| Popis: |
In this article we study the relationship between solutions to Cauchy problems for the abstract stochastic differential equation $dX(t)=AX(t)dt + BdW(t)$ and solutions to Cauchy problems (backward and forward) for the infinite dimensional deterministic partial differential equation $$ \pm\frac{\partial g}{\partial t}(t,x) + \frac{\partial g}{\partial x}(t,x)Ax + \frac{1}{2}\hbox{Tr}[(BQ^{1/2})^* \frac{\partial^2 g}{\partial x^2}(t,x) (BQ^{1/2})] = 0, $$ where g is the probability characteristic $g=\mathbb{E}^{t,x}[h(X(T))]$ in the backward case and $g=\mathbb{E}^{0,x}[h(X(t))]$ in the forward case. This relationship, that is the inifinite dimensional Feynman-Kac theorem, is proved in both directions: from stochastic to deterministic and from deterministic to stochastic. Special attention is given to the definition and interpretation of objects in the equations. |
