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

Abstract

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.

On the first-passage area of a L$\acute{\text{e}}$vy process

“…This is a continuation of the articles [1] and [2]; actually, in [1] we studied the distribution of the first-passage area (FPA) A(x) = τ (x) 0 X(t)dt, swept out by a one-dimensional jumpdiffusion process X(t), starting from x > 0, till its first-passage time (FPT) τ (x) below zero, while in [2] we examined the special case (without jumps) when X(t) is Brownian motion (BM) B t with negative drift −µ, that is, X(t) = x − µt + B t , studying in particular the joint distribution of τ (x) and A(x). In the present paper, we aim to investigate the joint distributions of τ (x) and A(x), in the case when X(t) is a Lévy process of the form…”

“…) is a polynomial of degree 4, otherwise it has polynomial growth of degree 4. Similar polynomial expressions can be obtained for any m and n, implying that the moments E [τ (x) m A(x) n ] are finite, for all m and n. As far as the form of the solution V m,n (x) of (3.19) is concerned, proceeding by induction, as done in [2], one gets: Theorem 3.4 For integers m, n ≥ 0, the solution of (3.19) vanishes at zero, and it is a polynomial of degree m + 2n if x ∈ N, otherwise it has a polynomial growth of degree m + 2n.…”

“…The aim of this subsection is to find the quantity V m,n (x) := E [τ (x) m A(x) n ] , with m, n ∈ N, following the same arguments used in [2] in the corresponding case. Taking the m-th partial derivative with respect to λ 1 and the n-th partial derivative with respect to λ 2 in the first equation of (3.8) and evaluating at λ 1 = λ 2 = 0, we obtain the following: ∈ , defining we can write…”

“…When x is an integer, by proceeding as done in [2] for the analogous situation, it is possible to obtain a compact closed form of V m,n (x); in fact, by using Theorem 3.4 for x, m, n ∈ N, we obtain that there exist a…”

Asymptotic results for certain first-passage times and areas of renewal processes

“…In some literature the results on the distribution of first-passage areas (possibly in terms of the joint distribution with first-passage times) concern Markov processes, and in particular some Lévy processes; see e.g. the jump-diffusion processes in [1] and the drifted Brownian motion [2]. This approach allows to consider suitable differential-difference equations (in terms of the generator of the Markov process) for the Laplace transform of (τ (x), A(x)) which can be solved.…”