Local martingale

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 τ_k = k. This sequence diverges almost surely, since τ_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}}}

here

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

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).