On the Joint Distribution of First-passage Time and First-passage Area of Drifted Brownian Motion

For drifted Brownian motion X(t) = x-µt + Bt (µ > 0) starting from x > 0, we study the joint distribution of the first-passage time below zero ,t(x), and the first-passage area ,A(x), swept out by X till the time t(x). In particular, we establish differential equations with boundary conditions for the joint moments E[t(x)mA(x)n], and we present an algorithm to find recursively them, for any m and n. Finally, the expected value of the time average of X till the time t(x) is obtained.