On Characterizing Integral Stopping Time Functionals on Diffusions as Solutions to Boundary Value Problems

Let T be the first exit time of a diffusion x(t) from a bounded domain Ohm subset of IRn. This paper demonstrates that certain integral functionals phi --> E[integral(0)(tau) phi(t)dt \x(0) = x], phi : [0, infinity) --> IR, may be characterized as solutions to elliptic boundary value problems. The result is established using probabilistic arguments together with results from the theory of partial differential equations. One particular functional, a stochastic analogue of the Fourier transform, is analyzed carefully. Its basic computational properties, including an inversion formula, are developed.