Extremal behavior of hitting a cone by correlated Brownian motion with drift

Abstract This paper derives an exact asymptotic expression for P x u { ∃ t ≥ 0 X ( t ) − μ t ∈ U } , as u → ∞ , where X ( t ) = ( X 1 ( t ) , … , X d ( t ) ) ⊤ , t ≥ 0 is a correlated d -dimensional Brownian motion starting at the point x u = − α u with α ∈ R d , μ ∈ R d and U = ∏ i = 1 d [ 0 , ∞ ) . The derived asymptotics depends on the solution of an underlying multidimensional quadratic optimization problem with constraints, which leads in some cases to dimension-reduction of the considered problem. Complementary, we study asymptotic distribution of the conditional first passage time to U , which depends on the dimension-reduction phenomena.

[1]  J. Hüsler,et al.  On asymptotics of multivariate integrals with applications to records , 2002 .

[2]  Michel Mandjes,et al.  Extremes of multidimensional Gaussian processes , 2010, 1006.0029.

[3]  Krzysztof Debicki,et al.  Parisian ruin of self-similar Gaussian risk processes , 2014, Journal of Applied Probability.

[4]  A. B. Dieker,et al.  Exact simulation of Brown-Resnick random fields at a finite number of locations , 2014, 1406.5624.

[5]  K. Joag-dev,et al.  Negative Association of Random Variables with Applications , 1983 .

[6]  Excursion probabilities of isotropic and locally isotropic Gaussian random fields on manifolds , 2015, 1504.08047.

[7]  B. Logan,et al.  DIFFUSION, CELL MOBILITY, AND BANDLIMITED FUNCTIONS* , 1984 .

[8]  J. Azaïs,et al.  Level Sets and Extrema of Random Processes and Fields , 2009 .

[9]  Enkelejd Hashorva,et al.  Asymptotics and Bounds for Multivariate Gaussian Tails , 2005 .

[10]  K. Dȩbicki,et al.  Extremes of $\gamma$-reflected Gaussian process with stationary increments , 2015, 1511.09234.

[11]  R. Adler,et al.  Random Fields and Geometry , 2007 .

[12]  Gaussian risk models with financial constraints , 2013, 1309.7621.

[13]  J. Pickands.  Upcrossing probabilities for stationary Gaussian processes , 1969 .

[14]  A limit theorem for the time of ruin in a Gaussian ruin problem , 2008 .

[15]  The exact asymptotic of the collision time tail distribution for independent Brownian particles with different drifts , 2007, 0704.0215.

[16]  Charles Knessl,et al.  Probability of Brownian Motion Hitting an Obstacle , 1999, SIAM J. Appl. Math..

[17]  Enkelejd Hashorva,et al.  Extremes of vector-valued Gaussian processes: exact asymptotics , 2015, 1505.06461.

[18]  Parisian ruin of self-similar Gaussian risk processes , 2015, Journal of Applied Probability.

[19]  Adam J. Harper.  Bounds on the suprema of Gaussian processes, and omega results for the sum of a random multiplicative function , 2010, 1012.0210.

[20]  James Pickands,et al.  Maxima of stationary Gaussian processes , 1967 .

[21]  T. Senior,et al.  Electromagnetic and Acoustic Scattering by Simple Shapes , 1969 .

[22]  Vladimir I. Piterbarg,et al.  Asymptotic Methods in the Theory of Gaussian Processes and Fields , 1995 .

[23]  Tomasz Rolski,et al.  Simulation of the Asymptotic Constant in Some Fluid Models , 2003 .

[24]  Gennady Samorodnitsky,et al.  Probability Tails of Gaussian Extrema , 1991 .

[25]  M. Lifshits.  Lectures on Gaussian Processes , 2012 .

[26]  K. Raschel,et al.  On the exit time from a cone for brownian motion with drift , 2013, 1311.1459.

[27]  Sebastian Engelke,et al.  Generalized Pickands constants and stationary max-stable processes , 2016, 1602.01613.