A Closed-Form Transition Density Expansion for Elliptic and Hypo-Elliptic SDEs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908541856514048 |
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| author | Iguchi, Yuga Beskos, Alexandros |
| author_facet | Iguchi, Yuga Beskos, Alexandros |
| contents | We introduce a closed-form expansion for the transition density of elliptic and hypo-elliptic multivariate Stochastic Differential Equations (SDEs), over a period $Δ\in (0,1)$, in terms of powers of $Δ^{j/2}$, $j\ge 0$. Our methodology provides approximations of the transition density, easily evaluated via any software that performs symbolic calculations. A major part of the paper is devoted to an analytical control of the remainder in our expansion for fixed $Δ\in(0,1)$. The obtained error bounds validate theoretically the methodology, by characterising the size of the distance from the true value. It is the first time that such a closed-form expansion becomes available for the important class of hypo-elliptic SDEs, to the best of our knowledge. For elliptic SDEs, closed-form expansions are available, with some works identifying the size of the error for fixed $Δ$, as per our contribution. Our methodology allows for a uniform treatment of elliptic and hypo-elliptic SDEs, when earlier works are intrinsically restricted to an elliptic setting. We show numerical applications highlighting the effectiveness of our method, by carrying out parameter inference for hypo-elliptic SDEs that do not satisfy stated conditions. The latter are sufficient for controlling the remainder terms, but the closed-form expansion itself is applicable in general settings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_07047 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Closed-Form Transition Density Expansion for Elliptic and Hypo-Elliptic SDEs Iguchi, Yuga Beskos, Alexandros Numerical Analysis Statistics Theory We introduce a closed-form expansion for the transition density of elliptic and hypo-elliptic multivariate Stochastic Differential Equations (SDEs), over a period $Δ\in (0,1)$, in terms of powers of $Δ^{j/2}$, $j\ge 0$. Our methodology provides approximations of the transition density, easily evaluated via any software that performs symbolic calculations. A major part of the paper is devoted to an analytical control of the remainder in our expansion for fixed $Δ\in(0,1)$. The obtained error bounds validate theoretically the methodology, by characterising the size of the distance from the true value. It is the first time that such a closed-form expansion becomes available for the important class of hypo-elliptic SDEs, to the best of our knowledge. For elliptic SDEs, closed-form expansions are available, with some works identifying the size of the error for fixed $Δ$, as per our contribution. Our methodology allows for a uniform treatment of elliptic and hypo-elliptic SDEs, when earlier works are intrinsically restricted to an elliptic setting. We show numerical applications highlighting the effectiveness of our method, by carrying out parameter inference for hypo-elliptic SDEs that do not satisfy stated conditions. The latter are sufficient for controlling the remainder terms, but the closed-form expansion itself is applicable in general settings. |
| title | A Closed-Form Transition Density Expansion for Elliptic and Hypo-Elliptic SDEs |
| topic | Numerical Analysis Statistics Theory |
| url | https://arxiv.org/abs/2502.07047 |