Functional convergence of Snell envelopes: Applications to American options approximations

The main result of the paper is a stability theorem for the Snell envelope under convergence in distribution of the underlying processes: more precisely, we prove that if a sequence $(X^n)$ of stochastic processes converges in distribution for the Skorokhod topology to a process $X$ and satisfies some additional hypotheses, the sequence of Snell envelopes converges in distribution for the Meyer-Zheng topology to the Snell envelope of $X$ (a brief account of this rather neglected topology is given in the appendix). When the Snell envelope of the limit process is continuous, the convergence is in fact in the Skorokhod sense. This result is illustrated by several examples of approximations of the American options prices; we give moreover a kind of robustness of the optimal hedging portfolio for the American put in the Black and Scholes model.