TY - JOUR

T1 - Compactness and continuity properties for a lévy process at a two-sided exit time

AU - Malle, Ross A.

AU - Mason, David M.

N1 - Publisher Copyright:
© 2020, Institute of Mathematical Statistics. All rights reserved.

PY - 2020

Y1 - 2020

N2 - We consider a Lévy process X = (X(t))t≥0 in a generalised Feller class at 0, and study the exit position, |X(T (r))|, as X leaves, and the position, |X(T (r)−)|, just prior to its leaving, at time T (r), a two-sided region with boundaries at ±r, r > 0. Conditions are known for X to be in the Feller class F C0 at zero, by which we mean that each sequence tk ↓ 0 contains a subsequence through which X(tk), after norming by a nonstochastic function, converges to an a.s. finite nondegenerate random variable. We use these conditions on X to characterise similar properties for the normed positions |X(T (r))| /r and |X(T (r)−)| /r, and also for the normed jump |∆X(T (r))/r| = |X(T (r)) − X(T (r)−)| /r (“the jump causing ruin"), as convergence takes place through sequences rk ↓ 0. We go on to give conditions for the continuity of distributions of the limiting random variables obtained in this way.

AB - We consider a Lévy process X = (X(t))t≥0 in a generalised Feller class at 0, and study the exit position, |X(T (r))|, as X leaves, and the position, |X(T (r)−)|, just prior to its leaving, at time T (r), a two-sided region with boundaries at ±r, r > 0. Conditions are known for X to be in the Feller class F C0 at zero, by which we mean that each sequence tk ↓ 0 contains a subsequence through which X(tk), after norming by a nonstochastic function, converges to an a.s. finite nondegenerate random variable. We use these conditions on X to characterise similar properties for the normed positions |X(T (r))| /r and |X(T (r)−)| /r, and also for the normed jump |∆X(T (r))/r| = |X(T (r)) − X(T (r)−)| /r (“the jump causing ruin"), as convergence takes place through sequences rk ↓ 0. We go on to give conditions for the continuity of distributions of the limiting random variables obtained in this way.

KW - Domain of partial attraction

KW - Generalised Feller class

KW - Lévy process

KW - Passage time distributions

KW - Stochastic compactness

KW - Two-sided exit problem

UR - http://www.scopus.com/inward/record.url?scp=85084703598&partnerID=8YFLogxK

U2 - 10.1214/20-EJP451

DO - 10.1214/20-EJP451

M3 - Article

SN - 1083-6489

VL - 25

JO - Electronic Journal of Probability

JF - Electronic Journal of Probability

M1 - 51

ER -