Path dependent Feynman–Kac formula for forward backward stochastic Volterra integral equations
Path dependent Feynman–Kac formula for forward backward stochastic Volterra integral equations
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DOI：
10.1214/21-aihp1158
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发表时间：
2020-04
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影响因子：
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通讯作者：
Hanxiao Wang;J. Yong;Jianfeng Zhang
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文献类型：
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作者：
Hanxiao Wang;J. Yong;Jianfeng Zhang
研究了正倒向随机沃尔泰拉积分方程（简称FBSVIEs）与时间非局部路径依赖偏微分方程（简称PPDE）之间的关系.由于沃尔泰拉型方程的性质，通常的流性质（或半群性质）不成立。受Viens-Zhang \cite{Viens-Zhang-2019}和Wang-Yong \cite{Wang-Yong-2019}的启发，引入辅助过程，使FBSVIEs自适应解的流动性质在适当的意义下得到恢复，从而使泛函Ito公式适用.在实现该阶段之后，找到自然的PPDE，使得后向SVIE的适应解允许根据前向SVIE的解经由PPDE的解的表示。另一方面，PPDE的解决方案承认的适应解决方案（路径依赖）FBSVIE，这是被称为费曼-卡茨公式的一个表示。这导致了一个经典的解决方案的PPDE的存在性和唯一性，光滑的FBSVIEs的系数条件下。进一步地，当光滑性条件放宽，FBSVIE的后向分量为一维时，引入了一个新的（合适的）PPDE粘性解的概念，建立了粘性解的比较原理，从而得到了粘性解的唯一性.最后，将已有的结果推广到耦合FBSVIE和II型BSVIE，并通过对线性FBSVIE的深入研究，得到了PPDE解的路径导数的表达式.
This paper is concerned with the relationship between forward-backward stochastic Volterra integral equations (FBSVIEs, for short) and a system of (non-local in time) path dependent partial differential equations (PPDEs, for short). Due to the nature of Volterra type equations, the usual flow property (or semigroup property) does not hold. Inspired by Viens-Zhang \cite{Viens-Zhang-2019} and Wang-Yong \cite{Wang-Yong-2019}, auxiliary processes are introduced so that the flow property of adapted solutions to the FBSVIEs is recovered in a suitable sense, and thus the functional Ito's formula is applicable. Having achieved this stage, a natural PPDE is found so that the adapted solution of the backward SVIEs admits a representation in terms of the solution to the forward SVIE via the solution to a PPDE. On the other hand, the solution of the PPDE admits a representation in terms of adapted solution to the (path dependent) FBSVIE, which is referred to as a Feynman-Kac formula. This leads to the existence and uniqueness of a classical solution to the PPDE, under smoothness conditions on the coefficients of the FBSVIEs. Further, when the smoothness conditions are relaxed with the backward component of FBSVIE being one-dimensional, a new (and suitable) notion of viscosity solution is introduced for the PPDE, for which a comparison principle of the viscosity solutions is established, leading to the uniqueness of the viscosity solution. Finally, some results have been extended to coupled FBSVIEs and type-II BSVIEs, and a representation formula for the path derivatives of PPDE solution is obtained by a closer investigation of linear FBSVIEs.