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Feynman--Kac Representation for Parabolic Equations with Memory: A New Class of BSDEs — arXiv:2611.01591

Source: arXiv (en)
Original: https://arxiv.org/abs/2611.01591
License (as archived): CC0-1.0
Archived: 2026-11-03T18:47:18.718Z
SHA-256 of the archived text: 86645b51efc8da2334b79039fad379ba9970b319fe6c71f063203278cc42628f
Stable URL of this page: https://metraipolis.com/documento/dc_625aa32019cb

How to cite

Feynman--Kac Representation for Parabolic Equations with Memory: A New Class of BSDEs — arXiv:2611.01591 — arXiv (en), https://arxiv.org/abs/2611.01591 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_625aa32019cb (Metraipolis archive dc_625aa32019cb, sha256 86645b51efc8da2334b79039fad379ba9970b319fe6c71f063203278cc42628f)

BibTeX

@misc{dc625aa32019cb,
  title = {Feynman--Kac Representation for Parabolic Equations with Memory: A New Class of BSDEs — arXiv:2611.01591},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2611.01591},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_625aa32019cb; sha256 86645b51efc8da2334b79039fad379ba9970b319fe6c71f063203278cc42628f},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2611.01591. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

This paper is concerned with the connection between parabolic equations with memory (MPEs, for short) and backward stochastic differential equations (BSDEs, for short). To provide a probabilistic representation of MPEs, we introduce a new class of BSDEs, called BSDEs with memory (MBSDEs, for short). The main feature of MBSDEs is that, at the current time $s$, the generator depends not only on the current value $Y^{t,x}(s)\equiv Y^{s,X^{t,x}(s)}(s)$ but also on the future path $r\mapsto Y^{r,X^{t,x}(s)}(r)$, $r\in[s,T]$. This structure reveals a fundamental difference from classical BSDEs: in the classical setting, the time and space variables are coupled and inseparable, whereas in MBSDEs they are completely independent. We prove the well-posedness of these MBSDEs and show that their adapted solutions provide a probabilistic representation for the solutions of MPEs. Furthermore, under a monotonicity condition, we establish a comparison principle for one-dimensional MBSDEs, which enables us to develop a viscosity solution theory for MPEs. This generalizes the celebrated nonlinear Feynman--Kac formula of Peng [25] and Pardoux--Peng [22] to the case with memory, in both the classical and viscosity solution senses.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.PR, nuovi lavori del giorno).
Categorie arXiv: Probability (math.PR).
Pagina del lavoro: https://arxiv.org/abs/2611.01591 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-03T18:15:07.710Z dalla pagina https://arxiv.org/list/math.PR/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).