For a one-dimensional jump-diffusion process X(t), starting from x > 0, it is studied the probability distribution of the area A(x) swept out by X(t) till its first-passage time below zero. In particular, it is shown that the Laplace transform and the moments of A(x) are solutions to certain partial differential-difference equations with outer conditions. The distribution of the maximum displacement of X(t) is also studied. Finally, some explicit examples are reported, regarding diffusions with and without ju

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For drifted Brownian motion X(t) = x − µt + B t (µ > 0) starting from x > 0, we study the joint distribution of the first-passage time below zero, τ (x), and the first-passage area, A(x), swept out by X till the time τ (x). In particular, we establish differential equations with boundary conditions for the joint moments E[τ (x) m A(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 τ (x) is obtained.

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By using the law of the excursions of Brownian motion with drift, we find the distribution of the n−th passage time of Brownian motion through a straight line S(t) = a+bt. In the special case when b = 0, we extend the result to a space-time transformation of Brownian motion.

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