A Weak Approximation for the Extrema's Distributions of Lévy Processes

Abstract

Suppose $X_t$ is a one-dimensional and real-valued Lévy process started from $X_0=0$, which ({\bf 1}) its nonnegative jumps measure $\nu$ satisfying ${\int }_{\mathbb{R}}\min \{1,x^2\}\nu (dx)<\infty$ and ({\bf 2}) its stopping time $\tau (q)$ is \emph{either} a geometric \emph{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 $L^{p^{\ast }}(\mathbb{R})$ (where $1/p^{\ast }+1/p=1$ and $1<p\le 2$), 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.

in Bulletin of the Iranian Mathematical Society 2017

References in corpus (2)

• Wiener-Hopf factorization and distribution of extrema for a family of Lévy processes
• On the Distribution of Extrema for a Class of Lévy Processes