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 Itō calculus, semimartingale, and Girsanov theorem).

Definition

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 ${\displaystyle X:[0,\infty )\times \Omega \rightarrow S}$ 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 ${\displaystyle \tau _{k}:\Omega \to [0,\infty )}$ such that

• the ${\displaystyle \tau _{k}}$ are almost surely increasing: ${\displaystyle P\left\{\tau _{k}<\tau _{k+1}\right\}=1}$;
• the ${\displaystyle \tau _{k}}$ diverge almost surely: ${\displaystyle P\left\{\lim _{k\to \infty }\tau _{k}=\infty \right\}=1}$;
• the stopped process
  $${\displaystyle X_{t}^{\tau _{k}}:=X_{\min\{t,\tau _{k}\}}}$$
  is an ${\displaystyle F_{*}}$-martingale for every ${\displaystyle k}$.

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

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

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

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

This process is not a martingale. However, it is a local martingale. A localizing sequence may be chosen as ${\displaystyle \tau _{k}=\min\{t:X_{t}=k\}}$ 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. ^{[details 1]}

Example 2

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:

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

where

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

The Dirac delta function ${\displaystyle \delta }$ (strictly speaking, not a function), being used in place of ${\displaystyle f,}$ leads to a process defined informally as ${\displaystyle Y_{t}=\operatorname {E} (\delta (W_{1})\mid F_{t})}$ 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 ${\displaystyle Y_{t}}$ is continuous almost surely (since ${\displaystyle W_{1}\neq 0}$ almost surely), nevertheless, its expectation is discontinuous,

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

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

Example 3

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

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

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,

${\displaystyle \operatorname {E} X_{t}\to \infty }$  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, ${\displaystyle \operatorname {E} |M_{t}^{\tau _{k}}-M_{t}|\to 0;}$ 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

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

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

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

is also sufficient.

Caution. The weaker condition

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

is not sufficient. Moreover, the condition

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

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

A special case:

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

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.^{[ vague ]}

References

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