Numerical solution for a class of SPDEs over bounded domains

Abstract

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 diffusion solves the normalized version of an equation of this type. We show that one can approximate the solution of the SPDE by the (unweighted) empirical measure of a finite system of interacting particle for the case when the diffusion evolves in a compact state space with reflecting boundary. This approximation differs from existing approximations where the particles are weighted and the particle interaction arises through the choice of the weights and not at the level of the particles' motion as it is the case in this work. The system of stochastic differential equations modelling the trajectories of the particles is approximated by the recursive projection scheme introduced by Pettersson [Stoch. Process. Appl. 59(2) (1995), pp. 295–308].

Keywords::

• stochastic partial differential equations
• interacting particle systems
• diffusions with reflecting boundary
• Kushner–FKK equation
• nonlinear filtering

MSC (2010) Classification::

• 60H15
• 60K35
• 35R60
• 93E11

Acknowledgements

This work was done during the second author's visits to Imperial College London. The hospitality of Imperial College London and the financial support from London Mathematical Society are gratefully acknowledged.

Notes

^1. The work of Dan Crisan was partially supported by the EPSRC Grant No. EP/H0005500/1.

^2. The work of Jie Xiong was partially supported by NSF DMS-0906907 and the London Mathematical Society (Scheme 2 Grant No. 2930).

^3. The same analysis applies for D being a closed convex domain.

^4. Here and later denotes the transpose of the matrix .

^5. As there are no observations available at time , the unconditional distribution and the conditional distribution of the coincide, hence we use the same notation for both. The law of is also the initial condition of the SPDE (1.2) and the approximating PDE (1.3). In particular, .

^6. In the following, if is a measure and is a test function, we denote .

^7. Better rates of convergence can be obtained under additional assumptions, see [7] and the references therein for details.

^8. Note that this partition may be different from that used to discretize the observation process Y, i.e. .

^9. If the observation path is not assumed fixed, then will be random and the random variables , () will be conditionally independent given.