First-passage times of two-dimensional Brownian motion

Abstract

First-passage times (FPTs) of two-dimensional Brownian motion have many applications in quantitative finance. However, despite various attempts since the 1960s, there are few analytical solutions available. By solving a nonhomogeneous modified Helmholtz equation in an infinite wedge, we find analytical solutions for the Laplace transforms of FPTs; these Laplace transforms can be inverted numerically. The FPT problems lead to a class of bivariate exponential distributions which are absolute continuous but do not have the memoryless property. We also prove that the density of the absolute difference of FPTs tends to ∞ if and only if the correlation between the two Brownian motions is positive.

Logarithmic Asymptotics for Probability of Component-Wise Ruin in a Two-Dimensional Brownian Model

“…Three different types of expansion are available for g(t * + t, s * + s), according to (ii.1) (t * + t, s * + s) ∈ A. Consider the two representations of g 3 (t, s) given in (12) and (13). Using the representation (12) we can show that…”

“…Due to its importance in, e.g., quantitative finance or ruin theory, the component-wise maxima have been studied extensively; see, e.g., [13,10,17,23,22]. In particular, some formulas for the joint distribution of (Q 1 (T ), Q 2 (T )) are known.…”

“…Interestingly, in [13] it was worked out a formula for joint survival function of (Q 1 (E p ), Q 2 (E p )), where E p is an independent exponential random variable with parameter p > 0. Vector (Q 1 (E p ), Q 2 (E p )) as well as (Q 1 , Q 2 ) have bivariate exponential distribution (BVE) in the sense of the terminology of Kou and Zhong [13], that is: (i) it has exponential marginals and (ii) it is absolute continuous with respect to two-dimensional Lebesgue measure. The later property for (Q 1 , Q 2 ) follows from Theorem 7.1 in [2] combined with the fact that P {Q j = 0} = 0, see also related Lemma 4.4 in [6].…”

Exact asymptotics of component-wise extrema of two-dimensional Brownian motion

“…have been studied extensively; see, e.g., [4,12,15,19,23,24]. In particular, some formulas for the joint distribution of (Q 1 (T ), Q 2 (T )) are known.…”

“…Interestingly, in [15] it was worked out a formula for joint survival function of (Q 1 (E p ), Q 2 (E p )), where E p is an independent exponential random variable with parameter p > 0. Vector (Q 1 (E p ), Q 2 (E p )) as well as (Q 1 , Q 2 ) have bivariate exponential distribution (BVE) in the sense of the terminology of Kou and Zhong [15], that is: (i) it has exponential marginals and (ii) it is absolute continuous with respect to two-dimensional Lebesgue measure. The later property for (Q 1 , Q 2 ) follows from Theorem 7.1 in [2] combined with the fact that P {Q j = 0} = 0, see also related Lemma 4.4 in [7].…”