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Simulation of elliptic and hypo-elliptic conditional diffusions

Part of: Markov processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  29 April 2020

Joris Bierkens ,
Frank van der Meulen  and
Moritz Schauer
Joris Bierkens*
Affiliation:
Vrije Universiteit Amsterdam
Frank van der Meulen*
Affiliation:
Delft Institute of Applied Mathematics
Moritz Schauer*
Affiliation:
Chalmers University of Technology and University of Gothenburg
*
*Postal address: Department of Mathematics, Vrije Universiteit Amsterdam, De Boelelaan 1081a, 1081HVAmsterdam, The Netherlands. Email address: joris.bierkens@tudelft.nl
**Postal address: Delft Institute of Applied Mathematics, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, Van Mourik Broekmanweg 6, 2628XEDelft, The Netherlands.
***Postal address: Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE-412 96Göteborg, Sweden.
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Abstract

Suppose X is a multidimensional diffusion process. Assume that at time zero the state of X is fully observed, but at time $T>0$ only linear combinations of its components are observed. That is, one only observes the vector $L X_T$ for a given matrix L. In this paper we show how samples from the conditioned process can be generated. The main contribution of this paper is to prove that guided proposals, introduced in [35], can be used in a unified way for both uniformly elliptic and hypo-elliptic diffusions, even when L is not the identity matrix. This is illustrated by excellent performance in two challenging cases: a partially observed twice-integrated diffusion with multiple wells and the partially observed FitzHugh–Nagumo model.

Keywords

Diffusion bridgeMonte Carlo methodFitzHugh–Nagumo modelguided proposalLangevin samplerpartially observed diffusiontwice-integrated diffusion

MSC classification

Primary: 60J60: Diffusion processes
Secondary: 65C30: Stochastic differential and integral equations 65C05: Monte Carlo methods
Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 1 , March 2020 , pp. 173 - 212
Copyright
© Applied Probability Trust 2020

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