Feynman--Kac Representation for Parabolic Equations with Memory: A New Class of BSDEs
Abstract
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 Yt,x(s)≡ Ys,Xt,x(s)(s) but also on the future path r↦ Yr,Xt,x(s)(r), r∈[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.
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BibTeX
@misc{wang2026feynmankac,
title = {{Feynman--Kac Representation for Parabolic Equations with Memory: A New Class of BSDEs}},
author = {Hanxiao Wang and Jiongmin Yong and Qi Zhang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01591},
eprint = {2611.01591},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01591},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Feynman--Kac Representation for Parabolic Equations with Memory: A New Class of BSDEs
AU - Hanxiao Wang
AU - Jiongmin Yong
AU - Qi Zhang
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01591
UR - https://arxiv.org/abs/2611.01591
AB - 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)≡ Y^s,X^t,x(s)(s) but also on the future path r↦ Y^r,X^t,x(s)(r), r∈[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.
ER -