Local martingale

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title: "Local martingale" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["martingale-theory"] description: "Stochastic process with sequence of stopping times so each stopped processes is martingale" topic_path: "general/martingale-theory" source: "https://en.wikipedia.org/wiki/Local_martingale" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Stochastic process with sequence of stopping times so each stopped processes is 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, 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,\infty)\times \Omega \rightarrow 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,\infty) such that

• the \tau_k are almost surely increasing: P\left{\tau_k ;
• the \tau_k diverge almost surely: P \left{\lim_{k\to\infty} \tau_k =\infty \right}=1;
• the stopped process X_t^{\tau_k} := X_{\min { t, \tau_k }} is an F_*-martingale for every k.

Examples

Example 1

::figure[src="https://upload.wikimedia.org/wikipedia/commons/4/41/Local_martingale.svg" caption="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 Wmin{ 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\left(\tfrac{t}{1-t},T\right)} &\text{for } 0 \le t -1 &\text{for } 1 \le t \end{cases}

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

: \displaystyle \operatorname{E} X_t = \begin{cases} 0 &\text{for } 0 \le t -1 &\text{for } 1 \le t \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 τk is a martingale. 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.

Example 2

Let W**t be the Wiener process and ƒ a measurable function such that \operatorname{E} |f(W_1)| Then the following process is a martingale: : X_t = \operatorname{E} ( f(W_1) \mid F_t ) = \begin{cases} f_{1-t}(W_t) &\text{for } 0 \le t f(W_1) &\text{for } 1 \le t \end{cases} where : f_s(x) = \operatorname{E} f(x+W_s) = \int f(x+y) \frac1{\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 = \operatorname{E} ( \delta(W_1) \mid F_t ) and formally as : Y_t = \begin{cases} \delta_{1-t}(W_t) &\text{for } 0 \le t 0 &\text{for } 1 \le t \end{cases} where : \delta_s(x) = \frac1{\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, : \operatorname{E} Y_t = \begin{cases} 1/\sqrt{2\pi} &\text{for } 0 \le t 0 &\text{for } 1 \le t \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 : 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, : \operatorname{E} X_t \to \infty as 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, \operatorname{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 \operatorname{E} \sup_k| M_t^{\tau_k} | for every t. Thus, Condition () is sufficient for a local martingale M_t being a martingale. A stronger condition : \textstyle (**) \quad \operatorname{E} \sup_{s\in[0,t]} |M_s| for every t is also sufficient.

Caution. The weaker condition : \textstyle \sup_{s\in[0,t]} \operatorname{E} |M_s| for every t is not sufficient. Moreover, the condition : \textstyle \sup_{t\in[0,\infty)} \operatorname{E} \mathrm{e}^{|M_t|} 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} + \frac12 \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 \varepsilon0 and t there exists C = C(\varepsilon,t) such that : \textstyle |f(s,x)| \le C \mathrm{e}^{\varepsilon x^2} for all s \in [0,t] and x \in \mathbb{R}.

Technical details

References

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