Relations between Stochastic and Partial Differential Equations in Hilbert Spaces

The aim of the paper is to introduce a generalization of the Feynman-Kac theorem in Hilbert spaces. Connection between solutions to the abstract stochastic differential equation and solutions to the deterministic partial differential (with derivatives in Hilbert spaces) equation for the probability characteristic is proved. Interpretation of objects in the equations is given. 1. Introduction The Feynman-Kac theorem in the numerical (vector) case relates solutions of the Cauchy problem for stochastic equations with a Brownian motion , : with solutions of the Cauchy problem for deterministic partial differential equations: for the probability characteristic with an arbitrary Borel function . Here means the mathematical expectation of a solution to (1.1) with initial value , . Study of the relationship between the problems (1.1) and (1.2) was initially caused by the needs of physics. For example, the process describes the random motion of particles in a liquid or gas, and is a probability characteristic such as temperature, determined by the Kolmogorov equation. In the last years, the importance of the relationship between stochastic (1.1) and deterministic (1.2) problems has become more acute with the development of numerical methods and applications in financial mathematics. Here , for example, stock price at time , then is the value of stock options, determined by the famous Black-Scholes equation [1, 2]. Moreover, there exist recent applications of infinite-dimensional stochastic equations in financial mathematics [3]. For example, consider function that is the price at time of the coupon bond with maturity date . Let be parametrized as for all and , be the forward curve; that is, . Then the Musiela reparametrization , , in the special case of zero HJM shift, satisfies the following equation in a Hilbert space of functions acting from to : where is the generator of the right-shifts semigroup in , is a -valued Wiener (in particular -Wiener) process, and is a random mapping from Hilbert space to . Here the value of bond options may be calculated, at least numerically, via defined for . Thereby it requires an infinite-dimensional analogue of the connection between problems (1.1) and (1.2), which the present paper is devoted to. Generalization of the Feynman-Kac theorem to the infinite-dimensional case raises many questions related with the very formulation of the problem in infinite-dimensional spaces, the definition of relevant objects and a rigorous rationale for the relationship between mentioned problems. The paper considers the stochastic Cauchy