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A branching particle system approximation for nonlinear stochastic filtering

Science China Mathematics 2013年第8期56卷 1521-1541页

作者： LIU HuiLi XIONG Jie Department of Mathematics Hebei Normal University Department of Mathematics and Statistics Concordia University Department of Mathematics University of Tennessee

The optimal filter 7r = {π,t ∈ [0, T]} of a stochastic signal is approximated by a sequence {Try} of measure-valued processes defined by branching particle systems in a random environment （given by the observation process）. The location and weight of each particle are governed by stochastic differential equations driven by the observation process, which is common for all particles, as well as by an individual Brownian motion, which applies to this specific particle only. The branching mechanism of each particle depends on the observation process and the path of this particle itself during its short lifetime δ = n-2α, where n is the number of initial particles and ~ is a fixed parameter to be optimized. As n → ∞, we prove the convergence of π to πt uniformly for t ∈ [0, T]. Compared with the available results in the literature, the main contribution of this article is that the approximation is free of any stochastic integral which makes the numerical implementation readily available.

关键词： optimal filter branching particle system uniform convergence numerical solution

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A branching particle system approximation for a class of FBSDEs

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Probability, Uncertainty and Quantitative Risk 2016年第1期1卷 305-338页

作者： Dejian Chang Huili Liu Jie Xiong School of Mathematics Shandong UniversityJinan 250100People’s Republic of China Department of Mathematics Hebei Normal UniversityShijiazhuang 050024People’s Republic of China Department of Mathematics University of MacaoAvenida da UniversidadeTaipaMacaoSpecial Administrative Region of China

In this paper,a new numerical scheme for a class of coupled forwardbackward stochastic differential equations(FBSDEs)is proposed by using branching particle systems in a random ***,by the four step scheme,we introduce a partial differential Eq.(PDE)used to represent the solution of the FBSDE ***,infinite and finite particle systems are constructed to obtain the approximate solution of the *** location and weight of each particle are governed by stochastic differential equations derived from the FBSDE ***,a branching particle system is established to define the approximate solution of the FBSDE *** branching mechanism of each particle depends on the path of the particle itself during its short lifetim ∈=n^(−2α),where n is the number of initial particles and α<1/2 is a fixed *** convergence of the scheme and its rate of convergence are obtained.

关键词： Forward-backward stochastic differential equation Partial differential equations branching particle system Numerical solution

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Some Limit Theorems for a particle system of Single Point Catalytic branching Random Walks

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Acta Mathematica Sinica,English Series 2007年第6期23卷 997-1012页

作者： Vladimir VATUTIN Jie XIONG Steklov Mathematical Institute Gubkin street 8 119991 Moscow Russia Department of Mathematics University of Tennessee Knoxville TN 37996 1300 USA Department of Mathematics Hebei Normal University Shijiazhuang 050016 P. R. China

We study the scaling limit for a catalytic branching particle system whose particles perform random walks on Z and can branch at 0 only. Varying the initial （finite） number of particles, we get for this system different limiting distributions. To be more specific, suppose that initially there are n^β particles and consider the scaled process Zt^n（·） = Znt（√n·）, where Zt is the measure-valued process 1 and to a representing the original particle system. We prove that Ztn converges to 0 when β 〈1/4 and to a nondegenerate discrete distribution when β=1/*** addition,if 1/4〈β〈1/2 then n-^（2β-1/2）Zt^n converges to a random limit,while if β 〉21then n^-βZtn converges to a deterministic limit.

关键词： Renewal equation branching particle system scaling limit

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Fluctuation limits of strongly degenerate branching systems

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Science China Mathematics 2011年第12期54卷 2669-2682页

作者： LI YuQiang School of Finance and Statistics East China Normal University Shanghai 200241 China

Functional limit theorems for scaled occupation time fluctuations of a sequence of generalized branching particle systems in Rd with anisotropic space motions and strongly degenerated splitting abilities are studied in the cases of critical and intermediate dimensions. The results show that the limit processes are time-independent measure-valued Wiener processes with simple spatial structure.

关键词： functional limit theorem occupation time fluctuation branching particle system

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