A weak approximation for the Extrema's distributions of Levy processes

Abstract

‎Suppose that $X_{t}$ is a one-dimensional and real-valued L'evy‎ ‎process started from $X_0=0$‎, ‎which ({bf 1}) its nonnegative‎ ‎jumps measure $nu$ satisfying $int_{Bbb‎ ‎R}min{1,x^2}nu(dx)<infty$ and ({bf 2}) its stopping time‎ ‎$tau(q)$ is either a geometric or an exponential‎ ‎distribution with parameter $q$ independent of $X_t$ and‎ ‎$tau(0)=infty.$ This article employs the Wiener-Hopf‎ ‎Factorization (WHF) to find‎, ‎an $L^{p^*}({Bbb R})$ (where‎ ‎$1/{p^*}+1/p=1$ and $1<pleq2$)‎, ‎approximation for the extrema's‎ ‎distributions of $X_{t}.$ Approximating the finite (infinite)-time‎ ‎ruin probability as a direct application of our findings has been‎ ‎given‎. ‎Estimation bounds‎, ‎for such approximation method‎, ‎along‎ ‎with two approximation procedures and‎ ‎several examples are explored.