On the optimal filtering of diffusion processes

Zakai, Moshe

doi:10.1007/bf00536382

Cited by 615 publications

(295 citation statements)

References 7 publications

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“…where ρ t is an Y t -adapted measure-valued process satisfying the following Zakai Equation (see [53]). …”

Section: The Filtering Problem and Key Resultsmentioning

confidence: 99%

Generalised particle filters with Gaussian mixtures

Crisan

2015

Stochastic Processes and their Applications

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Stochastic filtering is defined as the estimation of a partially observed dynamical system. A massive scientific and computational effort is dedicated to the development of numerical methods for approximating the solution of the filtering problem. Approximating the solution of the filtering problem with Gaussian mixtures has been a very popular method since the. Despite nearly fifty years of development, the existing work is based on the success of the numerical implementation and is not theoretically justified. This paper fills this gap and contains a rigorous analysis of a new Gaussian mixture approximation to the solution of the filtering problem. We deduce the L 2 -convergence rate for the approximating system and show some numerical example to test the new algorithm.

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“…where ρ t is an Y t -adapted measure-valued process satisfying the following Zakai Equation (see [53]). …”

Section: The Filtering Problem and Key Resultsmentioning

confidence: 99%

Generalised particle filters with Gaussian mixtures

Crisan

2015

Stochastic Processes and their Applications

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“…To start, a number of authors including Refs. [1][2][3] studied the nonlinear case in the setting of additional Gaussian noise. Other references in a similar framework are given, for instance, by Refs.…”

Section: Introductionmentioning

confidence: 99%

Recent Advances in Nonlinear Filtering with a Financial Application to Derivatives Hedging under Incomplete Information

Ceci¹,

Colaneri²

2017

Bayesian Inference

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In this chapter, we present some recent results about nonlinear filtering for jump diffusion signal and observation driven by correlated Brownian motions having common jump times. We provide the Kushner-Stratonovich and the Zakai equation for the normalized and the unnormalized filter, respectively. Moreover, we give conditions under which pathwise uniqueness for the solutions of both equations holds. Finally, we study an application of nonlinear filtering to the financial problem of derivatives hedging in an incomplete market with partial observation. Precisely, we consider the risk-minimizing hedging approach. In this framework, we compute the optimal hedging strategy for an informed investor and a partially informed one and compare the total expected squared costs of the strategies.

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“…However, (1) it is not easy to solve these equations, and (2) it is computationally 'expensive' due to the two stage calculation procedure (prediction and correction) in the real time. Later Zakai [27] obtained a simpler form of the stochastic differential equation for the posterior unnormalized conditional density Φ(t, x) = p(t, x|Z t ) for X t in the following form:…”

Section: Introductionmentioning

confidence: 99%

Fractional generalizations of filtering problems and their associated fractional Zakai equations

Umarov

Daum

Nelson

2014

Fract Calc Appl Anal

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In this paper we discuss fractional generalizations of the filtering problem. The "fractional" nature comes from time-changed state or observation processes, basic ingredients of the filtering problem. The mathematical feature of the fractional filtering problem emerges as the Riemann-Liouville or Caputo-Djrbashian fractional derivative in the associated Zakai equation. We discuss fractional generalizations of the nonlinear filtering problem whose state and observation processes are driven by time-changed Brownian motion or/and Lévy process.MSC 2010 : Primary 60H10; Secondary 35S10, 60G51, 60H05

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On the optimal filtering of diffusion processes

Cited by 615 publications

References 7 publications

Generalised particle filters with Gaussian mixtures

Generalised particle filters with Gaussian mixtures

Recent Advances in Nonlinear Filtering with a Financial Application to Derivatives Hedging under Incomplete Information

Fractional generalizations of filtering problems and their associated fractional Zakai equations

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