Weak martingale representation for continuous Markov processes and application to quadratic growth BSDEs

TL;DR

This paper establishes an exact martingale representation for functionals of continuous Markov martingales without regularity assumptions and applies it to solve quadratic growth BSDEs without orthogonal components.

Contribution

It proves a universal integral representation for Markov martingale functionals and extends this to solve quadratic growth BSDEs without orthogonal components.

Findings

Representation holds without regularity assumptions.

Quadratic growth BSDEs can be solved without orthogonal components.

Solutions exhibit differentiability and regularity properties.

Abstract

In this paper we prove that every random variable of the form $F(M_T)$ with $F:\real^d \to\real$ a Borelian map and $M$ a $d$-dimensional continuous Markov martingale with respect to a Markov filtration $\mathcal{F}$ admits an exact integral representation with respect to $M$, that is, without any orthogonal component. This representation holds true regardless any regularity assumption on $F$. We extend this result to Markovian quadratic growth BSDEs driven by $M$ and show they can be solved without an orthogonal component. To this end, we extend first existence results for such BSDEs under a general filtration and then obtain regularity properties such as differentiability for the solution process.