Particle method and quantization-based schemes for the simulation of the McKean-Vlasov equation
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arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866911877477433344 |
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| author | Liu, Yating |
| author_facet | Liu, Yating |
| contents | In this paper, we study three numerical schemes for the McKean-Vlasov equation \[\begin{cases} \;dX_t=b(t, X_t, μ_t) \, dt+σ(t, X_t, μ_t) \, dB_t,\: \\ \;\forall\, t\in[0,T],\;μ_t \text{ is the probability distribution of }X_t, \end{cases}\] where $X_0$ is a known random variable. Under the assumption on the Lipschitz continuity of the coefficients $b$ and $σ$, our first result proves the convergence rate of the particle method with respect to the Wasserstein distance, which extends a previous work [BT97] established in one-dimensional setting. In the second part, we present and analyse two quantization-based schemes, including the recursive quantization scheme (deterministic scheme) in the Vlasov setting, and the hybrid particle-quantization scheme (random scheme, inspired by the $K$-means clustering). Two examples are simulated at the end of this paper: Burger's equation and the network of FitzHugh-Nagumo neurons in dimension 3. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_14853 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Particle method and quantization-based schemes for the simulation of the McKean-Vlasov equation Liu, Yating Numerical Analysis Probability In this paper, we study three numerical schemes for the McKean-Vlasov equation \[\begin{cases} \;dX_t=b(t, X_t, μ_t) \, dt+σ(t, X_t, μ_t) \, dB_t,\: \\ \;\forall\, t\in[0,T],\;μ_t \text{ is the probability distribution of }X_t, \end{cases}\] where $X_0$ is a known random variable. Under the assumption on the Lipschitz continuity of the coefficients $b$ and $σ$, our first result proves the convergence rate of the particle method with respect to the Wasserstein distance, which extends a previous work [BT97] established in one-dimensional setting. In the second part, we present and analyse two quantization-based schemes, including the recursive quantization scheme (deterministic scheme) in the Vlasov setting, and the hybrid particle-quantization scheme (random scheme, inspired by the $K$-means clustering). Two examples are simulated at the end of this paper: Burger's equation and the network of FitzHugh-Nagumo neurons in dimension 3. |
| title | Particle method and quantization-based schemes for the simulation of the McKean-Vlasov equation |
| topic | Numerical Analysis Probability |
| url | https://arxiv.org/abs/2212.14853 |