Approximation of backward stochastic differential equations using Malliavin weights and least-squares regression

TL;DR

This paper introduces a numerical scheme combining Malliavin weights and least-squares regression to solve backward stochastic differential equations, providing tight error bounds and optimized complexity analysis.

Contribution

It presents a novel numerical method for backward SDEs that leverages Malliavin calculus and empirical regression, with rigorous error bounds and complexity optimization.

Findings

Error bounds depend on local regression errors and logarithmic factors.

Algorithm complexity is optimized based on dimension and smoothness.

Estimates incorporate terminal function regularity.

Abstract

We design a numerical scheme for solving a Dynamic Programming equation with Malliavin weights arising from the time-discretization of backward stochastic differential equations with the integration by parts-representation of the $Z$-component by (Ann. Appl. Probab. 12 (2002) 1390-1418). When the sequence of conditional expectations is computed using empirical least-squares regressions, we establish, under general conditions, tight error bounds as the time-average of local regression errors only (up to logarithmic factors). We compute the algorithm complexity by a suitable optimization of the parameters, depending on the dimension and the smoothness of value functions, in the limit as the number of grid times goes to infinity. The estimates take into account the regularity of the terminal function.