论文标题
依赖路径的Feynman-kac公式用于前向后的随机Volterra积分方程
Path Dependent Feynman-Kac Formula for Forward Backward Stochastic Volterra Integral Equations
论文作者
论文摘要
本文涉及前向后的随机Volterra积分方程（fbsvies，简称）与（时间非本地）路径依赖性偏微分方程（ppdes，简称为PPD）之间的关系。由于Volterra类型方程的性质，通常的流量属性（或Semigroup属性）不具有。受到Viens-Zhang \ Cite {Viens-Zhang-2019}和Wang-Yong \ Cite {Wang-Yong-2019}的启发，引入了Auxilariary流程，以便适用于FBSVIES的Flow flow属性在适当的意义上恢复了FBSVIES，从而在合适的意义上恢复了功能，因此它的功能是可起作用的。在实现了这个阶段后，找到了自然的PPDE，以便向后的SVIES的改编解决方案通过对PPDE的溶液来允许向前SVIE的解决方案表示表示。另一方面，PPDE的解决方案在适应于（路径依赖）FBSVIE的解决方案方面接受了表示，该解决方案被称为Feynman-kac公式。这导致在FBSVIES系数上的平滑度条件下，经典解针对PPDE的存在和独特性。此外，当FBSVie的向后分量是一维的，为PPDE引入了新的（且合适的粘度解决方案概念）时，为PPDE引入了粘度解决方案的比较原理，从而导致粘度解决方案的唯一性。最后，一些结果已扩展到耦合的FBSVIE和II型BSVIE，并且通过对线性FBSVIE的仔细研究获得了PPDE溶液路径衍生物的表示公式。
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 Itô'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.
