The Explicit Chaotic Representation of the powers of increments of Levy Processes

TL;DR

This paper derives explicit chaos representations for powers of Levy process increments, providing computational formulas and verifying their effectiveness through simulations, with applications in financial derivative modeling.

Contribution

It introduces explicit formulas for chaos expansions of Levy process increments and their powers, enhancing computational methods in stochastic analysis.

Findings

Explicit chaos expansion formulas for Levy increments

Verification of formulas through simulation results

Application to financial derivatives modeling

Abstract

An explicit formula for the chaotic representation of the powers of increments, (X_{t+t_0}-X_{t_0})^n, of a Levy process is presented. There are two different chaos expansions of a square integrable functional of a Levy process: one with respect to the compensated Poisson random measure and the other with respect to the orthogonal compensated powers of the jumps of the Levy process. Computationally explicit formulae for both of these chaos expansions of (X_{t+t_0}-X_{t_0})^n are given in this paper. Simulation results verify that the representation is satisfactory. The CRP of a number of financial derivatives can be found by expressing them in terms of (X_{t+t_0}-X_{t_0})^n using Taylor's expansion.