stopped process

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

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

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

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

 $X^{\tau-}\equiv\left\{\begin{array}[]{ll}X_{t},&\textrm{if }t<\tau,\\ X_{\tau-},&\textrm{if }t\geq\tau.\end{array}\right.$

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 $|X_{\tau}|\geq K$, 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.

Title stopped process StoppedProcess 2013-03-22 18:37:38 2013-03-22 18:37:38 gel (22282) gel (22282) 5 gel (22282) Definition msc 60G40 msc 60G05 optional stopping pre-stopped process prestopped process