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

We take advantage of recent (see Graf et al., 2008; Pages and Wilbertz, 2012) 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 continuous conditional densities in nonlinear filtering, the analysis of the error brings into play a new robustness result about optimal quantizers, the so-called distortion mismatch property: the $L^s$-mean quantization error induced by $L^r$-optimal quantizers of size $N$ converges at the same rate $N^{?\frac{1}{d}}$ for every $s\in (0,r+d)$.