Martingales etal.
ShotaGugushvili
Basic concepts
Stoppingtimes andmartingaletransforms
Martingales et al.
Shota Gugushvili
Leiden University
Amsterdam, 30 October 2013
Martingales etal.
ShotaGugushvili
Basic concepts
Stoppingtimes andmartingaletransforms
Outline
1 Basic concepts
2 Stopping times and martingale transforms
Martingales etal.
ShotaGugushvili
Basic concepts
Stoppingtimes andmartingaletransforms
Stochastic process
The probability space (Ω,F ,P) is fixed throughout and allthe quantities introduced are defined on this space.
Definition
A stochastic process X = (Xt)t∈T is a collection of randomvariables Xt indexed by a set T . A discrete time stochasticprocess is the one for which T is finite or countable.
The name ‘time’ (for t) refers to the fact that stochasticprocesses are used to model random dynamic phenomenaevolving over the course of time, e.g. asset prices.
A discrete time stochastic process is nothing else but asequence of random variables.
We concentrate on the case T = 0, 1, 2, . . . and writeX = (Xn)n≥0 for the process.
Martingales etal.
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Basic concepts
Stoppingtimes andmartingaletransforms
Filtration
Definition
Let F = (Fn)n≥0, where each Fn is a σ-algebra satisfyingFn ⊂ F , and moreover Fn ⊂ Fn+1, for all n ≥ 0. Thesequence F is called the filtration.
Definition
If X is a stochastic process, then the filtration FX := (FXn )n≥0
generated by X is defined by FXn := σ(X0, . . . ,Xn). FX is also
called the natural filtration associated with X .
Filtration can be thought of as a flow of information,where Fn represents all the information available up totime n.
Some notation: F∞ := σ(∪∞n=0Fn). Obviously, Fn ⊂ F∞.
Martingales etal.
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Basic concepts
Stoppingtimes andmartingaletransforms
Adapted process
Definition
Let a filtration F be given. A process Y is called F-adapted (oradapted to F, or just adapted), if for all n the random variableYn is Fn-measurable: Yn ∈ Fn.
Obviously, X is adapted to FX .
Martingales etal.
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Basic concepts
Stoppingtimes andmartingaletransforms
Martingale
Definition
A stochastic process M = (Mn)n≥0 is called a martingale (orF-martingale), if it is adapted to a filtration F, ifMn ∈ L1(Ω,F ,P) for all n ≥ 0 and if
E [Mn+1|Fn] = Mn a.s. (1)
Equation (1) is called the martingale property of M.
The equality (1) should be read in the sense that Mn is aversion of the conditional expectation E [Mn+1|Fn].
Martingales etal.
ShotaGugushvili
Basic concepts
Stoppingtimes andmartingaletransforms
Equivalent definition
Definition
A stochastic process M = (Mn)n≥0 is called a martingale (orF-martingale), if it is adapted to a filtration F, ifMn ∈ L1(Ω,F ,P) for all n ≥ 0 and if
E [Mm|Fn] = Mn a.s.
for all m ≥ n + 1.
Martingales etal.
ShotaGugushvili
Basic concepts
Stoppingtimes andmartingaletransforms
Examples
Example
Let X be a process consisting of independent random variablesXn, with n ≥ 1 and assume that Xn ∈ L1(Ω,F ,P) for all n.Put X0 = 0 and Sn =
∑nk=0 Xk =
∑nk=1 Xk . Take F = FX . X
is a martingale iff E [Xn] = 0 for all n.
Example
Let X be a process consisting of independent random variablesXn, with n ≥ 1 and assume that Xn ∈ L1(Ω,F ,P) for all n.Put X0 = 1 and Pn =
∏nk=0 Xk =
∏nk=1 Xk . Take F = FX . P
is a martingale iff EXn = 1 for all n.
Example
Let E [|Y |] <∞ and F be a filtration. The process M withMn = E [Y |Fn], n ≥ 0 is a martingale.
Martingales etal.
ShotaGugushvili
Basic concepts
Stoppingtimes andmartingaletransforms
Submartingale and supermartingale
Definition
A stochastic process X = (Xn)n≥0 is called a submartingale (orF-submartingale), if it is adapted to a filtration F, ifXn ∈ L1(Ω,F ,P) for all n ≥ 0 and if
E [Xn+1|Fn] ≥ Xn a.s. (2)
A stochastic process X = (Xn)n≥0 is called a supermartingale(or F-supermartingale), if −X is a submartingale.
Inequality (2), valid for all n ≥ 0 is called thesubmartingale property of X .
Martingales etal.
ShotaGugushvili
Basic concepts
Stoppingtimes andmartingaletransforms
Expectations
A martingale has a constant expectation (E [Xn] = const.for all n), a submartingale has an increasing expectation(E [Xm] ≥ E [Xn] for m ≥ n) and the supermartingale hasa decreasing expectation (E [Xm] ≤ E [Xn] for m ≥ n).
One can also talk informally about no trend, increasingtrend and decreasing trend for these three processes.
Martingales etal.
ShotaGugushvili
Basic concepts
Stoppingtimes andmartingaletransforms
Examples
Example
Let X be a process consisting of independent random variablesXn, with n ≥ 1 and assume that Xn ∈ L1(Ω,F ,P) for all n.Put X0 = 0 and Sn =
∑nk=0 Xk =
∑nk=1 Xk . Take F = FX . X
is a submartingale if E [Xn] ≥ 0 and a supermartingale ifE [Xn] ≤ 0 for all n.
Example
Let X be a process consisting of positive independent randomvariables Xn, with n ≥ 1 and assume that Xn ∈ L1(Ω,F ,P) forall n. Put X0 = 1 and Pn =
∏nk=0 Xk =
∏nk=1 Xk . Take
F = FX . P is a submartingale if EXn ≥ 1 and asupermartingale if EXn ≤ 1 for all n.
Martingales etal.
ShotaGugushvili
Basic concepts
Stoppingtimes andmartingaletransforms
Predictable process
Definition
Given a filtration F, a process Y = (Yn)n≥1 is calledF-predictable (or just predictable) if Yn ∈ Fn−1, n ≥ 1. Aconvenient additional convention is to set Y0 = 0.
Useful interpreation is this: a predictable process Y is a(trading or gambling) strategy. It tells you what youraction at time n is going to be, given that you use yourinformation available at time n − 1.
Martingales etal.
ShotaGugushvili
Basic concepts
Stoppingtimes andmartingaletransforms
Discrete time stochastic integral
Definition
Let Y and X be two stochastic processes. A stochastic processS = Y · X defined by
S0 = 0, Sn =n∑
i=1
Yi∆Xi , n ≥ 1
is called a discrete time stochastic integral.
The name derives from the analogy to∫ t
0YtdXt ,
a stochastic integral (whatever that is).
Martingales etal.
ShotaGugushvili
Basic concepts
Stoppingtimes andmartingaletransforms
Martingale transform
Theorem
Let X be an adapted process and Y a predictable process.Assume that the Xn are in L1(Ω,F ,P) as well as the Yn∆Xn.Let S = Y · X . The following results hold.
(i) If X is martingale, so is S .
(ii) If X is a submartingale (supermartingale) and if Y isnonnegative, also S is a submartingale (supermartingale).
In case (i) one calls S a martingale transform.
I have never seen anybody to use the term sub- orsupermartingale transform though (case (ii)).
Martingales etal.
ShotaGugushvili
Basic concepts
Stoppingtimes andmartingaletransforms
Stopping time
Definition
Let F be a filtration. A mapping T : Ω→ 0, 1, 2, . . . ∪ ∞is called a stopping time if for all n ∈ 0, 1, 2, . . . it holds thatT = n ∈ Fn.
T is a stopping time iff T ≤ n ∈ Fn is true for alln ∈ 0, 1, 2, . . .. Alternatively, iff T > n ∈ Fn is truefor all n ∈ 0, 1, 2, . . ..The event T =∞ can be written as (∪∞n=0T = n)c .Since T = n ∈ Fn ⊂ F∞, we have T =∞ ∈ F∞.Hence the requirement T = n ∈ Fn extends to n =∞.
Martingales etal.
ShotaGugushvili
Basic concepts
Stoppingtimes andmartingaletransforms
Example
Example
Let F be a filtration and X an adapted process. Let B ∈ B bea Borel set in R and let T = infn ≥ 0 : Xn ∈ B. Then T is astopping time.
Martingales etal.
ShotaGugushvili
Basic concepts
Stoppingtimes andmartingaletransforms
Stopped process
Definition
If X is an adapted process and T a stopping time, we definethe stopped process XT by XT
n (ω) := XT (ω)∧n(ω), n ≥ 0.
Tn(ω) := T (ω) ∧ n is a stopping time.
XT0 = X0 and XT
n (ω) = Xn(ω) for n ≤ T (ω). The latterexplains the name stopped process.
Martingales etal.
ShotaGugushvili
Basic concepts
Stoppingtimes andmartingaletransforms
Stopped processes and martingales et al.
Theorem
If X is an adapted process and T a stopping time, then XT isadapted too. Moreover, if X is a supermartingale, so is XT andthen EXT
n ≤ EX0. If X is a martingale, then XT is amartingale too and EXT
n = EX0.
Martingales etal.
ShotaGugushvili
Basic concepts
Stoppingtimes andmartingaletransforms
Doubling strategy
Martingales got their name from a gambling strategyconsisting in doubling the bet until the first win and thenimmediately stopping to play.
This also gives an example of a martingale S (and astopping time T ), such that E [Sn] 6= E [ST ].
Brief terminological remark: there exists a certain relationof martingales to harmonic functions. Sub- andsupermartingales are related to subharmonic andsuperharmonic functions, hence their names.
Martingales etal.
ShotaGugushvili
Basic concepts
Stoppingtimes andmartingaletransforms
Expectations
Theorem
Let X be supermartingale and T an a.s. finite stopping time.Then EXT ≤ EX0 under either of the assumptions
(i) X is a.s. bounded from below by random variableY ∈ L1(Ω,F ,P), or
(ii) T is bounded, i.e. there exists N <∞ such thatP(T ≤ N) = 1 or
(iii) The process ∆X is bounded by a constant C andET <∞.
If X is a martingale, then EXT = EX0 under (ii) and (iii) andalso under the assumption (iv) that X is bounded.
Martingales etal.
ShotaGugushvili
Basic concepts
Stoppingtimes andmartingaletransforms
Example
Example (Wald’s identity)
Let ξ1, ξ2 . . . be i.i.d. with E [|ξ1|] <∞. Then
E [ξ1 + . . . ξn] = nE [ξ1].
Wald’s identity generalises this to the case when n is replacedwith a stopping time T :
Let T ≥ 1 (with E [T ] <∞) be an Fξ = (Fξn)n≥0 stoppingtime with
Fξn = σ(ξ1, . . . , ξn).
ThenE [ξ1 + . . .+ ξT ] = E [ξ1]E [T ],
provided |ξ1| ≤ C (a weaker condition E [|ξ1|] <∞ suffices,but anyway).