Fast Filtering in Switching Approximations of Nonlinear Markov Systems With Applications to Stochastic Volatility

Abstract

We consider the problem of optimal statistical filtering in general non-linear non-Gaussian Markov dynamic systems. The novelty of the proposed approach consists in approximating the non-linear system by a recent Markov switching process, in which one can perform exact and optimal filtering with a linear time complexity. All we need to assume is that the system is stationary (or asymptotically stationary), and that one can sample its realizations. We evaluate our method using two stochastic volatility models and results show its efficiency.

Fast smoothing in switching approximations of non-linear and non-Gaussian models

“…We suggest to use a variant of the EM algorithm described in [7] to achieve the inference. The EM algorithm usually performs well in conditionally Gaussian switching models.…”

“…The EM algorithm is a great way to estimate the parameters of interest c i j , Υ i j , Ξ i j 1≤i, j≤K , however any alternative parameter estimation scheme may be used instead. Besides, for practical purposes, our original EM implementation proposed in [7] estimates directly A(r n+1 n ), Q(r n+1 n ), F(r n+1 n ), H(r n+1 n ) and G(r n+1 n ) for each value of pair r n+1 n instead of c i j , Υ i j , Ξ i j 1≤i, j≤K . Let us sum up our new smoothing method by the following algorithm, which thus contains two stages: parameter estimation (or identification of the Model 2) stage, and smoothing stage.…”

“…Therefore, the whole distribution p x N 1 , y N 1 derives from p x 2 1 , y 2 1 . Following [6,7], we propose to approximate the latter distribution by a Gaussian mixture of…”

“…Now, exact filtering and smoothing are computable in particular stationary Markov triplet models called "stationary conditionally Gaussian observed Markov switching models" (SCGOMSMs) (see e.g. [7,8]). Indeed, SCGOMSMs are particular "conditionally Markov switching models" (CMSHLMs) in which fast filtering is feasible [9], and we show in the paper that fast smoothing also is.…”

“…The interest of using (2) to approximate the general model (1) has already been demonstrated for filtering (c.f. [10,7]) and the aim of our paper is to show its interest for smoothing.…”