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).

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

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

The process ${\displaystyle X_{t}}$ is continuous almost surely; nevertheless, its expectation is discontinuous,

This process is not a martingale. However, it is a local martingale. A localizing sequence may be chosen as ${\displaystyle \tau _{k}=\min\{t:X_{t}=k\}}$ if there is such t, otherwise ${\displaystyle \tau _{k}=k}$. This sequence diverges almost surely, since ${\displaystyle \tau _{k}=k}$ 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.

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

where