Integration by parts formula for killed processes: A point of view from approximation theory

TL;DR

This paper develops probabilistic integration by parts formulas for killed diffusion processes, enabling unbiased Monte Carlo simulations and advancing the intersection of stochastic analysis and approximation theory.

Contribution

It introduces a novel probabilistic representation for integration by parts formulas for killed diffusions using Markovian perturbation and Malliavin calculus, with applications to Monte Carlo methods.

Findings

Derived Bismut-Elworthy-Li type formulas for killed processes

Established an unbiased Monte Carlo simulation method

Connected probabilistic representations with approximation theory

Abstract

In this paper, we establish a probabilistic representation for two integration by parts formulas, one being of Bismut-Elworthy-Li's type, for the marginal law of a one-dimensional diffusion process killed at a given level. These formulas are established by combining a Markovian perturbation argument with a tailor-made Malliavin calculus for the underlying Markov chain structure involved in the probabilistic representation of the original marginal law. Among other applications, an unbiased Monte Carlo path simulation method for both integration by parts formula stems from the previous probabilistic representations.