Diffusions, exit time moments and Weierstrass theorems
Victor de la Peña, Patrick McDonald · Proceedings of the American Mathematical Society · 2004
Let X t X_t be a one-dimensional diffusion with infinitesimal generator given by the operator L = 1 2 ( a ( x ) d d x ) 2 + b ( x ) d d x L = \frac 12 (a(x) \frac {d}{dx})^2 + b(x) \frac {d}{dx} where a ( x ) a(x) is a smooth, positive real-valued function and the ratio of a ( x ) a(x) and b ( x ) b(x) is a constant. Given a compact interval, we prove a Weierstrass-type theorem for the exit time moments of X t X_t and their corresponding (naturally weighted) first derivatives, and we provide an algorithm that produces uniform approximations of arbitrary continuous functions by exit time moments. We investigate analogues of these results in higher-dimensional Euclidean spaces. We give expansions for several families of special functions in terms of exit time moments.