Feynman-Kac Formulas

Consider the SDE \(\displaystyle dX_{t}=b(t,X_{t})dt+{\sigma }(t,X_{t})dW_{t} \) where W is a d-dimensional Brownian motion and \(\displaystyle \begin{aligned} b=b(t,x):\,]0,T[\times {\mathbb {R}}^{N}\longrightarrow {\mathbb {R}}^{N},\qquad {\sigma }={\sigma }(t,x):\,]0,T[\times {\mathbb {R}}^{N}\longrightarrow {\mathbb {R}}^{N\times d}. \end{aligned} \)