Martingales and Path-Dependent PDEs via Evolutionary Semigroups
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915369624535040 |
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| author | Denk, Robert Kunze, Markus Kupper, Michael |
| author_facet | Denk, Robert Kunze, Markus Kupper, Michael |
| contents | In this article, we develop a semigroup-theoretic framework for the analytic characterisation of martingales with path-dependent terminal conditions. Our main result establishes that a measurable adapted process of the form \[ V(t) - \int_0^tΨ(s)\, ds \] is a martingale with respect to an expectation operator $\mathbb{E}$ if and only if a time-shifted version of $V$ is a mild solution of a final value problem involving a path-dependent differential operator that is intrinsically connected to $\mathbb{E}$. We prove existence and uniqueness of strong and mild solutions for such final value problems with measurable terminal conditions using the concept of evolutionary semigroups. To characterise the compensator $Ψ$, we introduce the notion of $\mathbb{E}$-derivative of $V$, which in special cases coincides with Dupire's time derivative. We also compare our findings to path-dependent partial differential equations in terms of Dupire derivatives such as the path-dependent heat equation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_01845 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Martingales and Path-Dependent PDEs via Evolutionary Semigroups Denk, Robert Kunze, Markus Kupper, Michael Probability Analysis of PDEs Functional Analysis Primary: 60G48, 47D06, Secondary: 60J35, 35K10 In this article, we develop a semigroup-theoretic framework for the analytic characterisation of martingales with path-dependent terminal conditions. Our main result establishes that a measurable adapted process of the form \[ V(t) - \int_0^tΨ(s)\, ds \] is a martingale with respect to an expectation operator $\mathbb{E}$ if and only if a time-shifted version of $V$ is a mild solution of a final value problem involving a path-dependent differential operator that is intrinsically connected to $\mathbb{E}$. We prove existence and uniqueness of strong and mild solutions for such final value problems with measurable terminal conditions using the concept of evolutionary semigroups. To characterise the compensator $Ψ$, we introduce the notion of $\mathbb{E}$-derivative of $V$, which in special cases coincides with Dupire's time derivative. We also compare our findings to path-dependent partial differential equations in terms of Dupire derivatives such as the path-dependent heat equation. |
| title | Martingales and Path-Dependent PDEs via Evolutionary Semigroups |
| topic | Probability Analysis of PDEs Functional Analysis Primary: 60G48, 47D06, Secondary: 60J35, 35K10 |
| url | https://arxiv.org/abs/2507.01845 |