Local martingale

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, in general a local martingale is not 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 Itō calculus, semimartingale, Girsanov theorem.

Definition

Let (Ω, F, P) be a probability space; let F_∗ = { F_t | t ≥ 0 } be a filtration of F; let X : [0, +∞) × Ω → S be an F_∗-adapted stochastic process on set S. Then X is called an F_∗-local martingale if there exists a sequence of F_∗-stopping times τ_k : Ω → [0, +∞) such that

the τ_k are almost surely increasing: P[τ_k < τ_k+1] = 1;
the τ_k diverge almost surely: P[τ_k → +∞ as k → +∞] = 1;
the stopped process

X_{t}^{{\tau _{{k}}}}:=X_{{\min\{t,\tau _{k}\}}}

is an F_∗-martingale for every k.

Examples

Example 1

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

\displaystyle X_{t}={\begin{cases}W_{{\min({\tfrac {t}{1-t}},T)}}&{\text{for }}0\leq t<1,\\-1&{\text{for }}1\leq t<\infty .\end{cases}}

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

\displaystyle {\mathbb {E}}X_{t}={\begin{cases}0&{\text{for }}0\leq t<1,\\-1&{\text{for }}1\leq t<\infty .\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 τ_k = k. This sequence diverges almost surely, since τ_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.^{[details 1]}

Example 2

Let W_t be the Wiener process and ƒ a measurable function such that ${\mathbb {E}}|f(W_{1})|<\infty .$ Then the following process is a martingale:

\displaystyle X_{t}={\mathbb {E}}(f(W_{1})|F_{t})={\begin{cases}f_{{1-t}}(W_{t})&{\text{for }}0\leq t<1,\\f(W_{1})&{\text{for }}1\leq t<\infty ;\end{cases}}

here

\displaystyle f_{s}(x)={\mathbb {E}}f(x+W_{s})=\int f(x+y){\frac 1{{\sqrt {2\pi s}}}}{\mathrm {e}}^{{-y^{2}/(2s)}}.

The Dirac delta function $\delta$ (strictly speaking, not a function), being used in place of $f,$ leads to a process defined informally as $Y_{t}={\mathbb {E}}(\delta (W_{1})|F_{t})$ and formally as

\displaystyle Y_{t}={\begin{cases}\delta _{{1-t}}(W_{t})&{\text{for }}0\leq t<1,\\0&{\text{for }}1\leq t<\infty ,\end{cases}}

where

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

The process $Y_{t}$ is continuous almost surely (since $W_{1}\neq 0$ almost surely), nevertheless, its expectation is discontinuous,

\displaystyle {\mathbb {E}}Y_{t}={\begin{cases}1/{\sqrt {2\pi }}&{\text{for }}0\leq t<1,\\0&{\text{for }}1\leq t<\infty .\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, and

\displaystyle 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,

\displaystyle {\mathbb {E}}X_{t}\to \infty

t\to \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 $r\to \infty$ . (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 $k\to \infty$ ) for every t, that is, ${\mathbb {E}}|M_{t}^{{\tau _{k}}}-M_{t}|\to 0;$ here $M_{t}^{{\tau _{k}}}=M_{{t\wedge \tau _{k}}}$ is the stopped process. The given relation $\tau _{k}\to \infty$ implies that $M_{t}^{{\tau _{k}}}\to M_{t}$ almost surely. The dominated convergence theorem ensures the convergence in L¹ provided that

\textstyle (*)\quad {\mathbb {E}}\sup _{k}|M_{t}^{{\tau _{k}}}|<\infty

for every t.

Thus, Condition (*) is sufficient for a local martingale $M_{t}$ being a martingale. A stronger condition

\textstyle (**)\quad {\mathbb {E}}\sup _{{s\in [0,t]}}|M_{s}|<\infty

for every t

is also sufficient.

Caution. The weaker condition

\textstyle \sup _{{s\in [0,t]}}{\mathbb {E}}|M_{s}|<\infty

for every t

is not sufficient. Moreover, the condition

\textstyle \sup _{{t\in [0,\infty )}}{\mathbb {E}}{\mathrm {e}}^{{|M_{t}|}}<\infty

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

A special case:

\textstyle M_{t}=f(t,W_{t}),

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

{\Big (}{\frac {\partial }{\partial t}}+{\frac 12}{\frac {\partial ^{2}}{\partial x^{2}}}{\Big )}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 that

\textstyle |f(s,x)|\leq C{\mathrm {e}}^{{\varepsilon x^{2}}}

for all $s\in [0,t]$ and $x\in {\mathbb {R}}.$

Technical details

↑ 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

Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications (Sixth ed.). Berlin: Springer. ISBN 3-540-04758-1.

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