Antithetic Multilevel Particle Filters
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911372734889984 |
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| author | Jasra, Ajay Maama, Mohamed Ombao, Hernando |
| author_facet | Jasra, Ajay Maama, Mohamed Ombao, Hernando |
| contents | In this paper we consider the filtering of partially observed multi-dimensional diffusion processes that are observed regularly at discrete times. This is a challenging problem which requires the use of advanced numerical schemes based upon time-discretization of the diffusion process and then the application of particle filters. Perhaps the state-of-the-art method for moderate dimensional problems is the multilevel particle filter of \cite{mlpf}. This is a method that combines multilevel Monte Carlo and particle filters. The approach in that article is based intrinsically upon an Euler discretization method. We develop a new particle filter based upon the antithetic truncated Milstein scheme of \cite{ml_anti}. We show that for a class of diffusion problems, for $ε>0$ given, that the cost to produce a mean square error (MSE) in estimation of the filter, of $\mathcal{O}(ε^2)$ is $\mathcal{O}(ε^{-2}\log(ε)^2)$. In the case of multidimensional diffusions with non-constant diffusion coefficient, the method of \cite{mlpf} has a cost of $\mathcal{O}(ε^{-2.5})$ to achieve the same MSE. We support our theory with numerical results in several examples. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2301_12371 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Antithetic Multilevel Particle Filters Jasra, Ajay Maama, Mohamed Ombao, Hernando Numerical Analysis Computation In this paper we consider the filtering of partially observed multi-dimensional diffusion processes that are observed regularly at discrete times. This is a challenging problem which requires the use of advanced numerical schemes based upon time-discretization of the diffusion process and then the application of particle filters. Perhaps the state-of-the-art method for moderate dimensional problems is the multilevel particle filter of \cite{mlpf}. This is a method that combines multilevel Monte Carlo and particle filters. The approach in that article is based intrinsically upon an Euler discretization method. We develop a new particle filter based upon the antithetic truncated Milstein scheme of \cite{ml_anti}. We show that for a class of diffusion problems, for $ε>0$ given, that the cost to produce a mean square error (MSE) in estimation of the filter, of $\mathcal{O}(ε^2)$ is $\mathcal{O}(ε^{-2}\log(ε)^2)$. In the case of multidimensional diffusions with non-constant diffusion coefficient, the method of \cite{mlpf} has a cost of $\mathcal{O}(ε^{-2.5})$ to achieve the same MSE. We support our theory with numerical results in several examples. |
| title | Antithetic Multilevel Particle Filters |
| topic | Numerical Analysis Computation |
| url | https://arxiv.org/abs/2301.12371 |