The Sticky L\'evy Process as a solution to a Time Change Equation

TL;DR

This paper introduces a novel approach to modeling sticky Lévy processes using Time Change Equations, demonstrating advantages over traditional SDEs, including the existence of adapted solutions and strong approximation schemes.

Contribution

It presents the first representation of sticky Lévy processes as solutions to TCEs driven by Lévy processes, overcoming limitations of SDEs like non-adaptability.

Findings

Sticky Lévy processes can be uniquely represented as solutions to TCEs.

Strong approximation schemes for TCE solutions are feasible and effective.

TCEs offer advantages over SDEs in modeling certain stochastic processes.

Abstract

Stochastic Differential Equations (SDEs) were originally devised by It\^o to provide a pathwise construction of diffusion processes. A less explored approach to represent them is through Time Change Equations (TCEs) as put forth by Doeblin. TCEs are a generalization of Ordinary Differential Equations driven by random functions. We present a simple example where TCEs have some advantage over SDEs. We represent sticky L\'evy processes as the unique solution to a TCE driven by a L\'evy process with no negative jumps. The solution is adapted to the time-changed filtration of the L\'evy process driving the equation. This is in contrast to the SDE describing sticky Brownian motion, which is known to have no adapted solutions as first proved by Chitashvili. A known consequence of such non-adaptability for SDEs is that certain natural approximations to the solution of the corresponding SDE do not converge in probability, even though they do converge weakly. Instead, we provide strong approximation schemes for the solution of our TCE (by adapting Euler's method for ODEs), whenever the driving L\'evy process is strongly approximated.