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Hitting distributions of geometric Brownian motion

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Byczkowski T., Ryznar M.

Studia Mathematica

2006

tom 173

nr 1

19-38

Let τ be the first hitting time of the point 1 by the geometric Brownian motion X(t) = x exp(B(t) - 2μt) with drift μ ≥ 0 starting from x > 1. Here B(t) is the Brownian motion starting from 0 with EB²(t) = 2t. We provide an integral formula for the density function of the stopped exponential functional $A(τ) = ∫_0^τ X²(t)dt$ and determine its asymptotic behaviour at infinity. Although we basically rely on methods developed in [BGS], the present paper covers the case of arbitrary drifts μ ≥ 0 and provides a significant unification and extension of the results of the above-mentioned paper. As a corollary we provide an integral formula and give the asymptotic behaviour at infinity of the Poisson kernel for half-spaces for Brownian motion with drift in real hyperbolic spaces of arbitrary dimension.

Ograniczanie wyników

1 Studia Mathematica

1 Byczkowski T.

1 Ryznar M.

1 2006

Wyniki wyszukiwania

Hitting distributions of geometric Brownian motion