A high order time discretization of the solution of the non-linear filtering problem
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A high order time discretization of the solution of the non-linear filtering problem Dan Crisan1 · Salvador Ortiz-Latorre2 Received: 11 December 2017 / Revised: 20 October 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract The solution of the continuous time filtering problem can be represented as a ratio of two expectations of certain functionals of the signal process that are parametrized by the observation path. We introduce a class of discretization schemes of these functionals of arbitrary order. The result generalizes the classical work of Picard, who introduced first order discretizations to the filtering functionals. For a given time interval partition, we construct discretization schemes with convergence rates that are proportional with the m-power of the mesh of the partition for arbitrary m ∈ N. The result paves the way for constructing high order numerical approximation for the solution of the filtering problem. Keywords Non-linear filtering · Kallianpur–Striebel’s formula · High order time discretization Mathematics Subject Classification 60G35 · 60F05 · 60F25 · 60H35 · 60H07 · 93E11
The work of D. Crisan was partially supported by the EPSRC Grant EP/H0005500/1. The work of S. Ortiz-Latorre was partially supported by the “Beatriu de Pinós” postdoctoral grant 2009 BP-A 0237, funded by the Generalitat de Catalunya, and the project Energy Markets: Modeling, Optimization and Simulation (EMMOS), funded by the Norwegian Research Council under grant Evita/205328.
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Salvador Ortiz-Latorre [email protected] Dan Crisan [email protected]
1
Department of Mathematics, Imperial College London, Huxley’s Building, 180 Queen’s Gate, London SW7 2AZ, UK
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Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway
123
Stoch PDE: Anal Comp
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Basic framework and statement of the main result . . . . . . . . . 3 Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . 3.1 Iterated integrals . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Integrability of the likelihood functional and its discretizations 3.3 Proof of the Theorem 2.3 . . . . . . . . . . . . . . . . . . . . 4 Technical lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Malliavin calculus . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Martingale representations and Clark–Ocone formula . . . . . 4.3 Backward martingales estimates . . . . . . . . . . . . . . . . 4.4 Conditional expectation estimates . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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