The escape probability for integrated Brownian motion with non-zero drift

Abstract

We consider the two-dimensional process {X(t), V(t)} where {V(t)} is Brownian motion with drift, and {X(t)} is its integral. In this note we derive the joint density function of T and V(T) where T is the time at which the process {X(t)} first returns to its initial value. A series expansion of the marginal density of T is given in the zero-drift case. When V(0) and the drift are both positive there is a positive probability that {Χ (t)} never returns to its initial value. We show how this probability grows for small drift. Finally, using the Kontorovich–Lebedev transform pair we obtain the escape probability explicitly for arbitrary values of the drift parameter.