portfolio turnpikes in incomplete markets

7/29/2019 Portfolio turnpikes in incomplete markets

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Portfolio turnpikes in incomplete markets

Hao Xing

London School of Economics

joint work with

Paolo Guasoni, Kostas Kardaras, and Scott Robertson

Stochastic Analysis in Finance and Insurance, Ann Arbor, May 18, 2011

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What is a turnpike?
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What is a turnpike? Express way

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What is the portfolio turnpike?Mertons problem:

uT(x) = max

XadmissibleE[U(XT)].

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uT(x) = max


When U(x) = xp

/p, with p < 1, and the market is Black-Scholes,

=1

1 p1.

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uT(x) = max


When U(x) = xp


=1

1 p1.

For more general utility function and market structure:

Existence: Karatzas-Shreve (98), Kramkov-Schachermayer (99)Explicit solutions: Kallsen (00), Zariphopoulou (01), Goll-Kallsen (03),

Hu-Imkeller-Muller (05)

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uT(x) = max


When U(x) = xp


=1

1 p1.




Portfolio Turnpike:When investment horizon is distant and market grows indefinitely,

Optimal portfolios for generic utilities

is close to

Optimal portfolios of power or logarithmic utilities.

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uT(x) = max


When U(x) = x

p


=1

1 p1.




Portfolio Turnpike:When investment horizon is distant and market grows indefinitely,

Optimal portfolios for generic utilities

is close to

Optimal portfolios of power or logarithmic utilities.

For long term investments, follow the CRRA turnpike!8 / 6 2
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A more mathematical formulation

Consider problems

u0,T = supX admissible

EP[XpT/p], u

1,T = supX admissible

EP[U(XT)].

Suppose that

Distant horizon: T.

Market grows indefinitely: X s.t. limT XT = .

Similar utilities at large wealth levels:

limx

U(x)

xp1= 1.

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A more mathematical formulation

Consider problems

u0,T = supX admissible

EP[XpT/p], u

1,T = supX admissible

EP[U(XT)].

Suppose that

Distant horizon: T.

Market grows indefinitely: X s.t. limT XT = .

Similar utilities at large wealth levels:

limx

U(x)

xp1= 1.

Then 1,Ts and 0,Ts are similar for s [0, t] with t is fixed.

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Literature

Discrete time:

Mossin (68): U(x)/U

(x) = ax + b

Leland (72), Ross (74), Hakansson (74)

Huberman-Ross (83): U(x) is regular varying at :

limx

U(ax)

U(x)= ap1, for any a > 0.

Continuous time:

Cox-Huang (92)

Jin (98): include consumptions

Huang-Zariphopouou (99): viscosity solutions

Dybvig-Rogers-Back (99): NO independent returns, completemarkets with Brownian filtration

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Literature

Discrete time:

Mossin (68): U(x)/U

(x) = ax + b

Leland (72), Ross (74), Hakansson (74)

Huberman-Ross (83): U(x) is regular varying at :

limx

U(ax)

U(x)= ap1, for any a > 0.

Continuous time:

Cox-Huang (92)

Jin (98): include consumptions

Huang-Zariphopouou (99): viscosity solutions

Dybvig-Rogers-Back (99): NO independent returns, completemarkets with Brownian filtration

Assume independent return, except Dybvig et al. = DPP.

Assume market completeness = duality method.

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Incomplete markets with stochastic investmentopportunities

The portfolio choice problem is least tractable, turnpike theorems aremost needed.

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Incomplete markets with stochastic investmentopportunities

The portfolio choice problem is least tractable, turnpike theorems aremost needed.

We prove three kind of turnpike theorems:

Abstract Turnpike: minimum market assumptions, converge undermyopic probabilities except for the log utility;

Classical Turnpike: a class of diffusion models with stochastic driftand volatility. Convergence is shown under the physical measure.

Explicit Turnpike: finite horizon optimal portfolios for genericutilities converge to the long-run optimal portfolio for power utility.

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Admissible wealth processes

For each T R+, wealth process is chosen from a set of nonneg.

semimartingales X

T

, such thati) X0 = 1 for all X X

T;

ii) XT contains some strictly positive X;

iii) XT is convex;

iv) XT is stable under compounding: for any X, X XT and a [0, T]

valued stopping time ,

X = XI[[0,[[ + XXX

I[[,T]] XT.

One can switch to another strategy at any stopping time in aself-financing way.

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Admissible wealth processes

For each T R+, wealth process is chosen from a set of nonneg.

semimartingales X

T

, such thati) X0 = 1 for all X X

T;

ii) XT contains some strictly positive X;

iii) XT is convex;

iv) XT is stable under compounding: for any X, X XT and a [0, T]

valued stopping time ,

X = XI[[0,[[ + XXX

I[[,T]] XT.

One can switch to another strategy at any stopping time in aself-financing way.

These assumptions are satisfied in general frictionless market withsemimartingale dynamics.

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Assumptions on utilities

Let R(x) := U(x)/xp1.

limxR(x) = 1,

lim infx0R(x) > 0, for 0 = p < 1,

lim supx0R(x) < , for p < 1.

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limxR(x) = 1,

lim infx0R(x) > 0, for 0 = p < 1,


Assumptions in the literature:

1. limxU(x)xp1

= 1,

2. limxxU

(x)U(x) = 1 p,

3. limxU(ax)U(x) = a

p1 for any a > 0.

1. = 2. = 3., but not in the other direction.

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A i ili i
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limxR(x) = 1,

lim infx0R(x) > 0, for 0 = p < 1,


Assumptions in the literature:

1. limxU(x)xp1

= 1,

2. limxxU

(x)U(x) = 1 p,

3. limxU(ax)U(x) = a

p1 for any a > 0.

1. = 2. = 3., but not in the other direction.

Dybvig-Rogers-Back (99) does not have assumptions on R(x) at x = 0.

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Assumptions on portfolio choice problem

We assume the portfolio choice problem is well-posed for each horizon:

For all T > 0 and i = 0, 1,

< ui,T < ;

optimal wealth processes Xi,T exist.

Together with the admissible assumption, this implies the no arbitragecondition:

Xs = 0 = Xt = 0, for any t s.

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M i b biliti
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Myopic probabilities

Define the myopic probabilities (PT)T0 as

dPT

dP=

X

0,T

Tp

EP

X0,TT

p .

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M i b biliti
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dPT

dP=

X

0,T

Tp

EP

X0,TT

p .

Appear in Kramkov-Sirbu (06, 07);

PT = P when log utility is considered; Power optimal under P is log optimal under PT.

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M o ic obabilities
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dPT

dP=

X

0,T

Tp

EP

X0,TT

p .

Appear in Kramkov-Sirbu (06, 07);

PT = P when log utility is considered; Power optimal under P is log optimal under PT.

First order condition EP

X0,TT

p1(XT X

0,TT )

0 for any

X XT. Change the measure to PT,

EPT

XT/X0,TT

1, for any X XT.

Hence X0,TT has numeraire property under PT.

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Last assumption
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Last assumptionIn the Black-Scholes case:

dPT

dP= TX

0T,

where is the density of the unique martingale measure.

Note:

limT

dPT

dP= 0, P a.s. ifp = 0.

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Last assumption
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dPT

dP= TX

0T,


Note:

limT

dPT

dP= 0, P a.s. ifp = 0.

Assumption on market growth:There exists (XT)T0 such that X

T XT and

limT

PT(XTT N) = 1, for any N > 0.

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Last assumption
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dPT

dP= TX

0T,


Note:

limT

dPT

dP= 0, P a.s. ifp = 0.

Assumption on market growth:There exists (XT)T0 such that X

T XT and

limT

PT(XTT N) = 1, for any N > 0.

This condition is satisfied in

Safe rate bounded from below by a constant r > 0.

Black-Scholes market dSu/Su = du+ dWu,

X can be chosen as the power optimal X0,T when = 0.

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First turnpike theorem
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First turnpike theoremOptimal terminal wealths are unbounded as T.

Define

rT

u := X1,T

u /X0,T

u , T

u := u0 drTv /rTv, u [0,T].

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First turnpike theorem
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First turnpike theoremOptimal terminal wealths are unbounded as T.

Define

r

T

u := X1,T

u /X0,T

u , T

u := u0 drTv /rTv, u [0,T].Theorem (Abstract Turnpike)Under above assumptions, for any > 0,

a) limT PT

supu[0,T]rTu 1 = 0,

b) limT PT

T, TT

= 0.

Remark:

In the market dSu/Su = udu+ udWu,

T, T

=

0

1,Tu 0,Tu u2 du. When log utility is considered, the convergence is under P.

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An Example
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An ExampleIn Huberman-Ross (83), a discrete time market model with independentreturn was studied.

U is regular varying at is a necessary condition for turnpike.

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An Example
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An ExampleIn Huberman-Ross (83), a discrete time market model with independentreturn was studied.

U is regular varying at is a necessary condition for turnpike.

In continuous time with dependent return, this does not hold.

Consider a Black-Scholes model with r = 0 and dSt/St = dt+ dWt.

Let X be the log-optimal wealth process and n = inf{t 0 |Xt = n}.

For problem with horizon n, is the optimal wealth process for genericutility U.

E[n /Xn ] 1 = E[n ] n, for any wealth process .

E[U(n)] U(E[n ]) U(n) = E[U(Xn )].

Then change time from n to n, the return becomes dependent.

Turnpike trivially holds. But the generic utility may not be regularly

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Drawback
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Drawback

Convergence is under {PT}T0,

Limit of (PT)T0 is in general singular wrt P.

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Drawback
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Drawback



So we look at a time window [0, t]

limT

PT

sup

u[0,t]

rTu 1

= 0,

limTPT T, Tt = 0.

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Drawback
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So we look at a time window [0, t]

limT

PT

sup

u[0,t]

rTu 1

= 0,

limTPT T, Tt = 0.

Then consider the conditional densities

dPT

dP

Ft

.

Question:

Does limT dPT/dP|Ft exist?

Does the limit induce a measure equivalent to P on Ft?

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IID Myopic Turnpike
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y p p

Corollary

1. X0,Tt = X0,St Xt a.s. for all t S, T (myopic optimality);

2. X0,TT /X0,Tt andFt are independent for all t T (independent

returns).

then, for any > 0 and t 0:

a) limT P

supu[0,t]rTu 1

= 0,

b) limT P

T, Tt

= 0.

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IID Myopic Turnpike
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y p p

Corollary

1. X0,Tt = X0,St Xt a.s. for all t S, T (myopic optimality);

2. X0,TT /X0,Tt andFt are independent for all t T (independent

returns).

then, for any > 0 and t 0:

a) limT P

supu[0,t]rTu 1

= 0,

b) limT P

T, Tt

= 0.

Remark: This contains exponential Levy markets, Kallsen (00).

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Markets with stochastic investment opportunities
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pp

Consider a diffusion model, investment opportunities are driven by

dYt = b(Yt)dt+ a(Yt)dWt,

whose state space is E = (, ), with < .

The market includes a safe rate r(Yt) and d risky assets

dSit

Sit

= r(Yt)dt+ dRit, with

dRt = (Yt)dt+ (Yt)dZt,

dZi,Wt = i(Yt) dt.

DefineA

=a

2

, , =

, and = a

.

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Markets with stochastic investment opportunities
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Consider a diffusion model, investment opportunities are driven by

dYt = b(Yt)dt+ a(Yt)dWt,

whose state space is E = (, ), with < .

The market includes a safe rate r(Yt) and d risky assets

dSit

Sit

= r(Yt)dt+ dRit, with

dRt = (Yt)dt+ (Yt)dZt,

dZi,Wt = i(Yt) dt.

DefineA

=a

2

, , =

, and = a

.

We assume the martingale problem associated to (Y, R) is well posed.

Assumption on correlation: is a constant.

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Finite horizon problem for power
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Zariphopoulou (01) introduced the power transform. The nonlinear HJBequation is linearized.

uT

(x, y, t) =

xp

p vT(y, t) .vT solves the reduced HJB equation:

tv + Lv + cv = 0, v(T, y) = 1,

where L = 12 A2yy + (b q

1)y and c =1

(pr q2 1).

The optimal portfolio is

T =1

1 p1

+

vTy

vT

evaluated on (t,Yt).

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Finite horizon problem for power
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Zariphopoulou (01) introduced the power transform. The nonlinear HJBequation is linearized.

uT

(x, y, t) =

xp

p vT(y, t) .vT solves the reduced HJB equation:

tv + Lv + cv = 0, v(T, y) = 1,

where L = 12 A2yy + (b q

1)y and c =1

(pr q2 1).

The optimal portfolio is

T =1

1 p1

+

vTy

vT

evaluated on (t,Yt).

Under PT, the dynamics of (R, Y) is

dRt =1

1p

+

vTy

vT

(t,Yt) dt+ (Yt) dZt

dYt =

B+ A

vTy

vT

(t,Yt) dt+ a(Yt) d

Wt

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Long-run measure T
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Our goal: study the limit behavior of

dPT/dP|FtT0

.

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Long-run measure T
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dPT/dP|FtT0

.

Consider the ergodic HJB equation:

Lv+ cv = v.This relates to the risk sensitive control

max

limT

1

pTlog(E [U(XT)]) .

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Long-run measureO f

T / |
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dPT/dP|FtT0

.



max

limT

1

pTlog(E [U(XT)]) .

DefineP

as the solution to the following martingale problem: dRt =1

1p

+

vyv

(Yt) dt+ dZt

dYt =B+ A

vyv

(Yt) dt+ a dWt

Guasoni-Robertson (10) proposed that the long-run optimal strategy is

=1

1 p1

+

vyv

.

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Long-run measureO l d h li i b h i f

dPT /dP|
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dPT/dP|FtT0

.



max

limT

1

pTlog(E [U(XT)]) .

DefineP

as the solution to the following martingale problem: dRt =1

1p

+

vyv

(Yt) dt+ dZt

dYt =B+ A

vyv

(Yt) dt+ a dWt

Guasoni-Robertson (10) proposed that the long-run optimal strategy is

=1

1 p1

+

vyv

.

is the log optimal strategy under P.

Then dP/dP|Ft

is the natural candidate for limT

dPT/dP|Ft

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Classical Turnpike
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Proposition

When Y is positively recurrent under P, then P P on Ft. Moreover

P limT

dPT

dP|Ft = 1.

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Classical Turnpike
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Proposition

When Y is positively recurrent under P, then P P on Ft. Moreover

P limT

dPT

dP|Ft = 1.

This allows to replace PT in the abstract turnpike with P.

Theorem (Classical Turnpike)Under above assumptions, for 0 = p < 1 and any , t > 0:

a) limT P (supu[0,t]rTu 1

) = 0,

b) limT P

T, Tt

= 0.

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Explicit Turnpike
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Compare finite horizon optimal portfolio for a generic utility with thelong-run optimal for the power utility.

DefinerTu =

X1,Tu

Xu, Tu =

u0

drTv

rTv, for u [0,T].

Theorem (Explicit Turnpike)

Under above assumptions, for any , t > 0 and 0 = p < 1:a) limT P (supu[0,t]

rTu 1 ) = 0,b) limT P

T, T

t

= 0.

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Explicit Turnpike
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Compare finite horizon optimal portfolio for a generic utility with thelong-run optimal for the power utility.

DefinerTu =

X1,Tu

Xu, Tu =

u0

drTv

rTv, for u [0,T].

Theorem (Explicit Turnpike)

Under above assumptions, for any , t > 0 and 0 = p < 1:a) limT P (supu[0,t]

rTu 1 ) = 0,b) limT P

T, T

t

= 0.

Remark:

Finite horizon optimal for a generic utility long-run optimal forpower utility;

The long-run portfolio is a myopic strategy, expressed by thesolution of the ergodic HJB.

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Proof for the log caseThe first order condition:
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The first order condition:

E[U(XTT )(XT XTT )] 0, for any admissible X.

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Applying the first order condition to log x and U respectively.

EP[rT 1] 0,

EP[U(X1,TT )(X

0,TT X

1,TT )] 0.

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Applying the first order condition to log x and U respectively.

EP[rT 1] 0,

EP[U(X1,TT )(X

0,TT X

1,TT )] 0.

Sum previous two inequalities

EP

1

R(X1,TT )

rTT

(rTT 1)

0.

Lemma

EP1 R(X1,TT )rTT

(rTT 1)

2EP R(X1,TT ) 12 .R is bounded, P- limTR(X

1,TT ) = 1 = P limTr

TT = 1

= L1 limTrTT = 1.

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Convergence of portfolios
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Lemma (Kardaras (10))

Consider a sequence{YT

}T0 of cadlag processes and a sequence{PT}T0, such that:

i) For each T 0, YT0 = 1 and YT is strictly positive.

ii) Each YT is a PT-supermartingale.

iii) limT EPT

[|YT

T

1|] = 0.

Then, for any > 0,

a) limT PT(supu[0,T] |Y

Tu 1| ) = 0.

b) limT PT([LT, LT]T ) = 0, where Y

T is the stochasticlogarithm of YT.

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Convergence of portfolios
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Lemma (Kardaras (10))

Consider a sequence{YT

}T0 of cadlag processes and a sequence{PT}T0, such that:

i) For each T 0, YT0 = 1 and YT is strictly positive.

ii) Each YT is a PT-supermartingale.

iii) limT EPT

[|YT

T

1|] = 0.

Then, for any > 0,

a) limT PT(supu[0,T] |Y

Tu 1| ) = 0.

b) limT PT([LT, LT]T ) = 0, where Y

T is the stochasticlogarithm of YT.

Note that rT is a supermartingale and EP[|rTT 1|] = 0,

the abstract turnpike follows from the above lemma.

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Proof for the diffusion caseDefine

T
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hT(t, y) :=

vT(t, y)

ec(Tt)v(y).

It satisfies

thT + L hT = 0, (t, y) (0,T) E,

hT(T, y) = (v)1 (y), y E,

where

L := L + Avy

v

y.

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T ( )


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hT(t, y) :=

vT(t, y)

ec(Tt)v(y).

It satisfies

thT + L hT = 0, (t, y) (0,T) E,

hT(T, y) = (v)1 (y), y E,

where

L := L + Avy

v

y.

This is Doobs h-transform using v, the generator of Y under P.

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T ( )
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hT(t, y) :=

vT(t, y)

ec(Tt)v(y).

It satisfies

thT + L hT = 0, (t, y) (0,T) E,

hT(T, y) = (v)1 (y), y E,

where

L := L + Avy

v

y.


It has the stochastic representation hT(t, y) = EPyt

(v(YT))1

.

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T ( )
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hT(t, y) :=

vT(t, y)

ec(Tt)v(y).

It satisfies

thT + L hT = 0, (t, y) (0,T) E,

hT(T, y) = (v)1 (y), y E,

where

L := L + Avy

v

y.


It has the stochastic representation hT(t, y) = EPyt

(v(YT))1

.

vT(t, y) = ec(Tt)v(y)E

Py

t

(v(YT))

1

= EPy

t

exp

Tt

c(Ys)ds

.

This confirms a remark in Zariphopoulou(01) via h-transform.56/62

Convergence of densities
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dPT

dP

Ft

=hT(t,Yt)

hT(0, y) .

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Convergence of densities
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dPT

dP

Ft

=hT(t,Yt)

hT(0, y) .

When Y is positive ergodic under P,

(this can be characterized via the scale function of Y)

limT

hT(t, y) = limT

EPy

t

(v(YT))

1

= K, for all (t, y).

.

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Verification
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1. Assume Lv + cv = v has a strictly positive C2 solution v.

2. Define hT(t, y) = EPyt

(v(YT))1

.

3. Show hT is a C1,2 solution to the associated PDE.

No uniform ellipticity assumption.

4. Define vT(t, y) = ec(Tt)v(y)hT(t, y).

5. Verify that uT(t, x, y) = 1p

xp

vT(t, y)

.

uT(t, x, y) = 1p

xp

vT(t, y)

E[ETt ].

Show ET is a martingale.

We use the result in Cheriditio-Filipovic-Yor (05).

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Conclusion and future works
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We show three turnpike theorems in incomplete markets withstochastic investment opportunities.

The goal is to provide a tractable portfolio for a generic utility whenthe horizon is long.

This can be viewed as a stability result for portfolio choice problem

when the horizon is long.

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Conclusion and future works
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We show three turnpike theorems in incomplete markets withstochastic investment opportunities.

The goal is to provide a tractable portfolio for a generic utility whenthe horizon is long.

This can be viewed as a stability result for portfolio choice problem

when the horizon is long.

Future works:

Diffusion model with multiple state variables?

Market with transaction cost?

Forward utility?

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Thanks for your attention!

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portfolio turnpikes in incomplete markets

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