Local martingale

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

• 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 ${\displaystyle k}$.

Summarize Timeline Top Qs Fact Check

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]}

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 ${\displaystyle \tau _{k}=\min\{t:Y_{t}=k\}.}$

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).