Snell Envelope

The Snell envelope of an adapted stochastic process representing rewards is defined backward by

			(1)
			(2)

It is the smallest integrable stochastic process that dominates and satisfies . Under standard hypotheses, is the value of an optimal stopping problem, and the first stopping time at which is optimal. In particular, the discounted stochastic process giving the value of an American option is a Snell envelope.