A uniform convergence theorem for the numerical solving of the nonlinear filtering problem

1998 ◽  
Vol 35 (4) ◽  
pp. 873-884 ◽  
Author(s):  
P. Del Moral
Keyword(s):  
Monte Carlo ◽  
Uniform Convergence ◽  
Convergence Theorem ◽  
Nonlinear Filtering ◽  
Partial Information ◽  
Particle System ◽  
Convergence Result ◽  
Recursive Solution ◽  
Ergodic Condition ◽  
Filtering Problem

The filtering problem concerns the estimation of a stochastic process X from its noisy partial information Y. With the notable exception of the linear-Gaussian situation, general optimal filters have no finitely recursive solution. The aim of this work is the design of a Monte Carlo particle system approach to solve discrete time and nonlinear filtering problems. The main result is a uniform convergence theorem. We introduce a concept of regularity and we give a simple ergodic condition on the signal semigroup for the Monte Carlo particle filter to converge in law and uniformly with respect to time to the optimal filter, yielding what seems to be the first uniform convergence result for a particle approximation of the nonlinear filtering equation.

A uniform convergence theorem for the numerical solving of the nonlinear filtering problem

1998 ◽  
Vol 35 (04) ◽  
pp. 873-884 ◽  
Author(s):  
P. Del Moral
Keyword(s):  
Monte Carlo ◽  
Uniform Convergence ◽  
Convergence Theorem ◽  
Nonlinear Filtering ◽  
Partial Information ◽  
Particle System ◽  
Convergence Result ◽  
Recursive Solution ◽  
Ergodic Condition ◽  
Filtering Problem

The filtering problem concerns the estimation of a stochastic process X from its noisy partial information Y. With the notable exception of the linear-Gaussian situation, general optimal filters have no finitely recursive solution. The aim of this work is the design of a Monte Carlo particle system approach to solve discrete time and nonlinear filtering problems. The main result is a uniform convergence theorem. We introduce a concept of regularity and we give a simple ergodic condition on the signal semigroup for the Monte Carlo particle filter to converge in law and uniformly with respect to time to the optimal filter, yielding what seems to be the first uniform convergence result for a particle approximation of the nonlinear filtering equation.

Monte Carlo Methods for a Special Nonlinear Filtering Problem

2000 ◽  
Vol 33 (16) ◽  
pp. 347-352 ◽  
Author(s):  
O.A. Stepanov ◽  
V.M. Ivanov ◽  
M.L. Korenevski
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Methods ◽  
Nonlinear Filtering ◽  
Filtering Problem

scholarly journals Monte Carlo methods for backward equations in nonlinear filtering

2009 ◽  
Vol 41 (01) ◽  
pp. 63-100 ◽  
Author(s):  
G. N. Milstein ◽  
M. V. Tretyakov
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Methods ◽  
Nonlinear Filtering ◽  
Error Control ◽  
Variance Reduction ◽  
Stochastic Partial Differential Equation ◽  
Numerical Algorithms ◽  
Variance Reduction Techniques ◽  
Reduction Techniques ◽  
Filtering Problem

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 linear stochastic partial differential equation. We study convergence of the proposed numerical algorithms. The considered methods have such advantages as a capability in principle to solve filtering problems of large dimensionality, reliable error control, and recurrency. Their efficiency is achieved due to the numerical procedures which use effective numerical schemes and variance reduction techniques. The results obtained are supported by numerical experiments.

scholarly journals Monte Carlo methods for backward equations in nonlinear filtering

2009 ◽  
Vol 41 (1) ◽  
pp. 63-100 ◽  
Author(s):  
G. N. Milstein ◽  
M. V. Tretyakov
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Methods ◽  
Nonlinear Filtering ◽  
Error Control ◽  
Variance Reduction ◽  
Stochastic Partial Differential Equation ◽  
Numerical Algorithms ◽  
Variance Reduction Techniques ◽  
Reduction Techniques ◽  
Filtering Problem

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 linear stochastic partial differential equation. We study convergence of the proposed numerical algorithms. The considered methods have such advantages as a capability in principle to solve filtering problems of large dimensionality, reliable error control, and recurrency. Their efficiency is achieved due to the numerical procedures which use effective numerical schemes and variance reduction techniques. The results obtained are supported by numerical experiments.

scholarly journals Optimal switching problems under partial information

2015 ◽  
Vol 21 (2) ◽  
Author(s):  
Kai Li ◽  
Kaj Nyström ◽  
Marcus Olofsson
Keyword(s):  
Monte Carlo ◽  
Dynamic Programming ◽  
Value Function ◽  
Partial Information ◽  
Convergence Result ◽  
Optimal Switching ◽  
Filtering Theory ◽  
Stochastic Filtering ◽  
Observation Process ◽  
The Value Function

AbstractIn this paper, we formulate and study an optimal switching problem under partial information. In our model, the agent/manager/investor attempts to maximize the expected reward by switching between different states/investments. However, he is not fully aware of his environment and only an observation process, which contains partial information about the environment/underlying, is accessible. It is based on the partial information carried by this observation process that all decisions must be made. We propose a probabilistic numerical algorithm, based on dynamic programming, regression Monte Carlo methods, and stochastic filtering theory, to compute the value function. In this paper, the approximation of the value function and the corresponding convergence result are obtained when the underlying and observation processes satisfy the linear Kalman–Bucy setting. A numerical example is included to show some specific features of partial information.

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