On the Robustness of the Snell Envelope

We analyze the robustness properties of the Snell envelope backward evolution equation for the discrete time optimal stopping problem. We consider a series of approximation schemes, including cut-off-type approximations, Euler discretization schemes, interpolation models, quantization tree models, and the stochastic mesh method of Broadie and Glasserman. In each situation, we provide nonasymptotic convergence estimates, including $\mathbb{L}_p$-mean error bounds and exponential concentration inequalities. We deduce these estimates from a single and general robustness property of Snell envelope semigroups. In particular, this analysis allows us to recover existing convergence results for the quantization tree method and to improve significantly the rates of convergence obtained for the stochastic mesh estimator of Broadie and Glasserman. In the second part of the article, we propose a new approach based on a genealogical tree approximation model of the reference Markov process in terms of a neutral-type genetic model. In contrast to Broadie-Glasserman Monte Carlo models, the computational cost of this new stochastic approximation is linear in the number of particles. Some simulation results are provided and confirm the interest of this new algorithm.