Stopped process

Summary

The stopped process $X^\tau$ of a stochastic process $X$ at a stopping time $\tau$ follows $X$ up to time $\tau$. After time $\tau$ it stays frozen at the value $X_\tau$. Stopping preserves the martingale property, which is the heart of the optional stopping theorem. It is also the basic operation of localisation, used to define local martingales and stochastic integrals.