Optional Stopping Theorem

ProbabilityTheoryStochasticProcesses Given a continuous and Uniformly Integrable Martingale, and Stopping Times then

3. ,
4. and the stopped process is Uniformly Integrable

If is not necessarily Uniformly Integrable, but are bounded, then 1. holds. OST is useful when computing densities and expectations:

1. Take and define , then so is u.i. By OST: using the fact that partitions the sample space, we get