stopped process

A stochastic process ${(X_t)}_{t\in 𝕋}$ defined on a measurable space $(\mathrm{\Omega },ℱ)$ can be stopped at a random time $\tau :\mathrm{\Omega }\to 𝕋\cup \{\mathrm{\infty }\}$. The resulting stopped process is denoted by $X^{\tau }$,

$$X_t^{\tau }\equiv X_{\min (t,\tau )}.$$

The random time $\tau$ used is typically a stopping time.

If the process $X_t$ has left limits (http://planetmath.org/CadlagProcess) for every $t\in 𝕋$, then it can alternatively be stopped just before the time $\tau$, resulting in the pre-stopped process

Stopping is often used to enforce boundedness or integrability constraints on a process.For example, if $B$ is a Brownian motion and $\tau$ is the first time at which $|B_{\tau }|$ hits some given positive value, then the stopped process $B^{\tau }$ will be a continuous and bounded martingale.It can be shown that many properties of stochastic processes, such as the martingale property, are stable under stopping at any stopping time $\tau$. On the other hand, a pre-stopped martingale need not be a martingale.

For continuous processes, stopping and pre-stopping are equivalent procedures.If $\tau$ is the first time at which $|X_{\tau }|\ge K$, for any given real number $K$, then the pre-stopped process $X^{\tau -}$ will be uniformly bounded.However, for some noncontinuous processes it is not possible to find a stopping time $\tau >0$ making $X^{\tau }$ into a uniformly bounded process. For example, this is the case for any Levy process (http://planetmath.org/LevyProcess) with unbounded jump distribution.

Stopping is used to generalize properties of stochastic processes to obtain the related localized property. See, for example, local martingales.