An infinite-dimensional approach to path-dependent Kolmogorov equations

Abstract

In this paper, a Banach space framework is introduced in order to deal with finite-dimensional path-dependent stochastic differential equations. A version of Kolmogorov backward equation is formulated and solved both in the space of $L^p$ paths and in the space of continuous paths using the associated stochastic differential equation, thus establishing a relation between path-dependent SDEs and PDEs in analogy with the classical case. Finally, it is shown how to establish a connection between such Kolmogorov equation and the analogue finite-dimensional equation that can be formulated in terms of the path-dependent derivatives recently introduced by Dupire, Cont and Fourni\'{e}.Comment: Published at http://dx.doi.org/10.1214/15-AOP1031 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org