On the rates of convergence of simulation-based optimization algorithms for optimal stopping problems

Denis Belomestny

doi:10.1214/10-aap692

Abstract

In this paper, we study simulation-based optimization algorithms for solving discrete time optimal stopping problems. Using large deviation theory for the increments of empirical processes, we derive optimal convergence rates for the value function estimate and show that they cannot be improved in general. The rates derived provide a guide to the choice of the number of simulated paths needed in optimization step, which is crucial for the good performance of any simulation-based optimization algorithm. Finally, we present a numerical example of solving optimal stopping problem arising in finance that illustrates our theoretical findings.

Highlights

Let us consider a discrete time optimal stopping problem of the form: (1.1)V ∗ = sup E[Zτ ], 1≤τ ≤K where τ is a stopping time taking values in the set {1, . . . , K} and (Zk)k≥0 is a Markov chain
We study a simulation-based approach to the optimal stopping problem (1.1)
The way of estimating the optimal value function V ∗ presented in Section 2 suggests that one can use the simulation-based optimization algorithm to estimate the boundaries of stopping regions as well

Summary

Weierstrass Institute

We study simulation-based optimization algorithms for solving discrete time optimal stopping problems. Using large deviation theory for the increments of empirical processes, we derive optimal convergence rates for the value function estimate and show that they cannot be improved in general. The rates derived provide a guide to the choice of the number of simulated paths needed in optimization step, which is crucial for the good performance of any simulation-based optimization algorithm. We present a numerical example of solving optimal stopping problem arising in finance that illustrates our theoretical findings

Introduction

Discussion

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