Observing that the recent developments of the recursive (product) quantization method induces a family of Markov chains which includes all standard discretization schemes of diffusions processes , we propose to compute a general error bound induced by the recursive quantization schemes using this generic markovian structure. Furthermore, we compute a marginal weak error for the recursive quantization. We also extend the recursive quantization method to the Euler scheme associated to diffusion processes with jumps, which still have this markovian structure, and we say how to compute the recursive quantization and the associated weights and transition weights.
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 $N^{-frac 1d}$, $0<s< r+d$.
Let $P$ be a probability distribution on $mathbb{R}^d$ (equipped with an Euclidean norm $|cdot|$). Let $ r> 0 $ and let $(alpha_n)_{n geq1}$ be an (asymptotically) $L^r(P)$-optimal sequence of $n$-quantizers. We investigate the asymptotic behavior of the maximal radius sequence induced by the sequence $(alpha_n)_{n geq1}$ defined for every $n geq1$ by $rho(alpha_n) = max{|a|, a inalpha_n}$. When $card(supp(P))$ is infinite, the maximal radius sequence goes to $sup{|x|, x inoperatorname{supp}(P)}$ as $n$ goes to infinity. We then give the exact rate of convergence for two classes of distributions with unbounded support: distributions with hyper-exponential tails and distributions with polynomial tails. In the one-dimensional setting, a sharp rate and constant are provided for distributions with hyper-exponential tails.
We quantize a multidimensional $SDE$ (in the Stratonovich sense) by solving the related system of $ODE$s in which the $d$-dimensional Brownian motion has been replaced by the components of functional stationary quantizers. We make a connection with rough path theory to show that the solutions of the quantized solutions of the $ODE$ converge toward the solution of the $SDE$. On our way to this result we provide convergence rates of optimal quantizations toward the Brownian motion for $frac 1q$-H older distance, $q>2$, in $L^p(P)$.
We build a sequence of empirical measures on the space D(R_+,R^d) of R^d-valued c`adl`ag functions on R_+ in order to approximate the law of a stationary R^d-valued Markov and Feller process (X_t). We obtain some general results of convergence of this sequence. Then, we apply them to Brownian diffusions and solutions to Levy driven SDEs under some Lyapunov-type stability assumptions. As a numerical application of this work, we show that this procedure gives an efficient way of option pricing in stochastic volatility models.
