Parameter Estimation in Nonlinear Multivariate Stochastic Differential Equations Based on Splitting Schemes
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866908321081982976 |
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| author | Pilipovic, Predrag Samson, Adeline Ditlevsen, Susanne |
| author_facet | Pilipovic, Predrag Samson, Adeline Ditlevsen, Susanne |
| contents | The likelihood functions for discretely observed nonlinear continuous-time models based on stochastic differential equations are not available except for a few cases. Various parameter estimation techniques have been proposed, each with advantages, disadvantages, and limitations depending on the application. Most applications still use the Euler-Maruyama discretization, despite many proofs of its bias. More sophisticated methods, such as Kessler's Gaussian approximation, Ozaki's Local Linearization, Aït-Sahalia's Hermite expansions, or MCMC methods, might be complex to implement, do not scale well with increasing model dimension, or can be numerically unstable. We propose two efficient and easy-to-implement likelihood-based estimators based on the Lie-Trotter (LT) and the Strang (S) splitting schemes. We prove that S has $L^p$ convergence rate of order 1, a property already known for LT. We show that the estimators are consistent and asymptotically efficient under the less restrictive one-sided Lipschitz assumption. A numerical study on the 3-dimensional stochastic Lorenz system complements our theoretical findings. The simulation shows that the S estimator performs the best when measured on precision and computational speed compared to the state-of-the-art. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_11884 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Parameter Estimation in Nonlinear Multivariate Stochastic Differential Equations Based on Splitting Schemes Pilipovic, Predrag Samson, Adeline Ditlevsen, Susanne Methodology Statistics Theory 62F12, 62H12, 62M99, 37M15, 60G65 The likelihood functions for discretely observed nonlinear continuous-time models based on stochastic differential equations are not available except for a few cases. Various parameter estimation techniques have been proposed, each with advantages, disadvantages, and limitations depending on the application. Most applications still use the Euler-Maruyama discretization, despite many proofs of its bias. More sophisticated methods, such as Kessler's Gaussian approximation, Ozaki's Local Linearization, Aït-Sahalia's Hermite expansions, or MCMC methods, might be complex to implement, do not scale well with increasing model dimension, or can be numerically unstable. We propose two efficient and easy-to-implement likelihood-based estimators based on the Lie-Trotter (LT) and the Strang (S) splitting schemes. We prove that S has $L^p$ convergence rate of order 1, a property already known for LT. We show that the estimators are consistent and asymptotically efficient under the less restrictive one-sided Lipschitz assumption. A numerical study on the 3-dimensional stochastic Lorenz system complements our theoretical findings. The simulation shows that the S estimator performs the best when measured on precision and computational speed compared to the state-of-the-art. |
| title | Parameter Estimation in Nonlinear Multivariate Stochastic Differential Equations Based on Splitting Schemes |
| topic | Methodology Statistics Theory 62F12, 62H12, 62M99, 37M15, 60G65 |
| url | https://arxiv.org/abs/2211.11884 |