefore the gainfrom this strategy also shrinks to zero, and it must be suboptimal. This demonstrates that nonstationaryarbitrage strategies are suboptimal. The optimal strategy must be stationary, and this in turn implies thatconditional expected period profits are bounded. Therefore the second term in (B.1) converges to zero inthe limit as s approaches infinity.

(ii) Convergence. We now approximate the optimal solution xi,t+τ by the optimal solution to a finite-horizon model. We assume that the price function is the same in both the infinite horizon case and the finitehorizon case. This is valid because we are assuming that the strategies xj,t+τ of the other traders are thesame in both the finite and infinite-horizon cases: we are only changing the strategy of speculator i.

Denote by xs,it the solution of the finite-horizon optimization problem from the first term of (B.1):

maxxit

E

[

s∑

τ=0


∣

∣

∣ei,t, (Xt + ut)t

]

(B.2)

In the terminal period t + s the finite-horizon strategy xs,it has the property that it is independent of(Xt+ut)

t+s. This is because at the terminal period t+s the optimization is independent of any future outcomeor action, along with our previous argument about independence of the term E[vt+s|(Xt + ut)

t+s, ei,t+s] inthe first part of this proof. At time t + s − 1, the original argument goes through, and in addition there isno dependence that can arise from the dependence of the future xi,t+s. This logic holds recursively for eachprior period. Thus xi,t is independent of (Xt + ut)

t.Now we argue that lims→0 xx,t = xt. Having established the boundedness of expected period utility, the

second term in (B.1) tends to zero in the limit, and therefore the finite-horizon discounted expected utility

E

[

s∑

τ=0

βτ (vt+τ − pt+τ )xs,i,t+τ

∣

∣

∣ei,t, (Xt + ut)t

]

must approach the first term in (B.1). Because the quadratic objective is concave, xs,it converges to xit.Therefore we have approximated xit by xs,i,t, which is independent of (Xt + ut)

t. This completes the proof.

LEMMA B.1: Let 0 < β < 1, 0 < a < 1, and define

MT ≡

1 0 0 0 . . . 0βa β 0 0 . . . 0...

...βtat βtat−1 . . . βt . . . 0

......

βT aT βT aT−1 βT aT−2 . . . . . . βT

Define AT = 12 (MT + M ′

T ). Then AT is positive definite.

29


PROOF: We explicitly consider only the undiscounted case with β ≡ 1 here. The argument in thediscounted case is virtually identical. Without discounting we can write AT explicitly as

2 a a2 a3 . . . at+T

a 2 a a2 . . . at+T−1

......

at at−1 . . . 2 . . . aT

......

at+T at+T−1 at+T−2 . . . . . . 2

We can write this as

AT = I + BT = I +

1 a a2 a3 . . . at+T

a 1 a a2 . . . at+T−1

......

at at−1 . . . 1 . . . aT

......

at+T at+T−1 at+T−2 . . . . . . 1

We now establish that BT is a positive definite matrix. B is positive definite if B−1 is positive definite, andB−1 is positive definite if its eigenvalues are all strictly positive because B−1 is symmetric. It can be verifiedby direct multiplication that

B−1 =1

1 − a2

1 −a 0 0 . . . 0 0 0−a 1 + a2 −a 0 . . . 0 0 00 −a 1 + a2 −a . . . 0 0 0...

...0 0 0 0 . . . −a 1 + a2 −a0 0 0 0 . . . 0 −a 1

It suffices to show that the eigenvalues of the numerator matrix must be positive. The characteristic equationis

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

1 − λ −a 0 0 . . . 0 0 0−a 1 + a2 − λ −a 0 . . . 0 0 00 −a 1 + a2 − λ −a . . . 0 0 0...

...0 0 0 0 . . . −a 1 + a2 − λ −a0 0 0 0 . . . 0 −a 1 − λ

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

= 0

Define the determinant

DT ≡

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

1 + a2 − λ −a 0 0 . . . 0 0 0−a 1 + a2 − λ −a 0 . . . 0 0 00 −a 1 + a2 − λ −a . . . 0 0 0...

...0 0 0 0 . . . −a 1 + a2 − λ −a0 0 0 0 . . . 0 −a 1 − λ

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

Then the characteristic equation can be written more succinctly as

(1−λ)DT−1+a

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

−a −a 0 0 . . . 0 0 00 1 + a2 − λ −a 0 . . . 0 0 00 −a 1 + a2 − λ −a . . . 0 0 0...

...0 0 0 0 . . . −a 1 + a2 − λ −a0 0 0 0 . . . 0 −a 1 − λ

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

= (1−λ)DT−1−a2DT−2

30


Thus the characteristic equation is(1 − λ)DT−1 − a2DT−2 = 0

It is also immediate thatDt = (1 + a2 − λ)Dt−1 − a2Dt−2

This is a second-order linear difference equation. It is more useful to express it in first-order nonlinear form:

Dt

Dt−1= (1 + a2 − λ) −

a2

Dt−1

Dt−2

or

xt = (1 + a2 − λ) −a2

xt−1

The terminal condition is thenDT−1

DT−2= xT−1 =

a2

1 − λ

There is an initial condition as well:

D1 = 1 − λ D2 = (1 + a2 − λ)(1 − λ) − a2

and therefore

x2 =(1 + a2 − λ)(1 − λ) − a2

1 − λ

We now demonstrate that λ cannot be negative. Suppose λ = −b, b > 0. Then

x2 = 1 + a2 + b −a2

1 + b> 1

Similarly,

xt = 1 + a2 + b −a2

xt−1> 1

if xt−1 > 1. Therefore xT−1 > 1. But the terminal condition is then

xT−1 =a2

1 + b< 1

which contradicts xT−1 > 1 from the previous step.Because the sum of positive definite matrices is positive definite, it follows that AT = I +BT is positive

definite.LEMMA B.2: Denote the kth eigenvalue of ξj by δk(ξj). Then for each k, k = 1, . . . ,M , δk(ξ0) >

δk(ξ1) > . . . > 0. Moreover, the δk(ξj) are proportional to sums of geometric series.PROOF: Examining the definition of q(L), we have

q(L)λ(L) ≡ (I + γi)−1(I +

∑

j

γj(L))λ(L) = (I − γi(L))−1µ(L)

In the symmetric setting the poles of (1 + γi)−1 and (I +

∑

j γj(L)) cancel, leaving a ratio of polynomialsof identical order. Because λ consists of a single pole term, the partial fractions expansion is a sum of polesform. Moreover, because the constituent terms are required to be invertible and analytic, we are guaranteedthat the poles are outside the unit circle. The pole terms can then be expanded in geometric series.

The remaining task is to demonstrate that I −γi(0) and µ(0) are positive definite. We show this for thesymmetric case. Because ξ(L) ≡ q(L)λ(L), we just need to demonstrate that q(0)λ(0) is positive definite.We therefore just need to demonstrate that I − γi(0) and µ(0) are positive definite. We show this for thesymmetric case.

31


From equation (5.6) the formula for µ′ is

µ′ = J−1[

J∗′−1b∗A∗ΣeΦ

′]

+

(i) The J function is formed from a factorization, therefore J0 is a standard deviation, which is positive.(The same then applies to J−1.)

(ii) The Φ function is an autoregressive process, and Φ0 is positive by assumption.(iii) From the definition of J , We can write

b∗A∗ΣeA′b′ = J∗J − Σ2

u

Because of the definition of J∗J , this expression is a Hermitian positive-definite rational matrix. Therefore

it can be factored and has a positive coefficient b∗A∗Σ1/2e .

(iv) From Lemma C.19, we have that ‖γi‖ ≤ 1N and therefore I + γi(0) is positive definite, and finally

(I + γi(0))−1 is positive definite.

We now turn to the first-order conditions for the frequency-domain problems in section 5.The first-order condition for bi. The variational method described in Whiteman (1985, pp. 235-7) will

be applied to find the first order conditions. These conditions will then be analyzed using functional analysismethods like those in Futia (1982). Consider a that is the optimum, bj , and a variation αη(z), where η isanalytic on the β−1/2 disk and α is a real scalar. Substitute the sum bj + αη(z) into the objective, take thederivative with respect to α, then set α to zero. The resulting variational first order condition is

0 =1

2πi

∮

tr

Φ′ −

N∑

k=1

(

bkeAk + γk

N∑

`=1

bèA`

)′

λ′

(

η∗A∗j + γ∗

j η∗A∗j

)

Σe

−

(

ηAj +

N∑

k=1

γkηAj

)′

λ′

(

b∗jeA∗j + γ∗

j

N∑

k=1

b∗keA∗k

)

Σe

dz

z

Observe that the noise-trade terms drop out completely from this expression. The next step is to useβ-symmetry:

0 =1

2πi

∮

tr

Φ′ −

N∑

k=1

(

bkeAk + γk

N∑

`=1

bèA`

)′

λ′

(

η∗A∗j + γ∗

j η∗A∗j

)

Σe

−

(

η∗A∗j +

N∑

k=1

γ∗kη∗A∗

j

)′

λ∗′

(

bjeAj + γj

N∑

k=1

bkeAk

)

Σe

dz

z

Next, commute under the trace operator in order to isolate the η∗ terms:

0 =1

2πi

∮

tr

A∗jΣe

Φ′ −N∑

k=1

(

bkeAk + γk

N∑

`=1

bèA`

)′

λ′

(

I + γ∗j

)

η∗

−η∗′

(

I +

N∑

k=1

γ∗k

)′

λ∗′

(

bjeAj + γj

N∑

k=1

bkeAk

)

ΣeA∗j′

dz

z

The transpose operator is the identity under the trace:

0 =1

2πi

∮

tr

A∗jΣe

Φ′ −N∑

k=1

(

bkeAk + γk

N∑

`=1

bèA`

)′

λ′

(

I + γ∗j

)

η∗

−A∗jΣe

(

bjeAj + γj

N∑

k=1

bkeAk

)′

λ∗

(

I +

N∑

k=1

γ∗k

)

η∗

dz

z

32


The bracketed term must therefore satisfy

A∗jΣe

Φ′ −

N∑

k=1

(

bkeAk + γk

N∑

`=1

bèA`

)′

λ′

(

I + γ∗j

)

−A∗jΣe

(

bjeAj + γj

N∑

k=1

bkeAk

)′

λ∗

(

I +

N∑

k=1

γ∗k

)

=

−1∑

−∞

(B.3)

where∑−1

−∞ is an M × 1 vector. Observe that the noise process does not enter into this first-order conditionat all. The A∗

j coefficient cannot be removed by inversion since it is noninvertible by construction. Theexpression can be rewritten

A∗jΣe

N∑

k=1

(

bkeAk + γk

N∑

`=1

bèA`

)′

λ′(

I + γ∗j

)

+A∗jΣe

(

bjeAj + γj

N∑

k=1

bkeAk

)′

λ∗

(

I +N∑

k=1

γ∗k

)

= A∗jΣeΦ

′(I + γ∗j ) +

−1∑

−∞

(B.4)

which is the same as (5.2).

First-order condition for γi. Let γi = αη. Taking the derivative of the objective and setting it to zeroyields

0 =1

2πi

∮

tr

(

Φ′ −∑N

k=1

(

bkeAk + γk

∑N`=1 bèA`

)′

λ′

−(∑N

k=1 γk + IM )′λ′

)

(

η∗j

∑Nk=1 b∗keA

∗k η∗

j

)

(

Σe 00 Σu

)

dz

z

−1

2πi

∮

tr

(

∑N`=1 (bèA`)

′η′

jλ′

η′jλ

′

)

(

b∗jeA∗j + γ∗

j

∑Nk=1 b∗keA

∗k γ∗

j

)

(

Σe 00 Σu

)

dz

z

Now impose β-symmetry:

0 =1

2πi

∮

tr

(

Φ′ −∑N

k=1

(

bkeAk + γk

∑N`=1 bèA`

)′

λ′

−(∑N

k=1 γk + IM )′λ′

)

(

η∗j

∑Nk=1 b∗keA

∗k η∗

j

)

(

Σe 00 Σu

)

dz

z

−1

2πi

∮

tr

(∑N

`=1 (b∗èA∗` )

′η∗

j′λ∗′

η∗j′λ∗′

)

(

bjeAj + γj

∑Nk=1 bkeAk γj

)

(

Σe 00 Σu

)

dz

z

or

0 =1

2πi

∮

tr

(

Φ′ −∑N

k=1

(

bkeAk + γk

∑N`=1 bèA`

)′

λ′

−(∑N

k=1 γk + IM )′λ′

)

η∗j

(

∑Nk=1 b∗keA

∗k I

)

(

Σe 00 Σu

)

dz

z

−1

2πi

∮

tr

(

∑N`=1 (b∗èA

∗` )

′

I

)

η∗j′λ∗′

(

bjeAj + γj

∑Nk=1 bkeAk γj

)

(

Σe 00 Σu

)

dz

z

The transpose is the identity under the trace:

0 =1

2πi

∮

tr

(

Φ′ −∑N

k=1

(

bkeAk + γk

∑N`=1 bèA`

)′

λ′

−(∑N

k=1 γk + IM )′λ′

)

η∗j

(

∑Nk=1 b∗keA

∗k I

)

(

Σe 00 Σu

)

dz

z

−1

2πi

∮

tr

(

Σe 00 Σu

)

((

bjeAj + γj

∑Nk=1 bkeAk

)′

γ′j

)

λ∗η∗j

(

∑N`=1 (b∗èA

∗` ) I

)

dz

z

33


Now commute under the trace:

0 =1

2πi

∮

tr

η∗j

(

∑Nk=1 b∗keA

∗k I

)

(

Σe 00 Σu

)

(

Φ′ −∑N

k=1

(

bkeAk + γk

∑N`=1 bèA`

)′

λ′

−(∑N

k=1 γk + IM )′λ′

)

dz

z

−1

2πi

∮

tr

η∗j

(

∑N`=1 (b∗èA

∗` ) I

)

(

Σe 00 Σu

)

((

bjeAj + γj

∑Nk=1 bkeAk

)′

γ′j

)

λ∗

dz

z

The resulting Wiener-Hopf equation is

(

∑Nk=1 b∗keA

∗k I

)

(

Σe 00 Σu

)

(

Φ′ −∑N

k=1

(

bkeAk + γk

∑N`=1 bèA`

)′

λ′

−(∑N

k=1 γk + IM )′λ′

)

−(

∑N`=1 (b∗èA

∗` ) I

)

(

Σe 00 Σu

)

((

bjeAj + γj

∑Nk=1 bkeAk

)′

γ′j

)

λ∗ =−1∑

−∞

(B.5)

Defining

Γ ≡ I +

N∑

k

γk

and

bA ≡ ( b1 b2 . . . bN ) (A1 A2 . . . AN ) ≡ bA

the condition can be written

( b∗A∗ I )

(

Σe 00 Σu

)(

Φ′ − A′b′Γ′λ′

−Γ′λ′

)

− ( b∗A∗ I )

(

Σe 00 Σu

)(

A′jb

′j + A′b′γ′

j

γ′j

)

λ∗ =

−1∑

−∞

or

( b∗A∗ I )

(

Σe 00 Σu

)(

A′b′

I

)

Γ′λ′ + ( b∗A∗ I )

(

Σe 00 Σu

)(

A′jb

′j + A′b′γ′

j

γ′j

)

λ∗

= ( b∗A∗ I )

(

Σe 00 Σu

)(

Φ′

0

)

+

−1∑

−∞

This can be simplified even further:

( b∗A∗ I )

(

Σe 00 Σu

)(

A′b′

I

)

(

Γ′λ′ + γ′jλ

∗)

=( b∗A∗ I )

(

Σe 00 Σu

)((

Φ′

0

)

−

(

A′jb

′j

0

))

+

−1∑

−∞

which is (5.4).

Derivations for the market-maker’s problem. The objective of the market maker is

maxλ

1

2πi

∮

tr

(

Φ′ −∑N

k=1

(

bkeAk + γk

∑N`=1 bèA`

)′

λ′

−(∑N

k=1 γk + IM )′λ′

)

×(

Φ∗ −∑N

k=1

(

b∗keA∗k + γ∗

k

∑N`=1 b∗èA

∗`

)

λ∗ −(∑N

k=1 γ∗k + IM )λ∗

)

(

Σe 00 Σu

)

dz

z

34


The variational condition is

0 =2

2πi

∮

tr

(

Φ′ −∑N

k=1

(

bkeAk + γk

∑N`=1 bèA`

)′

λ′

−(∑N

k=1 γk + IM )′λ′

)

×(

−η∗∑N

k=1

(

b∗keA∗k + γ∗

k

∑N`=1 b∗èA

∗`

)

−η∗(∑N

k=1 γ∗k + IM )

)

(

Σe 00 Σu

)

dz

z

Commuting under the trace yields

0 =2

2πi

∮

tr

(

−∑N

k=1

(

b∗keA∗k + γ∗

k

∑N`=1 b∗èA

∗`

)

−(∑N

k=1 γ∗k + IM )

)

×

(

Σe 00 Σu

)

(

Φ′ −∑N

k=1

(

bkeAk + γk

∑N`=1 bèA`

)′

λ′

−(∑N

k=1 γk + IM )′λ′

)

η∗ dz

z

so the bracketed term must satisfy

(

−∑N

k=1

(

b∗keA∗k + γ∗

k

∑N`=1 b∗èA

∗`

)

−(∑N

k=1 γ∗k + IM )

)

×

(

Σe 00 Σu

)

(

Φ′ −∑N

k=1

(

bkeAk + γk

∑N`=1 bèA`

)′

λ′

−(∑N

k=1 γk + IM )′λ′

)

=

−1∑

−∞

Observe that elements of this expression are identical to those in the first-order condition for γi, (5.4).Defining

Γ ≡ I +N∑

k

γk

and

bA ≡ ( b1 b2 . . . bN )

A1

A2...

AN

≡ bA

the condition can be written

(−Γ∗b∗A∗ −Γ∗ )

(

Σe 00 Σu

)(

Φ′ − A′b′Γ′λ′

−Γ′λ′

)

=

−1∑

−∞

or

Γ∗ ( b∗A∗ I )

(

Σe 00 Σu

)(

A′b′

I

)

Γ′λ′ = Γ∗ ( b∗A∗ I )

(

Σe 00 Σu

)(

Φ′

0

)

+

−1∑

−∞

which is (5.5).

35


APPENDIX C: PROOF OF EXISTENCE

In this appendix we prove existence. Our proof makes use of a contraction-mapping property to generatea fixed point. The contraction argument is non-standard in that it does not spring directly from an opti-mization problem such as consumer-surplus maximization, but rather relies on iteration of best responses, asexpressed in (6.3). In addition, since it depends on an initial guess that has a specific functional structure,uniqueness does not automatically follow, although we are confident that uniqueness does in fact hold.

Our proof develops a fixed point for the mapping implicit in the formulas for bi, γi, and µ. In orderto do this we first note that the functions we are examining have the property that they are analytic onthe disk with radius β1/2, and they are also square-integrable along the boundary of that domain. That is,functions with inner product and norm

⟨

f, g⟩

≡1

2πi

∫

|z|=β1/2

trf(z)g(z−1)′dz

z‖f‖

2≡

1

2πi

∫

|z|=β1/2

trf(z)f(z−1)′dz

z

where z ≡ β1/2z. They therefore are elements of the space H2, which is a Banach space (see Rudin, 1974,p. 364). A contraction argument therefore generates a fixed point.

H2 is a subspace of L2 (square-integrable functions with domain the unit circle), which has the same

inner product and norm, but in which poles are allowed. We can also say L2 = H2 ⊕ H2⊥.We need to extend the norm concepts to norms of the operators we are using. With the exception of

the annihilator operator, the operators we use have two essential properties: (i) they are multiplicative, i.e.,if x is an element of H2, then we consider mappings T : H2 → H2, with T [x] defined as f(z)x(z); (ii) f isitself an element of the appropriate H∞ space, which is a subset of H2. The question then becomes, whatis the relationship between the operator norm, i.e.,

‖f‖op ≡ sup‖x‖

2=1

‖fx‖2 ≡ sup‖x‖6=0

‖fx‖

‖x‖

and the 2-norm‖f‖2

which can be calculated because f is also an element of H2. A moment’s thought suggests that

‖f‖op = sup|z|=1

|f |

because an x can be found that “picks out” the supremum of f . Because ‖f‖2 is an average of |f | on theunit circle, we can then conclude that

‖f‖2 ≤ ‖f‖op

Our focus will be on the operator norm.The operator norm can be defined in the matrix context. Let f be a matrix operator f , that is a matrix

of analytic functions. The operator norm is

‖f‖op = supz:|z|=1

|λi(z)|ni=1

where |λi(z)ni=1 is the collection of singular values of f . [Note: this fact is demonstrated in Conway,

Theorem 1.5, p. 28.] For a matrix A, the singular values are the square roots of the eigenvalues λiNi=1 of

the product matrix A∗′A. The singular values coincide with absolute values of the eigenvalues of A whenA is positive semi-definite, or Hermitian. (References include Kailath, Sayed and Hassibi, p. 734, andSanchez-Pena and Sznaier, p. 460.)

A number of lemmas are needed for the proof. The first two lemmas establish some broad algebraicfacts.

LEMMA C.1: Let f and g be elements of H2 and define T : H2 → H2 by T [x] ≡ fgx, i.e., themultiplication operator. Then

‖T‖op ≤ ‖f‖op ‖g‖op

36


[Note: This is also a result in Conway, Proposition 1.2, p. 27.]PROOF: We have from the definition of the operator norm,

‖T‖ ≡ supx∈H2

‖(fg)[x]‖

‖x‖≤ sup

x∈H2

‖|f ||g[x]|‖

‖x‖≤ sup

y∈H2

‖(f)[y]‖

‖y‖sup

x∈H2

‖g[x]‖

‖x‖

The second inequality follows because y is not constrained to equal g[x].

LEMMA C.2: Let f be an element of H2, and define Tf as the associated multiplication operator,Tf : H2 → H2. Then

‖Tf‖op ≤ sup|z|=1

|f |

PROOF: Calculating the norm, we have

‖Tf‖2≡ sup

x∈H2:‖x‖=1

1

2πi

∮

|f |2|x|2dz

z≤ sup

‖x‖=1

1

2πi

∮

( sup|z|=1

|f |)2|x|2dz

z

= ( sup|z|=1

|f |)2 sup‖x‖=1

1

2πi

∮

|x|2dz

z

= ( sup|z|=1

|f |)2

LEMMA C.3: Let f ∈ H2. Then∥

∥[f ]+∥

∥

2≤ ‖f‖2.

PROOF: Write f =∑∞

−∞ fizi. Then

∥

∥[f ]+∥

∥ =

∞∑

i=0

|fi|2 ≤

∞∑

i=−∞

|fi|2

COROLLARY C.4: The operator norm of the annihilator is unity, i.e.∥

∥[·]+∥

∥

op = 1.

PROOF: This is because we can use an analytic x to calculate [x]+, so that [x]+ = x. Keep in mind thedistinction between H2, which is the square-summable analytic functions and the larger class L2 of squaresummable functions that includes poles. This lemma applies to this larger class.

COROLLARY C.5: Let f , g be elements of H∞. Then ‖f∗g‖op ≤ ‖f‖op ‖g‖op.PROOF: This is an application of Cauchy-Schwarz. That is,

‖f∗g‖ = sup|z|=1

|f(z)g(z−1)| ≤ sup|z|=1

|f(z)||g(z−1)|

by Cauchy-Schwarz, and we can then continue the inequality with

≤ sup|z|=1

|f(z)| sup|z|=1

|g(z−1)| ≡ ‖f‖op ‖g‖op .

Alternatively, ‖f∗g‖op ≤ ‖f∗‖op ‖g‖op by Lemma C.1, and then note that ‖f∗‖op = ‖f‖op directlyfrom the definition of the norm.

COROLLARY C.6: Let f , g be elements of H2. Then

|⟨

f, g⟩

2| =

∣

∣

∣

∣

∣

1

2πi

∫

|z|=1

fg∗dz

z

∣

∣

∣

∣

∣

≤ ‖f‖2 ‖g‖2 .

37


PROOF: We can simply invoke Cauchy-Schwarz on L2.

LEMMA C.7: Let x, g and h ∈ H2. Define the mapping T : H2 → H2 by T [x] = hx + g, where h is themultiplication operator. Let ‖h‖ ≤ ρ < 1 and let ‖g‖ < ∞. Then T is a contraction.

PROOF:‖T [f1] − T [f2]‖ = ‖h(f1 − f2)‖ ≤ ‖h‖ ‖f1 − f2‖2 ≤ ρ ‖f1 − f2‖2

where ‖h‖ is the operator norm. The first inequality uses Lemma C.1.

We also need a standard result from complex analysis:LEMMA C.8: Let f be a function of a complex variable that is analytic on the disk z

∣

∣|z| ≤ β1/2.Then f attains its maximum on the boundary of the disk.

We next state some lemmas that establish norm bounds on the functions in the mapping.LEMMA C.9: Let a matrix A have eigenvalues that are positive fractions. Then the eigenvalues of I−A

are positive fractions.PROOF:

I − A = I − V DV −1 = V (V −1IV − D)V −1 = V (I − D)V −1

and the last entry has eigenvalues I − D.COROLLARY C.10: Let A be positive definite so that its singular values are identical with its eigenvalues,

and let ‖A‖ < 1. Then ‖I − A‖ < 1.PROOF: If A has norm less than one then its maximal eigenvalue is also less than one, and therefore

all its eigenvalues are less than one. Therefore Lemma C.9 applies.

We next state some matrix simplifications that hold in the symmetric case. Define

p ≡

(

0 θθ 0

)

P ≡

0 0 0 θ0 0 θ 00 θ 0 0θ 0 0 0

Also, define bi as the representative trader’s trading intensity. Recall that

Φ(z) =

(

1 1 0 00 0 1 1

)

φ(z)

in the two-asset, two-trader case for example, where φ is the common autoregressive term in the valueprocess— Then we have the following relations:

LEMMA C.11: In the symmetric case,

(i) AΣeA′ = I + P

(ii) AiΣeA′i = I and

∑

i6=j

A∗i ΣeA

′j =

(

0 θθ 0

)

≡ p and A∗i ΣeΦ

′ =

(

1 θθ 1

)

φ = (I + p)φ

As a result,

(iii) b∗AΣeA′b′ = Nb∗i (I + p)b′i and b∗AΣeA

′ib

′i = b∗i (I + p)b′i

(iv) J∗′J = Nb∗i (I + p)b′i + Σu

(v) µ′ = J−1[

J∗′−1Nb∗i (I + p)φ

]

+

38


(vi) γ′ = J−1[

J∗′−1b∗i (I + p)b′i

]

+

PROOF: For (i) and (ii) calculate directly:

(A∗ΣeA′) =

1 0 0 00 0 1 00 1 0 00 0 0 1

1 0 0 θ0 1 θ 00 θ 1 0θ 0 0 1

1 0 0 00 0 1 00 1 0 00 0 0 1

= I + P

et cetera. For (iii), direct calculation:

b∗A∗ΣeA′b′ = b∗i ( I . . . I ) AΣeA

′

I...I

b′i

= b∗i ( I . . . I ) (I + P )

I...I

b′i

= b∗i (( I · · · I )) + ( p · · · p ))

I...I

b′i

= b∗i N(I + p)b′i

et cetera.

Now we are ready to discuss the specifics of the mapping. We begin by noting that the mapping mapsbetween appropriate spaces.

LEMMA C.12: Define the mapping T by the iteration in (6.5), with µ and γi defined as in (5.6) and(5.7). Then T : H2(Dβ) → H2(Dβ).

PROOF: The constituent elements in (6.5) are analytic, and analyticity is preserved under the operationsof multiplication, summation, factorization, and annihilation that comprise the mapping implicit in theformulas for γi, µ, and bi.

Next, we manipulate the mapping and use symmetry to simplify it.

THEOREM C.13: For sufficiently small values of θ and ρ, a stationary linear equilibrium exists.The proof uses a modified version of equation (6.4). First multiply both side of (6.4) by b∗:

b∗(A∗ΣeA′)b′ = b∗

[

∑

j 6=1 A∗1ΣeA

′jb

′jµ

∗G∗1′−1]

+G−1

1

...[

∑

j 6=N A∗NΣeA

′jb

′jµ

∗G∗N

′−1]

+G−1

N

+ b∗

[

A∗1ΣeΦ

′(I + γ∗1 )G∗

1′−1]

+G−1

1

...[

A∗1ΣeΦ

′(I + γ∗N )G∗

N′−1]

+G−1

N

As a practical matter the information filter terms Ai are scalar matrices and therefore we can bring themout of the annihilator operators:

b∗(A∗ΣeA′)b′ = b∗

∑

j 6=1 A∗1ΣeA

′j

[

b′jµ∗G∗

1′−1]

+G−1

1

...∑

j 6=N A∗NΣeA

′j

[

b′jµ∗G∗

N′−1]

+G−1

N

+ b∗

A∗1Σe

[

Φ′(I + γ∗1 )G∗

1′−1]

+G−1

1

...A∗

NΣe

[

Φ′(I + γ∗N )G∗

N′−1]

+G−1

N

39


Because of the symmetry, and using Lemma C.11, we can write each block row of the recursion in terms ofidentical generic trading intensity functions bi. Dividing by N then yields the following equation for the ithtrader:

b∗i (I + p)b′i = b∗i p[

b′iµ∗G∗

i′−1]

+G−1

i + b∗i (I + p)[

φ(I + γ∗i )G∗

i′−1]

+G−1

i (C.10)

The left hand side of the equation is the generalization of a quadratic form in bi, and it is an element of L2,the square-integrable functions. We will further embellish this recursion as a recursion of b∗i (I + p)b′i. Wewill demonstrate a contraction mapping in this recursion. bi can then be recovered from the fixed point byfactorization. In particular, because of the factorization step, bi is invertible (see Lemma C.14 below). 10

Integrating the left hand side of (C.10) around the unit circle, we obtain the squared 2-norm of (I +p)1/2b′i; thus dividing the integral of the right hand side by this norm and finding the supremum over z onthe unit circle yields the operator norm for T . We need to demonstrate the contraction property for thisnorm.

We can write the first term in C.10 as follows:

1

2πi

∮

b∗i p[

b′iµ∗G∗

i′−1]

+G−1

i

dz

z=

1

2πi

∮

b∗i (I + p)b′ip(I + p)−1b′i−1[

b′iµ∗G∗

i′−1]

+G−1

i

dz

z

≤∥

∥b∗i (I + p)−1∥

∥

2∣

∣

∣

∣

1

2πi

∮

p(I + p)−1b′i−1[

b′iµ∗G∗

i′−1]

+G−1

i

dz

z

∣

∣

∣

∣

so that we must demonstrate∣

∣

∣

∣

1

2πi

∮

p(I + p)−1b′i−1[

b′iµ∗G∗

i′−1]

+G−1

i

dz

z

∣

∣

∣

∣

is appropriately bounded.We will take the tack of demonstrating separately that

∥

∥

∥

∥

(I + p)−1/2pb′i−1[

b′iµ∗G∗

i′−1]

+G−1

i

∥

∥

∥

∥

op< α (C.11)

and that the second term in (C.10) satisfies the inequality∥

∥

∥

∥

b∗(I + p)[

φ(I + γ∗i )G∗

i′−1]

+G−1

i

∥

∥

∥

∥

op≤ α

1

N‖Nb∗i (I + p)b′i + Σu‖ (C.12)

where α is a positive fraction that will be determined. The two inequalities in (C.11) and (C.12) thengenerate a contraction.

We will return to the proof after developing a number of lemmas. The following lemmas are essential.LEMMA C.14: Let H(z) be a rational Hermitian matrix function (that is, such that H∗′ = H). Then

there is a factorization of H, FF ∗′, such that F is invertible on the unit disk, has no poles inside the unitdisk, and has real coefficients.

PROOF: This is the standard Rozanov theorem (Rozanov, 1967).LEMMA C.15: Let the coefficients of bi, γi, and µ be real. Then in the symmetric case the coefficients

of the iterated values bi, γi, and µ are real.PROOF: The standard Rozanov theorem on factorization yields real coefficients for the iterated value

of bi on the left hand side of (C.10). The iterated values of J and µ are constructed from J and Gi, whichare also obtained from factorization, and so they also have real coefficients.

LEMMA C.16: Let f be an analytic function that is the sum of pole terms, that is,

f(z) =k∑

i=1

ci

1 − aiz

10 There is one more detail: since I + p is positive definite, it has an invertible factorization and we can therefore construct

an invertible factor bi that satisfies b∗i (I + p)b′i = b∗i (I + p)b′i.

40


with ci, ai real. (Analyticity on D means that |ai| < 1.) Then arg supz|Re[f ]| = ±1 and arg infz|Re[f ]| = ±1.PROOF: We can consider f as a first-order pole and c1 = 1 without loss of generality.

2Re[f ] =1

1 − a1z+

1

1 − a1z−1=

∞∑

j=1

ai1(z

j + z−j) =

∞∑

j=1

ai12 cos(jθ)

which is convergent, and which takes maxima and minima at θ ∈ 0, π or z ∈ −1, 1.LEMMA C.17: Let f be an analytic function that is the sum of pole terms, that is,

f(z) =

k∑

i=1

ci

1 − aiz

with ci, ai real and |ai| < 1. Also, let h be an analytic function on D. Then arg infz|f + hh∗

f∗| = ±1 and

also infz|f + hh∗

f∗| ≥ 2infz|Re[f ]|.

PROOF: The coefficient hh∗

is a Blaschke factor and therefore∣

∣

hh∗

∣

∣ = 1. The effect of h/h∗ is to rotate the

value of h∗. However, at z = ±1, there is no rotation and |f+ hh∗

f∗|z=±1 = |Re[f ]|z=±1. Suppose the infimumis at z = −1. Therefore fore every z 6= −1, |Re[f(−1)]| < |Re[f(z)]| and |Re[f(z)]| < |Re[f(z)] + Im[f(z)]| =|f(z)|; the last inequality holds by the Pythagorean theorem.

We begin with the analysis of (C.11).

LEMMA C.18:∥

∥J−1J(0)−1Σu

∥

∥ < 1.PROOF:

∥

∥J−1J(0)−1Σu

∥

∥ ≤∥

∥J−1∥

∥

∥

∥J(0)−1∥

∥ ‖Σu‖

Recalling that∥

∥[·]+∥

∥ = 1 from Lemma C.4,∥

∥J−1∥

∥

∥

∥J(0)−1∥

∥ ‖Σu‖ =∥

∥J−1∥

∥

∥

∥

∥

[

J(z)−1]

+

∥

∥

∥‖Σu‖ ≤

∥

∥J−1∥

∥

∥

∥J(z)−1∥

∥ ‖Σu‖ =∥

∥J−1∥

∥

2‖Σu‖

Because J∗′J is Hermitian, the singular values of J and the square roots of the eigenvalues of ‖J‖2

are

identical, and therefore we can phrase the argument in terms of eigenvalues. Because ‖J‖2

is the sum oftwo positive definite matrices, one of which is Σu, then the last expression is a positive fraction. This latterassertion holds by application of Proposition 64 of Dhrymes, p. 74, setting B ≡ J ∗′J and A ≡ Σu.

The following two lemmas put norm bounds on coefficients in the recursion for the trading intensityfilter b.

LEMMA C.19: Assume symmetry across assets and agents. Then ‖γi‖ ≤ 1N .

PROOF:

‖γ′i‖ =

∥

∥

∥

∥

∥

−J−1

[

J∗′−1( b∗A∗ I )

(

Σe 00 Σu

)(

A′ib

′i

0

)]

+

∥

∥

∥

∥

∥

=

∥

∥

∥

∥

J−1[

J∗′−1b∗A∗ΣeA

′ib

′i

]

+

∥

∥

∥

∥

=

∥

∥

∥

∥

∥

J−1

[

J∗′−1b∗A∗Σe

1

NA′b′

]

+

∥

∥

∥

∥

∥

using symmetry assumption so bi = bj

=1

N

∥

∥

∥

∥

J−1[

J∗′−1(J∗′J − Σu)

]

+

∥

∥

∥

∥

definition of J

=1

N

∥

∥

∥

∥

J−1[

(JJ−1)J∗′−1(J∗′J − Σu)

]

+

∥

∥

∥

∥

=1

N

∥

∥

∥

∥

J−1([J ]+ −[

J∗′−1Σu

]

+)

∥

∥

∥

∥

(J∗′J)−1 = J−1J∗′−1

=1

N

∥

∥

∥I − J−1J(0)′−1

Σu

∥

∥

∥

≤1

NLemma C.18

41


We can make the latter argument by noting that

∥

∥

∥

∥

[

J∗′−1Σu

]

+

∥

∥

∥

∥

=

∥

∥

∥

∥

[

JJ−1J∗′−1Σu

]

+

∥

∥

∥

∥

≤∥

∥[J ]+∥

∥

∥

∥

∥

∥

[

J−1J∗′−1Σu

]

+

∥

∥

∥

∥

≤ ‖J‖

The symmetry across assets is also needed, so that

b∗A∗ΣeA′ib

′i = b∗i ( I I ) AΣeA

′ib

′i = Nb∗i

(

1 θθ 1

)

b′i

When the bi are not symmetric then 1N will be replaced by a looser bound.

COROLLARY C.20: Let γi be in sum of poles form with real coefficients. Then

‖I + γi‖ ≥N − 1

N

PROOF: Recall that γi is subtracted, and by the previous lemma ‖γi‖ ≤ 1N .

LEMMA C.21: The coefficient in equation (C.11) satisfies

∥

∥

∥

∥

(I + p)−1/2pb′i−1[

b′iµ∗G∗

i′−1]

+G−1

i

∥

∥

∥

∥

op≤ (I + p)−1/2p

PROOF:

∥

∥

∥

∥

(I + p)−1/2pb′i−1[

b′iµ∗G∗

i′−1]

+G−1

i

∥

∥

∥

∥

=

∥

∥

∥

∥

(I + p)−1/2pb′i−1[

b′iµ∗G∗

i′−1

G−1i Gi

]

+G−1

i

∥

∥

∥

∥

=∥

∥

∥(I + p)−1/2pb′i

−1[b′iµ

∗(µ′(I + γ∗) + (I + γ′)µ∗)−1Gi

]

+G−1

i

∥

∥

∥

Next, we can extract the norm of the internal term µ∗(µ′(I + γ∗) + (I + γ′)µ∗)−1:

∥

∥

∥(I + p)−1/2pb′i

−1[b′iµ

∗(µ′(I + γ∗) + (I + γ′)µ∗)−1Gi

]

+G−1

i

∥

∥

∥

≤∥

∥µ∗(µ′(I + γ∗) + (I + γ′)µ∗)−1∥

∥

∥

∥

∥(I + p)−1/2pb′i−1

[b′iGi]+G−1i

∥

∥

∥

≤∥

∥

∥(µ′µ∗−1(I + γ∗) + (I + γ′))−1∥

∥

∥

∥

∥

∥(I + p)−1/2pb′i−1

b′iGiG−1i

∥

∥

∥

≤∥

∥

∥(µ′µ∗−1(I + γ∗) + (I + γ′))−1

∥

∥

∥

∥

∥

∥(I + p)−1/2p

∥

∥

∥

Next, from C.23 below, the structure of γ is sum-of-poles with real coefficients, so we can invoke LemmaC.16:

≤‖2Re[I + γ′]‖∥

∥

∥(I + p)−1/2p∥

∥

∥ Lemma C.16

≤1

2(I + p)−1/2p

∥

∥(I + γ′)−1∥

∥

≤∥

∥

∥(I + p)−1/2p∥

∥

∥ Corollary C.19

These arguments have demonstrated the inequality in (C.11).

Our next step is to develop the inequality in (C.12).LEMMA C.22: Let 0 < a < 1, and let b(z) be a partial fractions expression with all coefficients ai such

that 0 < ai < a. Then∥

∥b(z−1)b(a)−1∥

∥ < 1 + a

42


PROOF: The partial fractions are a convex combination of individual pole terms, so we can considerone pole term without loss of generality. Thus

∥

∥b(z−1)b(a)−1∥

∥ =1 − aia

1 − ai<

1 − a2i

1 − ai= 1 + ai < 1 + a

where the constant terms have cancelled.LEMMA C.23: Let b take the form of sum of pole terms. Then γi, µ and T [b] are also sums of pole

terms.PROOF: There are two terms in T . Each term has an annihilate. The first annihilate,

[

b′iµ∗G∗

i′−1]

+,

is the sum of pole terms because b′i is the sum of pole terms and we can apply Proposition A.1 to each pole

term separately. The second annihilate,[

φ(I + γ∗i )G∗

i′−1]

+, is a single pole term because φ is a single pole

term by assumption, and we can again apply Proposition A.1.The remaining task is to demonstrate that G−1 is a ratio of polynomials of the same order. When

multiplied by a sum of pole terms, the result is a sum of pole terms.G is the factor of µ′(I + γ∗

i ) + (I + γ′i)µ

∗. We first characterize µ and γi. Both have annihilate termsand by similar reasoning the annihilates are in pole or sum of pole form. Each is multiplied by J−1; Bythe definition of J , J is a ratio of polynomials of the same order, and therefore J−1 is also the ratio ofpolynomials of the same order. When multiplied by the annihilates, which are pole terms, the product isthe sum of pole terms. Finally, the term I + γi is the ratio of polynomials of the same order. Therefore, theterm µ′(I + γ∗

i ) + (I + γ′i)µ

∗ is a ratio of polynomials of the same order in both z and z−1. Therefore G isthe ratio of polynomials of the same order.

LEMMA C.24: Let bi be in sum-of-pole terms with real coefficients form. Then∥

∥

∥

∥

b∗i (I + p)[

φ(I + γ∗1 )G∗

i′−1]

+G−1

i

∥

∥

∥

∥

≤1 + ρ

2

1

N

1 + ρ

1 − ρsup|Nb∗i (I + p)b′i + Σu|

PROOF: We first note that∥

∥

∥

∥

b∗i (I + p)[

φ(I + γ∗1 )G∗

i′−1]

+G−1

i

∥

∥

∥

∥

=

∥

∥

∥

∥

b∗i (I + p)[

φ(I + γ∗1 )G∗

i′−1

G−1i Gi

]

+G−1

i

∥

∥

∥

∥

=∥

∥

∥b∗i (I + p)[

φ(I + γ∗1 )(µ′(I + γ∗) + (I + γi)µ

∗)−1Gi

]

+G−1

i

∥

∥

∥

The term (I + γ∗1 )(µ′(I + γ∗) + (I + γi)µ

∗)−1 can be extracted:

≤∥

∥(I + γ∗1 )(µ′(I + γ∗) + (I + γi)µ

∗)−1∥

∥

∥

∥b∗i (I + p)[φGi]+G−1i

∥

∥

=∥

∥(I + γ∗1 )(µ′(I + γ∗) + (I + γi)µ

∗)−1∥

∥ ‖b∗i (I + p)φ‖

=∥

∥(µ′ + (I + γi)(I + γ∗i )−1µ∗)−1

∥

∥ ‖b∗i (I + p)φ‖

Because of the total symmetry µ = µ′, and therefore µ and (I + γi) commute in the expression (µ′(I + γ∗)+(I + γi)µ

∗)−1. Invoking Lemma C.17, we can write

≤∥

∥(2Re[µ])−1∥

∥ ‖b∗i (I + p)φ‖

Because µ is in sum-of-pole terms form, and all the coefficients are real by Lemma C.15, then by LemmaC.16 the supremum

∥

∥(Re[µ])−1∥

∥ will be attained at ±1, and therefore we can write∥

∥(2Re[µ])−1∥

∥ = 12

∥

∥µ−1∥

∥.Now we can write

1

2‖b∗i (I + p)φ‖

∥

∥

∥

∥

(J−1[

J∗′−1Nb∗i (I + p)φ

]

+)−1

∥

∥

∥

∥

=1

2

1

N‖b∗i (I + p)φ‖

∥

∥φ−1(I + p)−1bi(βρ)−1J(βρ)′J∥

∥

≤1

2

1

N

∥

∥b∗i bi(βa)−1∥

∥ ‖φ‖∥

∥φ−1∥

∥ ‖J(βa)′J‖

≤1

2

1

N

∥

∥b∗i bi(βa)−1∥

∥ ‖φ‖∥

∥φ−1∥

∥ ‖J‖2

Lemma C.17

43


Now apply Lemma C.21 to∥

∥b∗b(βa)−1∥

∥, yielding

≤1 + ρ

2

1 + ρ

1 − ρ

1

N‖J‖

2

Now recall that J∗′J ≡ Nb∗i (I + p)b′i + Σu.

We can now return to Proposition C.13. Choose an initial guess of b in sum-of-poles form. By LemmaC.23, all subsequent iterations are also of pole form, as are the iterates of γi and µ. From the definition ofp, ‖p‖ = θ. Therefore, the norm bound established in Lemma C.21 is a fraction:

∥

∥(I + p)−1/2p∥

∥ ≤ θ. Thenorm bound in Lemma C.23 can be made small by choosing ρ small. Therefore the sum of the bounds inLemmas C.21 and C.24 can be made to sum to a fraction. This establishes Proposition C.13.

The bounds in the theorem are somewhat restrictive; in practice they will be lower because the tradingintensity function b(z) will have an average pole that is smaller than ρ. Also, it is restrictive to seed theiteration with a sum-of-pole-terms form, because uniqueness can no longer be asserted.

The norm bounds are thus less than one when ρ and θ are sufficiently small. As we increase ρ, themodel has no obvious discontinuities, and so one should expect the existence result to still hold, although wedo not have a formal proof. Also, the result, and uniqueness as well, goes through more robustly in specialcases that we have developed elsewhere, so we are confident that the result is in fact general.

44


APPENDIX D: ADDITIONAL NOTES ON NUMERICAL METHODS

In this appendix we discuss an important detail of our numerical approach, namely the generation ofthe Popov form, which is a preliminary step in the factorizations of GiG

∗i′ and J∗′J .

In order to calculate the factorization GiG∗i′, we first transform

µ′(I + γ∗) + (I + γ′)µ∗

(We remind readers that our convention in this setting the star and transposition operations are kept separate,unlike the usual convention in which the star operation includes transposition.) It turns out that this is aparticularly convenient form for the factorization needed to produce G. In the state-space formulation ofthe factorization problem, the expression is first decomposed into the Popov form as follows (see Kailath,Sayed and Hassibi, p. 295):

( H(zI − F )−1 I )

(

Q SS∗′ R

)(

(z−1I − F ∗′)−1H∗′

I

)

(D.1)

where H,F , Q, S, and R are matrices, with

(

Q SS∗′ R

)

and R positive definite. With the expression in this

form, then by Theorem 8.3.2 of (see Kailath, Sayed and Hassibi, p. 277), the solution of the factorizationproblem is

G(z) =(

I + H(zI − F )−1Kp

)

R1/2e

Kp ≡ (FPH∗′ + GS)R−1e

Re ≡ R + HPH∗′

P = FPF ∗′ + GQG∗′ − KpReK∗p′

Note that the last equation is the Riccati equation.Software can find the decomposition in (D.1), but numerical accuracy can be increased if the Popov

decomposition can be expressed naturally and analytically, and in this case it can be done. Writing

Q =

(

0 II 0

)

S =

(

I0

)

R = ( 0 )

we can write

GiG∗i′ = ( (µ′ γ′ ) I )

(

0 II 0

) (

I0

)

( I 0 ) ( 0 )

(

µ∗

γ∗

)

I

(D.2)

Performing the multiplication,

= ( ( γ′ µ′ ) + ( I 0 ) µ′ )

(

µ∗

γ∗

)

I

= µ′γ∗ + γ′µ∗ + µ′ + µ∗

as desired.The agenda is to convert the expression in (D.2) into an expression directly corresponding to (D.1).

First of all, the concatenated matrix (µ′ γ′) can be used to determine F and H directly. (Actually, theconcatenated matrix (µ∗ γ∗) due to the pole reversal in the economics model.) The internal Q, S, andR terms will be multiplied by coefficients in this concatenated matrix, which will complicate those termsslightly. That is, we can write

GiG∗′ = ( H(zI − F )−1B + D I )

(

Q SS∗′ R

)(

B∗′(z−1I − F ∗′)−1H∗′ + D∗′

I

)

45


= ( H(zI − F )−1 I )

(

BQB∗′ BQD∗′ + BSDQB∗′ + S∗′B∗′ DQD∗′ + DS + S∗′D∗′ + R

)(

(z−1I − F ∗′)−1H∗′

I

)

( H(zI − F )−1 I )

B

(

0 II 0

)

B∗′ B

(

0 II 0

)

D∗′ + B

(

I0

)

D

(

0 II 0

)

B∗′ + ( I 0 ) B∗′ D

(

0 II 0

)

D∗′ + ( I 0 ) D∗′ + D

(

I0

)

+ ( 0 )

(

(z−1I − F ∗′)−1H∗′

I

)

The equivalent terms for expression (D.1) are then

H(zI − F )−1B + D = ( µ∗ γ∗ )

and the central matrix terms are

Q = B

(

0 II 0

)

B∗′ S = B

(

0 II 0

)

D∗′+B

(

I0

)

R = D

(

0 II 0

)

D∗′+( I 0 ) D∗′+D

(

I0

)

+( 0 )

In the textbook formulation of the general model, the poles of an analytic function are presumed to lie insidethe unit circle, which is the opposite of economic models. Thus, the factorization needs to be of the starred(∗) version of the matrix, with the star operation re-applied to the resulting factor.

The J∗′J matrix can also be decomposed directly in a similar and even simpler manner. That matrix is

J∗J = ( b∗A∗ I )

(

Σe 00 Σu

)(

A′b′

I

)

This expression is close to Popov form, except that just as in the GiG∗i′ case the G matrix (or the B-matrix if

we think in terms of A,B,C,D formulation) must be multiplied times the middle

(

Q SS∗ R

)

matrix. In this

case the numerical routine will find J∗ directly, and J can be found from that by taking the star-transposeoperation. Thus,

H(zI − F )−1B + D = ( b∗A∗ )

and the central matrix becomes(

BΣeB∗′ BΣeD

∗′

DΣeB∗′ DΣeD

∗′ + Σu

)

In both cases, the result of the Popov form generation is fed to a Riccati solver to complete the factorization.

46

eml.berkeley.educshannon/debreu/taub.pdf · the dynamics of strategic information flows/p. seiler,...

Documents