The Feynman-Kac formula and decomposition of Brownian paths
Let \(B_t\) be a one-dimensional Wiener process, \(f:\mathbb{R}\mapsto\mathbb{R}^+\) a locally bounded Borel function, and consider the problem of computing the law of \(A^f_t:=\int_0^t ds f(B_s)\). Let \(g_t\) denote the last time the Brownian path hits level 0 before time \(t\), and denote by \(\theta_k\) an exponentially distributed variable, with rate \(k^2/2\), independent of \(B_t\). A decomposition of the Brownian path as independent processes before and after \(g_{\theta_k}\) allows to express \(\text{E}[\exp (-A^{f}_{\theta_k})]\) in terms of the inverse local time of \(B_t\) at zero and the first hitting time of level zero of a Brownian family. On the other hand, Sturm-Liouville equations \(\frac{1}{2}F''(x)=m(x)F(x)\) have probabilistic interpretations which can be used to derive an explicit expression for \(\text{E}[\exp (-A^{f}_{\theta_k})]\). Thirdly, consider the following version of the Feynman-Kac formula for one-dimensional Brownian motion: If \(q:\mathbb{R}\rightarrow\mathbb{R}^+\) is a Borel function and \(k>0\), then \[ \int_0^{\infty} dt e^{-(k^2/2)t} \text{E}[q(B_t)\exp(-A^f_t)]= \int_{-\infty}^{\infty} dx q(x)U(k,x), \] where \(U\) solves the Sturm--Liouville equation with \(m=k^2/2+f(x)\) and suitable boundary conditions. Using the value of \(\text{E}[\exp (-A^{f}_{\theta_k})]\) one can derive an expression for the function \(U\). The paper discusses these relations between Sturm-Liouville equations, Feynman-Kac formulae and decompositions of the Wiener process, and apply them to the explicit computation of the law of \(A^f_t\), for several classes of functions \(f\): constant, Heavyside, absolute value, linear, quadratic, potential, exponential and hyperbolic cotangent.
scientific article; zbMATH DE number 1022357