@article {IOPORT.05840697,
    author       = {Sellami, Afef},
    title        = {Quantization based filtering method using first order approximation.},
    year         = {2010},
    journal      = {SIAM Journal on Numerical Analysis},
    volume       = {47},
    number       = {6},
    issn         = {0036-1429},
    pages        = {4711-4734},
    publisher    = {Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA},
    doi          = {10.1137/060652580},
    abstract     = {The quantization-based filtering method is a grid-based approximation method to solve nonlinear filtering problems with discrete time observations (see [{\it G. Pag\`es} and {\it H. Pham}, Bernoulli 11, No.~5, 893--932 (2005; Zbl 1084.62095)] and [{\it G. Pag\`es, H. Pham} and {\it J. Printems}, in: Handbook of computational and numerical methods in finance. Boston, MA: Birkh\"auser, 253--297 (2004; Zbl 1138.91467)]). In quantization filtering schemes, a signal process is an $\mathbb R^d$-valued discrete time hidden Markov chain $(X_k)_{0\leq k\leq n}$ defined by the signal equation $X_{k+1} = F_{k+1}(X_k,\epsilon_{k+1}),\ 0\leq k\leq n-1,$ where $F_k:\mathbb R^d\times\mathbb R^q\to\mathbb R^d$ is a Borel function and $(\epsilon_{k})_{1\leq k\leq n}$ is a sequence of i.i.d. $\mathbb R^q$-valued random variables, independent of $X_0$. The distribution $\mu_0$ of $X_0$ is supposed to be known. The dynamics of the $\mathbb R^d$-valued noisy observation process $(Y_k)_{0\leq k\leq n}$ is defined by the equation $Y_{k} = G_{k}(Y_{k-1},X_k,\eta_{k}),\ 1\leq k\leq n$, where $(\eta_k)$ is a sequence of i.i.d. random variables, independent of $\sigma(X_0,\epsilon_k, k \geq 1)$. The problem is to determine $\operatorname{E}[f(X_n)|Y_1=y_1,\dots,Y_n=y_n]$ for a reasonable Borel function $f:\mathbb R^d\to\mathbb R$ and a given realization $(Y_1=y_1,\dots,Y_n=y_n)$ of the observation process. Using the Kallianpur-Striebel formula [{\it G. Kallianpur} and {\it C. Striebel}, Ann. Math. Stat. 39, 785--801 (1968; Zbl 0174.22102)], the problem can be reduced to the computation of the unnormalized filter $\pi_n$ defined by $\pi_nf=\operatorname{E}\left[ f(X_n)\Pi_{k=1}^ng_k(y_{k-1},X_k,y_k)\right]$. By introducing the operators $(H_k)_{0\leq k\leq n}$ by the relations $H_kf(x)=g_k(x)\operatorname{E}[f(X_{k+1})|X_k=x]$, $0\leq k\leq {n-1}$, $H_n^nf(x)=g_n(x)f(x)$, a sequential definition of the unnormalized filter $\pi_n$ can be given by the formula $\pi_nf=\mu_0\circ H_0\circ\cdots\circ H_n^nf$. Consequently, we can write sequentially, either in the forward way, $U_0=\mu_o\circ H_0, U_k=U_{k-1}\circ H_k$, $0\leq k\leq {n-1}$, or in the backward way, $R_n=H_n^n, R_k=H_k\circ R_{k+1}$, $0\leq k\leq {n-1}$, and $\pi_nf=\mu_0\circ R_0f=U_{n-1}\circ H_n^nf$. From the recursive definition of either $U_k$ or $R_k$, it is clear that it will be useful to approximate $X_k$ by a random variable $\hat{X}_k$ taking a finite number of values, in order to transform conditional expectations into finite weighted sums. This operation is commonly called quantization and is extensively used in signal processing fields (see [{\it S. Graf} and {\it H. Luschgy}, Foundations of quantization for probability distributions. Lecture Notes in Mathematics. 1730. Berlin: Springer (2000; Zbl 0951.60003); {\it V. Bally} and {\it G. Pag\`es}, Bernoulli 9, No. 6, 1003--1049 (2003; Zbl 1042.60021); {\it G. Pag\`es} and {\it J. Printems}, Monte Carlo Methods Appl. 9, No.2, 135--165 (2003; Zbl 1029.65012)]). Pag\`es and Pham [2005, loc.\,cit.] used these quantizations $\hat{X}_k$ to produce a piecewise constant approximation of $R_k$ and the natural approximation procedure by quantization appeared as a zero order scheme. In this paper, the author proposes an improvement of this method. He deals with the first order approximation using optimal or at least stationary quantizers to estimate the required probability density function. Convergence results are given and numerical results are presented for the particular cases of linear Gaussian model and stochastic volatility models.},
    reviewer     = {Mikhail P. Moklyachuk (Ky\"\i v)},
    identifier   = {05840697},
}