# 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, FP) be a probability space; let F = { Ft | 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 τk : Ω → [0, +∞) such that

${\displaystyle X_{t}^{\tau _{k}}:=X_{\min\{t,\tau _{k}\}}}$
is an F-martingale for every k.

## Examples

### Example 1

Let Wt be the Wiener process and T = min{ t : Wt = −1 } the time of first hit of −1. The stopped process Wmin{ tT } 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

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

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

${\displaystyle \displaystyle \mathbb {E} X_{t}={\begin{cases}0&{\text{for }}0\leq t<1,\\-1&{\text{for }}1\leq t<\infty .\end{cases}}}$

This process is not a martingale. However, it is a local martingale. A localizing sequence may be chosen as ${\displaystyle \tau _{k}=\min\{t:X_{t}=k\}}$  if there is such t, otherwise τk = k. This sequence diverges almost surely, since τk = 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.[details 1]

### Example 2

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

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

here

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

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

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

where

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

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

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

This process is not a martingale. However, it is a local martingale. A localizing sequence may be chosen as ${\displaystyle \tau _{k}=\min\{t:Y_{t}=k\}.}$

### Example 3

Let ${\displaystyle Z_{t}}$  be the complex-valued Wiener process, and

${\displaystyle \displaystyle X_{t}=\ln |Z_{t}-1|\,.}$

The process ${\displaystyle X_{t}}$  is continuous almost surely (since ${\displaystyle Z_{t}}$  does not hit 1, almost surely), and is a local martingale, since the function ${\displaystyle u\mapsto \ln |u-1|}$  is harmonic (on the complex plane without the point 1). A localizing sequence may be chosen as ${\displaystyle \tau _{k}=\min\{t:X_{t}=-k\}.}$  Nevertheless, the expectation of this process is non-constant; moreover,

${\displaystyle \displaystyle \mathbb {E} X_{t}\to \infty }$    as ${\displaystyle t\to \infty ,}$

which can be deduced from the fact that the mean value of ${\displaystyle \ln |u-1|}$  over the circle ${\displaystyle |u|=r}$  tends to infinity as ${\displaystyle r\to \infty }$ . (In fact, it is equal to ${\displaystyle \ln r}$  for r ≥ 1 but to 0 for r ≤ 1).

## Martingales via local martingales

Let ${\displaystyle M_{t}}$  be a local martingale. In order to prove that it is a martingale it is sufficient to prove that ${\displaystyle M_{t}^{\tau _{k}}\to M_{t}}$  in L1 (as ${\displaystyle k\to \infty }$ ) for every t, that is, ${\displaystyle \mathbb {E} |M_{t}^{\tau _{k}}-M_{t}|\to 0;}$  here ${\displaystyle M_{t}^{\tau _{k}}=M_{t\wedge \tau _{k}}}$  is the stopped process. The given relation ${\displaystyle \tau _{k}\to \infty }$  implies that ${\displaystyle M_{t}^{\tau _{k}}\to M_{t}}$  almost surely. The dominated convergence theorem ensures the convergence in L1 provided that

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

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

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

is also sufficient.

Caution. The weaker condition

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

is not sufficient. Moreover, the condition

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

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

A special case:

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

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

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

However, this PDE itself does not ensure that ${\displaystyle M_{t}}$  is a martingale. In order to apply (**) the following condition on f is sufficient: for every ${\displaystyle \varepsilon >0}$  and t there exists ${\displaystyle C=C(\varepsilon ,t)}$  such that

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

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

## Technical details

1. ^ 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.