Local martingale explained

In mathematics, a local martingale is a type of stochastic process, satisfying the localized version of the martingale property. Every martingale is a local martingale; every bounded local martingale is a martingale; in particular, every local martingale that is bounded from below is a supermartingale, and every local martingale that is bounded from above is a submartingale; however, a local martingale is not in general a martingale, because its expectation can be distorted by large values of small probability. In particular, a driftless diffusion process is a local martingale, but not necessarily a martingale.

Local martingales are essential in stochastic analysis (see Itô calculus, semimartingale, and Girsanov theorem).

Definition

Let

(\Omega,F,P) be a probability space; let F _{*=\{F} _{t\mid} t\geq 0 \} be a filtration of F ; let X\colon [ 0 , inf ty) x \Omega → S be an F _{*} -adapted stochastic process on the set S . Then X is called an F _{*} -local martingale if there exists a sequence of F _{*} -stopping times \tau _{k} \colon \Omega \to [ 0 , inf ty) such that

• the

\tau _{k} are almost surely increasing: P\left\{\tau _{k} < \tau _{k+1} \right\} =1 ;

• the

\tau _{k} diverge almost surely: P \left\{\lim _{k\toinfty} \tau _{k} =inf ty \right\} =1 ;

• the stopped process $$X\_t^\; :=\; X\_$$ is an

F _{*} -martingale for every k .

Examples

Example 1

Let W_t be the Wiener process and T = min the time of first hit of -1. The stopped process W_min is a martingale. Its expectation is 0 at all times; nevertheless, its limit (as t → ∞) is equal to -1 almost surely (a kind of gambler's ruin). A time change leads to a process

\displaystyle X _{t} = \begin{cases} W _{min\left(\tfrac{t} { 1- t},T\right)} & for 0 \le t < 1 ,\\ -1 & for 1 \le t < inf ty. \end{cases}

The process

X _{t} is continuous almost surely; nevertheless, its expectation is discontinuous,\displaystyle \operatorname{E} X _{t} = \begin{cases} 0 & for 0 \le t < 1 ,\\ -1 & for 1 \le t < inf ty. \end{cases}

This process is not a martingale. However, it is a local martingale. A localizing sequence may be chosen as

\tau _{k} = min \{ t : X _{t} = k \} if there is such t, otherwise \tau _{k} = k . This sequence diverges almost surely, since \tau _{k} = k for all k large enough (namely, for all k that exceed the maximal value of the process X). The process stopped at τ_k is a martingale.^[1]

Example 2

Let W_t be the Wiener process and ƒ a measurable function such that

\operatorname{E} |f(W _{1)|} < inf ty. Then the following process is a martingale:X _{t} = \operatorname{E} ( f(W _{1)} \mid F _{t} ) = \begin{cases} f _{1-t} (W _{t)} & for 0 \le t < 1 ,\\ f(W _{1)} & for 1 \le t < inf ty; \end{cases} wheref _{s(x)} = \operatorname{E} f(x + W _{s)} = \int f(x + y) (1 {\sqrt{ 2 \pi)/(s}}) e ^{- y ^{2/(2s)}} dy. The Dirac delta function \delta (strictly speaking, not a function), being used in place of f, leads to a process defined informally as Y _{t} = \operatorname{E} ( \delta(W _{1)} \mid F _{t} ) and formally asY _{t} = \begin{cases} \delta _{1-t} (W _{t)} & for 0 \le t < 1 ,\\ 0 & for 1 \le t < inf ty, \end{cases} where\delta _{s(x)} = (1 {\sqrt{ 2 \pi)/(s}}) e ^{- x ^{2/(2s)}} . The process Y _{t} is continuous almost surely (since W _{1} \ne 0 almost surely), nevertheless, its expectation is discontinuous,\operatorname{E} Y _{t} = \begin{cases} 1 /\sqrt{ 2 \pi} & for 0 \le t < 1 ,\\ 0 & for 1 \le t < inf ty. \end{cases} This process is not a martingale. However, it is a local martingale. A localizing sequence may be chosen as \tau _{k} = min \{ t : Y _{t} = k \}.

Example 3

Let

Z _{t} be the complex-valued Wiener process, andX _{t} = ln | Z _{t} - 1 | . The process X _{t} is continuous almost surely (since Z _{t} does not hit 1, almost surely), and is a local martingale, since the function u \mapsto ln |u -1 | is harmonic (on the complex plane without the point 1). A localizing sequence may be chosen as \tau _{k} = min \{ t : X _{t} = - k \}. Nevertheless, the expectation of this process is non-constant; moreover,\operatorname{E} X _{t} \to inf ty   as t \to inf ty, which can be deduced from the fact that the mean value of ln |u -1 | over the circle |u| = r tends to infinity as r \to inf ty . (In fact, it is equal to ln r for r ≥ 1 but to 0 for r ≤ 1).

Martingales via local martingales

Let

M _{t} be a local martingale. In order to prove that it is a martingale it is sufficient to prove that M ^{\tau _{k}} _{t} \to M _{t} in L¹ (as k \to inf ty ) for every t, that is, \operatorname{E} | M ^{\tau _{k}} _{t} - M _{t} | \to 0 ; here M ^{\tau _{k}} _{t} = M _{t\wedge \tau _{k}} is the stopped process. The given relation \tau _{k} \to inf ty implies that M ^{\tau _{k}} _{t} \to M _{t} almost surely. The dominated convergence theorem ensures the convergence in L¹ provided thatstyle (*) \operatorname{E} \sup _{k|} M ^{\tau _{k}} _{t} | < inf ty    for every t.Thus, Condition (*) is sufficient for a local martingale M _{t} being a martingale. A stronger conditionstyle (**) \operatorname{E} \sup _{s\in[0,t]} |M _{s|} < inf ty    for every tis also sufficient.

Caution. The weaker condition

style \sup _{s\in[0,t]} \operatorname{E} |M _{s|} < inf ty    for every tis not sufficient. Moreover, the conditionstyle \sup _{t\in[0,infty)} \operatorname{E} e ^{|M _{t|}} < inf ty is still not sufficient; for a counterexample see Example 3 above.

A special case:

style M _{t} = f(t,W _{t),} where W _{t} is the Wiener process, and f : [ 0 , inf ty) x R \to R is twice continuously differentiable. The process M _{t} is a local martingale if and only if f satisfies the PDE( (\partial)/(\partial t) + (12)/((\partial ^{2})/(\partial x ^{2})) ) f(t,x) = 0 . However, this PDE itself does not ensure that M _{t} is a martingale. In order to apply (**) the following condition on f is sufficient: for every \varepsilon> 0 and t there exists C = C(\varepsilon,t) such thatstyle |f(s,x)| \le C e ^{\varepsilon x ^{2}} for all s \in [ 0 ,t] and x \in R .

Technical details

1. For the times before 1 it is a martingale since a stopped Brownian motion is. After the instant 1 it is constant. It remains to check it at the instant 1. By the bounded convergence theorem the expectation at 1 is the limit of the expectation at (n-1)/n (as n tends to infinity), and the latter does not depend on n. The same argument applies to the conditional expectation.

References

• Book: Øksendal, Bernt K. . Bernt Øksendal

. Bernt Øksendal . Stochastic Differential Equations: An Introduction with Applications . Sixth . Springer . Berlin . 2003 . 3-540-04758-1.