Multilevel approximation of backward stochastic differential equations

TL;DR

This paper introduces a multilevel method for efficiently solving backward stochastic differential equations, reducing computational complexity significantly compared to existing approaches.

Contribution

The authors develop a fully implementable multilevel scheme with explicit error estimates, improving computational efficiency for BSDE solutions.

Findings

Reduces computational complexity by nearly one order in precision ε

Provides explicit, non-asymptotic error bounds for the multilevel scheme

Demonstrates practical efficiency improvements through computational examples

Abstract

We develop a multilevel approach to compute approximate solutions to backward differential equations (BSDEs). The fully implementable algorithm of our multilevel scheme constructs sequential martingale control variates along a sequence of refining time-grids to reduce statistical approximation errors in an adaptive and generic way. We provide an error analysis with explicit and non-asymptotic error estimates for the multilevel scheme under general conditions on the forward process and the BSDE data. It is shown that the multilevel approach can reduce the computational complexity to achieve precision $\epsilon$, ensured by error estimates, essentially by one order (in $\epsilon^{-1}$) in comparison to established methods, which is substantial. Computational examples support the validity of the theoretical analysis, demonstrating efficiency improvements in practice.