Kernel-based collocation methods for Zakai equations

We examine an application of the kernel-based interpolation to numerical solutions for Zakai equations in nonlinear filtering, and aim to prove its rigorous convergence. To this end, we find the class of kernels and the structure of collocation points explicitly under which the process of iterative interpolation is stable. This result together with standard argument in error estimation shows that the approximation error is bounded by the order of the square root of the time step and the error that comes from a single step interpolation. Our theorem is well consistent with the results of numerical experiments.

[1]  M. Aschwanden.  Statistics of Random Processes , 2021, Biomedical Measurement Systems and Data Science.

[2]  István Gyöngy,et al.  Localization errors in solving stochastic partial differential equations in the whole space , 2017, Math. Comput..

[3]  Ling Xu,et al.  On Galerkin Approximations for the Zakai Equation with Diffusive and Point Process Observations , 2013, SIAM J. Numer. Anal..

[4]  Mtw,et al.  Stochastic flows and stochastic differential equations , 1990 .

[5]  Robert Schaback,et al.  Unsymmetric meshless methods for operator equations , 2010, Numerische Mathematik.

[6]  Robert Schaback,et al.  The meshless Kernel-based method of lines for parabolic equations , 2014, Comput. Math. Appl..

[7]  István Gyöngy,et al.  On the solvability of degenerate stochastic partial differential equations in Sobolev spaces , 2014, 1404.4401.

[8]  Alʹbert Nikolaevich Shiri︠a︡ev,et al.  Statistics of random processes , 1977 .

[9]  Dan Crisan,et al.  Convergence of a Branching Particle Method to the Solution of the Zakai Equation , 1998, SIAM J. Appl. Math..

[10]  B. Rozovskii,et al.  Nonlinear Filtering Revisited: A Spectral Approach , 1997 .

[11]  M. Urner.  Scattered Data Approximation , 2016 .

[12]  R. Glowinski,et al.  Approximation of the Zakai¨ equation by the splitting up method , 1990 .

[13]  Paul Kergin,et al.  A natural interpolation of Ck functions , 1980 .

[14]  B. Rozovskii.  Stochastic Evolution Systems , 1990 .

[15]  国田 寛.  Stochastic flows and stochastic differential equations , 1990 .

[16]  Yumiharu Nakano,et al.  Convergence of meshfree collocation methods for fully nonlinear parabolic equations , 2014, Numerische Mathematik.

[17]  E. Stein.  Singular Integrals and Di?erentiability Properties of Functions , 1971 .

[18]  D. Crisan,et al.  Fundamentals of Stochastic Filtering , 2008 .

[19]  A. Bensoussan.  Stochastic Control of Partially Observable Systems , 1992 .

[20]  A. Shiryayev,et al.  Statistics of Random Processes I: General Theory , 1984 .

[21]  E. Kansa.  MULTIQUADRICS--A SCATTERED DATA APPROXIMATION SCHEME WITH APPLICATIONS TO COMPUTATIONAL FLUID-DYNAMICS-- II SOLUTIONS TO PARABOLIC, HYPERBOLIC AND ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS , 1990 .

[22]  V. Benes.  Exact finite-dimensional filters for certain diffusions with nonlinear drift , 1981 .

[23]  N. U. Ahmed,et al.  A Powerful Numerical Technique Solving Zakai Equation for Nonlinear Filtering , 1997 .

[24]  William F. Moss,et al.  Decay rates for inverses of band matrices , 1984 .

[25]  Hyek Yoo,et al.  Semi-discretization of stochastic partial differential equations on R1 by a finite-difference method , 2000, Math. Comput..