On the Finite Dimensional Joint Characteristic Function of L\'{e}vy's Stochastic Area Processes

TL;DR

This paper derives a formula for the finite dimensional joint characteristic function of a coupled process involving Brownian motion and Lévy's stochastic area process, using Riccati equations and matrix ODEs, with detailed analysis in 2D.

Contribution

It provides a new explicit formula for the joint characteristic function of Lévy's stochastic area process coupled with Brownian motion, accommodating a general matrix A.

Findings

Derived a recursive system of symmetric matrix Riccati equations.

Reduced the problem to solving independent first order linear matrix ODEs.

Detailed analysis of the 2D Lévy's stochastic area process.

Abstract

The goal of this paper is to derive a formula for the finite dimensional joint characteristic function (the Fourier transform of the finite dimensional distribution) of the coupled process ${(W_{t},L_{t}^{A}):t\in \lbrack 0,\infty)}$, where $\{W_{t}:t\in \lbrack 0,\infty)}$ is a $d$-dimensional Brownian motion and $\{L_{t}^{A}:t\in \lbrack 0,\infty)}$ is the generalized $d$-dimensional L$\acute{e}$vy's stochastic area process associated to a $d\times d$ matrix $A.$ Here $A$ need not be skew-symmetric, and in our computation we allow $A$ to vary. The problem finally reduces to the solution of a recursive system of symmetric matrix Riccati equations and a system of independent first order linear matrix ODEs. As an example, the two dimensional L\'{e}vy's stochastic area process is studied in detail.