SDEs Driven by SDE Solutions

TL;DR

This paper investigates SDEs driven by solutions of other SDEs, revealing their non-Markovian nature but establishing Markov properties for combined processes, with explicit generator calculations and simulation insights.

Contribution

It introduces the concept of iterated SDEs driven by Levy processes, analyzing their properties and providing explicit generator and semimartingale characteristics.

Findings

Solutions are non-Markovian but combined processes are Markov and Feller.

Explicit formulas for generators and semimartingale characteristics.

Simulation study illustrating theoretical results.

Abstract

We consider stochastic differential equations (SDEs) driven by Feller processes which are themselves solutions of multivariate Levy driven SDEs. The solutions of these 'iterated SDEs' are shown to be non-Markovian. However, the process consisting of the driving process and the solution is Markov and even Feller in the case of bounded coefficients. The generator as well as the semimartingale characteristics of this process are calculated explicitly and fine properties of the solution are derived via the stochastic symbol. A short simulation study and an outlook in the direction of stochastic modeling round out the paper.