# 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, 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: $P\left\{\tau _{k}<\tau _{k+1}\right\}=1$;
* 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

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
    $\displaystyle X_{t}={\begin{cases}W_{\min \left({\tfrac {t}{1-t}},T\right)}&{\text{for }}0\leq t<1,\\-1&{\text{for }}1\leq t<\infty .\end{cases}}$
The process $X_{t}$ is continuous almost surely; nevertheless, its expectation is discontinuous,
    $\displaystyle \operatorname {E} X_{t}={\begin{cases}0&{\text{for }}0\leq t<1,\\-1&{\text{for }}1\leq t<\infty .\end{cases}}$
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 $\tau _{k}=k$. 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 $\operatorname {E} |f(W_{1})|<\infty .$ Then the following process is a martingale:
    $X_{t}=\operatorname {E} (f(W_{1})\mid F_{t})={\begin{cases}f_{1-t}(W_{t})&{\text{for }}0\leq t<1,\\f(W_{1})&{\text{for }}1\leq t<\infty ;\end{cases}}$
where
    $f_{s}(x)=\operatorname {E} f(x+W_{s})=\int f(x+y){\frac {1}{\sqrt {2\pi s}}}\mathrm {e} ^{-y^{2}/(2s)}\,dy.$
The Dirac delta function $\delta$ (strictly speaking, not a function), being used in place of $f,$ leads to a process defined informally as $Y_{t}=\operatorname {E} (\delta (W_{1})\mid F_{t})$ and formally as
    $Y_{t}={\begin{cases}\delta _{1-t}(W_{t})&{\text{for }}0\leq t<1,\\0&{\text{for }}1\leq t<\infty ,\end{cases}}$
where
    $\delta _{s}(x)={\frac {1}{\sqrt {2\pi s}}}\mathrm {e} ^{-x^{2}/(2s)}.$
The process $Y_{t}$ is continuous almost surely (since $W_{1}\neq 0$ almost surely), nevertheless, its expectation is discontinuous,
    $\operatorname {E} Y_{t}={\begin{cases}1/{\sqrt {2\pi }}&{\text{for }}0\leq t<1,\\0&{\text{for }}1\leq t<\infty .\end{cases}}$
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
    $X_{t}=\ln |Z_{t}-1|\,.$
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 L$^1$ (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 L$^1$ provided that
    $\textstyle (*)\quad \operatorname {E} \sup _{k}|M_{t}^{\tau _{k}}|<\infty$ for every t.
Thus, Condition (*) is sufficient for a local martingale $M_{t}$ being a martingale. A stronger condition
    $\textstyle (**)\quad \operatorname {E} \sup _{s\in [0,t]}|M_{s}|<\infty$ for every t
is also sufficient.
Caution. The weaker condition
    $\textstyle \sup _{s\in [0,t]}\operatorname {E} |M_{s}|<\infty$ for every t
is not sufficient. Moreover, the condition
    $\textstyle \sup _{t\in [0,\infty )}\operatorname {E} \mathrm {e} ^{|M_{t}|}<\infty$
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
    ${\Big{\frac {\partial }{\partial t}}+{\frac {1}{2}}{\frac {\partial ^{2}}{\partial x^{2}}}{\Big )}f(t,x)=0.$
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.