Some variations on Kac's formula for Brownian motion
Let f be a bounded, uniformly continuous function mapping \({\mathbb{R}}^ 4\to {\mathbb{R}}\); let \(\alpha \in C^ 2({\mathbb{R}})\), \(\beta \in C^ 2({\mathbb{R}}^ 3)\), and denote by \(X_ t\) the Brownian motion on \({\mathbb{R}}\). In this short note a generalization of the Feynman-Kac-formula of the form \[ E_ xf\{a+\int^{t}_{0}\beta [b+\int^{s}_{0}\alpha (X_ r)dr,\quad c+s,\quad X_ s]ds,\quad b+\int^{t}_{0}\alpha (X_ s)ds,\quad c+t,\quad X_ t\} \] is considered.
generalization of the Feynman-Kac-formula
scientific article; zbMATH DE number 4111738