Improved error bounds for quantization based numerical schemes for BSDE and nonlinear filtering

TL;DR

This paper leverages recent advances in optimal quantization theory to enhance error bounds in numerical schemes for BSDEs and nonlinear filtering, introducing new theoretical insights and robustness results.

Contribution

It introduces improved quadratic error bounds for quantization-based schemes in BSDEs and nonlinear filtering, utilizing a Pythagoras-like theorem and a robustness property of optimal quantizers.

Findings

Enhanced error bounds for BSDE numerical schemes

Robustness of optimal quantizers in different L^s norms

Application of a Pythagoras-like theorem to conditional expectation

Abstract

We take advantage of recent and new results on optimal quantization theory to improve the quadratic optimal quantization error bounds for backward stochastic differential equations (BSDE) and nonlinear filtering problems. For both problems, a first improvement relies on a Pythagoras like Theorem for quantized conditional expectation. While allowing for some locally Lipschitz functions conditional densities in nonlinear filtering, the analysis of the error brings into playing a new robustness result about optimal quantizers, the so-called distortion mismatch property: $L^r$-quadratic optimal quantizers of size $N$ behave in $L^s$ in term of mean error at the same rate $N^{-\frac 1d}$, $0<s< r+d$.