  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 $N^{-\frac 1d}$, $0<s<
r+d$.
