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

Document Type : Research Paper

Authors

1 Department of Mathematical Sciences‎, ‎Shahid Beheshti‎ ‎University‎, ‎G.C‎. ‎Evin‎, ‎1983963113‎, ‎Tehran‎, ‎Iran.

2 Department of Mathematics and Statistics‎, ‎University of New‎ ‎Brunswick‎, ‎Fredericton‎, ‎N.B‎. ‎Canada E3B 5A3.

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 $1<p\leq2$)‎, ‎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.

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### History

• Receive Date: 26 November 2015
• Revise Date: 27 October 2016
• Accept Date: 28 October 2016
• First Publish Date: 30 November 2017