# A Weak Approximation for the Extrema's Distributions of L\'evy Processes

## TL;DR

This paper develops an $L^{p^*}$ approximation method for the distributions of extrema of Lévy processes using Wiener-Hopf factorization, with applications to ruin probabilities and estimation bounds.

## Contribution

It introduces a novel approximation approach for extrema distributions of Lévy processes employing Wiener-Hopf factorization, including bounds and practical procedures.

## Key findings

- Provides $L^{p^*}$ approximation for extrema distributions
- Establishes estimation bounds for the approximation method
- Demonstrates applications to ruin probability calculations

## Abstract

Suppose $X_{t}$ 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 \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 $L^{p^*}({\Bbb R})$ (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.