Numerical Solutions for a Class of SPDEs with Application to Filtering

@inproceedings{Kurtz2001NumericalSF,
  title={Numerical Solutions for a Class of SPDEs with Application to Filtering},
  author={Thomas G. Kurtz and Jie Xiong},
  year={2001}
}
A simulation scheme for a class of nonlinear stochastic partial differential equations is proposed and error bounds for the scheme are derived. The scheme is based on the fact that the solutions of the SPDEs can be represented by the weighted empirical measure of an infinite system of interacting particles. There are two sources of error in the scheme, one due to finite sampling of the infinite collection of particles and the other due to the Euler scheme used in the simulation of the… 
View via Publisher
math.wisc.edu
39 Citations
Highly Influential Citations
1
Background Citations
14
Methods Citations
10
Results Citations
5

39 Citations

Numerical solution for a class of SPDEs over bounded domains

Parabolic stochastic partial differential Equations (SPDEs) with multiplicative noise play a central rôle in nonlinear filtering. More precisely, the conditional distribution of a partially observed…

Numerical solutions for a class of SPDEs over bounded domains

The optimal filter for a bounded signal with reflecting boundary is approximated by the (un-weighted) empirical measure of a finite interacting particle system. The main motivation of this…

Numerical Solution of the Dirichlet Problem for Linear Parabolic SPDEs Based on Averaging over Characteristics

Numerical methods for the Dirichlet problem for linear parabolic stochastic partial differential equations are constructed. The methods are based on the averaging-over-characteristic formula and the…

Monte Carlo methods for backward equations in nonlinear filtering

We consider Monte Carlo methods for the classical nonlinear filtering problem. The first method is based on a backward pathwise filtering equation and the second method is related to a backward…

The Numerical Approximation of Stochastic Partial Differential Equations

The numerical solution of stochastic partial differential equations (SPDEs) is at a stage of development roughly similar to that of stochastic ordinary differential equations (SODEs) in the 1970s,…

A stochastic evolution equation arising from the fluctuations of a class of interacting particle systems

In an earlier paper, we studied the approximation of solutions V (t) to a class of SPDEs by the empirical measure V n (t) of a system of n interacting difiusions. In the present paper, we consider a…

A branching particle system approximation for a class of FBSDEs

In this paper, a new numerical scheme for a class of coupled forward-backward stochastic differential equations (FBSDEs) is proposed by using branching particle systems in a random environment.…

Higher Order Pathwise Numerical Approximations of SPDEs with Additive Noise

A new numerical scheme for the time and space discretization of semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive noise is proposed, and this scheme is shown to converge for such SPDEs faster than standard numerical schemes.

Particle Approximations for a Class of Stochastic Partial Differential Equations

The paper presents a particle approximation for a class of nonlinear stochastic partial differential equations. The work is motivated by and applied to nonlinear filtering. The new results permit the…

PARTICLE APPROXIMATIONS TO THE FILTERING PROBLEM IN CONTINUOUS TIME

In this chapter, we survey some recent results on the particle system approximations to stochastic filtering problems in continuous time. First, a weighted particle system representation of the…

References

SHOWING 1-10 OF 22 REFERENCES

A particle approximation of the solution of the Kushner–Stratonovitch equation

Abstract. We construct a sequence of branching particle systems αn convergent in measure to the solution of the Kushner–Stratonovitch equation. The algorithm based on this result can be used to solve…

The numerical solution of stochastic differential equations

  • P. KloedenE. Platen
  • Mathematics
    The Journal of the Australian Mathematical Society. Series B. Applied Mathematics
  • 1977
Abstract A method is proposed for the numerical solution of Itô stochastic differential equations by means of a second-order Runge–Kutta iterative scheme rather than the less efficient Euler…

An Approximation for the Zakai Equation

A polygonal approximation to the unnormalized conditional measure of a filtering problem, which is the solution of the Zakai stochastic differential equation on measure space, is considered.

Convergence of a Branching Particle Method to the Solution of the Zakai Equation

A sequence of branching particle systems Un convergent in distribution to the solution of the Zakai equation is constructed, which can be used to solve numerically the filtering problem.

An approximation for the nonlinear filtering problem, with error bound †

The approximate filter is given by the solution to an appropriate discrete-time approximating filtering problem that can be explicitly solved by a finite-dimensional procedure.

Time-discretization of the Zakai equation for diffusion processes observed in correlated noise

A time discretization scheme is provided for the Zakai equation, a stochastic PDE which gives the conditional law of a diffusion process observed in white-noise, and an error estimate of order √β is proved for the overall numerical scheme.

Interacting particle systems approximations of the Kushner-Stratonovitch equation

In this paper we consider the continuous-time filtering problem and we estimate the order of convergence of an interacting particle system scheme presented by the authors in previous works. We will…

Nonlinear Filtering Revisited: A Spectral Approach

The objective of this paper is to develop an approach to nonlinear filtering based on the Cameron--Martin version of Wiener chaos expansion. This approach gives rise to a new numerical scheme for…

A Robust Discrete State Approximation to the Optimal Nonlinear Filter for a Diffusion.

The approximation is robust in the sense that it is locally Lipschitz continuous in the data y( °) (sup norm) uniformly in h and it converges to the optimal filter for the diffusion.

Approximation of nonlinear filtering problems and order of convergence

In this paper, we consider a filtering problem where the observation is a function of a diffusion corrupted by an independent white noise. We estimate the error caused by a discretization of the time…