First-Passage Distributions of Bidimensional Processes

Abstract

Bidimensional processes defined by dx(t) = ρ(x, y) dt and dy(t) = f(y) dt + σ(y) dW(t), where W(t) is a Wiener process, are considered. Let T(x, y, ξ) = inf[t ≥ 0: x(t] = ξ| x(0) = x, y(0) = y). Explicit expressions for the moment generating function of T(x, y, 0) and for the characteristic function of y(T(x, y, ξ)) are obtained in two special cases. The method of similarity solutions is used. Applications to optimal control problems are presented.