Local martingale

Template:Short description 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 $(\mathrm{\Omega },F,P)$ be a probability space; let $F_{*}=\{F_t\mid t\ge 0\}$ be a filtration of $F$; let $X:[0,\mathrm{\infty })\times \mathrm{\Omega }\to S$ 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 ${\tau }_k:\mathrm{\Omega }\to [0,\mathrm{\infty })$ such that

• the ${\tau }_k$ are almost surely increasing: $P\{{\tau }_k<{\tau }_{k+1}\}=1$;
• the ${\tau }_k$ diverge almost surely: $P\{\underset{k\to \mathrm{\infty }}{\lim }{\tau }_k=\mathrm{\infty }\}=1$;
• the stopped process $$X_t^{{\tau }_k}:=X_{\min \{t,{\tau }_k\}}$$ is an $F_{*}$-martingale for every $k$.

Examples

Example 1

Illustration for local martingale. Up Panel: Multiple simulated paths of the process $X_t$ which is stopped upon hitting $-1$. This shows gambler's ruin behavior, and is not a martingale. Down Panel: Paths of $X_t$ with an additional stopping criterion: the process is also stopped when it reaches a magnitude of $k=2.0$. This no longer suffers from gambler's ruin behavior, and is a martingale.

Let W_t be the Wiener process and T = min{ t : W_t = −1 } the time of first hit of −1. The stopped process W_{min{ t, T }} 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

$X_t=\{\begin{array}{cc}{W}_{\min (\frac{t}{1-t},T)}& \text{for }0\le t<1,\\ -1& \text{for }1\le t<\mathrm{\infty }.\end{array}$

The process $X_t$ is continuous almost surely; nevertheless, its expectation is discontinuous,

$\mathrm{E}{X}_t=\{\begin{array}{cc}0& \text{for }0\le t<1,\\ -1& \text{for }1\le t<\mathrm{\infty }.\end{array}$

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.

Example 2

Let W_t be the Wiener process and ƒ a measurable function such that $\mathrm{E}|f(W_1)|<\mathrm{\infty }.$ Then the following process is a martingale:

$X_t=\mathrm{E}(f(W_1)\mid F_t)=\{\begin{array}{cc}{f}_{1-t}(W_t)& \text{for }0\le t<1,\\ f(W_1)& \text{for }1\le t<\mathrm{\infty };\end{array}$

where

$f_s(x)=\mathrm{E}f(x+W_s)=\int f(x+y)\frac{1}{\sqrt{2\pi s}}{\mathrm{e}}^{-y^2/(2s)}dy.$

The Dirac delta function $\delta$ (strictly speaking, not a function), being used in place of $f,$ leads to a process defined informally as $Y_t=\mathrm{E}(\delta (W_1)\mid F_t)$ and formally as

$Y_t=\{\begin{array}{cc}{\delta }_{1-t}(W_t)& \text{for }0\le t<1,\\ 0& \text{for }1\le t<\mathrm{\infty },\end{array}$

where

${\delta }_s(x)=\frac{1}{\sqrt{2\pi s}}{\mathrm{e}}^{-x^2/(2s)}.$

The process $Y_t$ is continuous almost surely (since $W_1\ne 0$ almost surely), nevertheless, its expectation is discontinuous,

$\mathrm{E}{Y}_t=\{\begin{array}{cc}1/\sqrt{2\pi }& \text{for }0\le t<1,\\ 0& \text{for }1\le t<\mathrm{\infty }.\end{array}$

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, and

$X_t=\ln |Z_t-1|.$

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,

$\mathrm{E}{X}_t\to \mathrm{\infty }$   as $t\to \mathrm{\infty },$

which can be deduced from the fact that the mean value of $\ln |u-1|$ over the circle $|u|=r$ tends to infinity as $r\to \mathrm{\infty }$. (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 $M_t^{{\tau }_k}\to M_t$ in L¹ (as $k\to \mathrm{\infty }$) for every t, that is, $\mathrm{E}|M_t^{{\tau }_k}-M_t|\to 0;$ here $M_t^{{\tau }_k}=M_{t\wedge {\tau }_k}$ is the stopped process. The given relation ${\tau }_k\to \mathrm{\infty }$ implies that $M_t^{{\tau }_k}\to M_t$ almost surely. The dominated convergence theorem ensures the convergence in L¹ provided that

$(*)\mathrm{E}\underset{k}{\sup }|M_t^{{\tau }_k}|<\mathrm{\infty }$    for every t.

Thus, Condition (*) is sufficient for a local martingale $M_t$ being a martingale. A stronger condition

$(**)\mathrm{E}\underset{s\in [0,t]}{\sup }|M_s|<\mathrm{\infty }$    for every t

is also sufficient.

Caution. The weaker condition

$\underset{s\in [0,t]}{\sup }\mathrm{E}|M_s|<\mathrm{\infty }$    for every t

is not sufficient. Moreover, the condition

$\underset{t\in [0,\mathrm{\infty })}{\sup }\mathrm{E}{\mathrm{e}}^{|M_t|}<\mathrm{\infty }$

is still not sufficient; for a counterexample see Example 3 above.

A special case:

$M_t=f(t,W_t),$

where $W_t$ is the Wiener process, and $f:[0,\mathrm{\infty })\times ℝ\to ℝ$ is twice continuously differentiable. The process $M_t$ is a local martingale if and only if f satisfies the PDE

$(\frac{\partial }{\partial t}+\frac{1}{2}\frac{{\partial }^2}{\partial x^2})f(t,x)=0.$

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 $\epsilon >0$ and t there exists $C=C(\epsilon ,t)$ such that

$|f(s,x)|\le C{\mathrm{e}}^{\epsilon x^2}$

for all $s\in [0,t]$ and $x\in ℝ.$

References

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