 Snell envelope

The finite-horizon Snell envelope of rewards $Z_n$ is defined backward by $V_N=Z_N$ and $V_n=\max \{Z_n,\mathbb{E}[V_{n+1}\mid {\mathcal{F}}_n]\}$. It is the least supermartingale dominating $Z$.

The Snell envelope is a concept used primarily in the fields of stochastic control and optimal stopping theory. It provides a way to characterize the value of optimal stopping problems, particularly in scenarios where a decision-maker can stop a stochastic process at various times to maximize their expected payoff. Mathematically, the Snell envelope is defined as the least upper bound of the expected values of stopping times given a stochastic process. Formally, if \( X_t \) is a stochastic process (e.g.