Differentiability of quadratic BSDEs generated by continuous martingales

TL;DR

This paper investigates the differentiability of quadratic BSDEs driven by continuous martingales, establishing a representation formula for the control component crucial for optimal hedging in financial markets.

Contribution

It proves the differentiability of FBSDEs with quadratic drivers driven by continuous martingales and derives a representation formula for the control component, advancing stochastic control methods.

Findings

Markov property of FBSDEs with strong Markov martingales established

Differentiability of FBSDEs with respect to initial conditions proven

Representation formula for the control component derived

Abstract

In this paper we consider a class of BSDEs with drivers of quadratic growth, on a stochastic basis generated by continuous local martingales. We first derive the Markov property of a forward--backward system (FBSDE) if the generating martingale is a strong Markov process. Then we establish the differentiability of a FBSDE with respect to the initial value of its forward component. This enables us to obtain the main result of this article, namely a representation formula for the control component of its solution. The latter is relevant in the context of securitization of random liabilities arising from exogenous risk, which are optimally hedged by investment in a given financial market with respect to exponential preferences. In a purely stochastic formulation, the control process of the backward component of the FBSDE steers the system into the random liability and describes its optimal derivative hedge by investment in the capital market, the dynamics of which is given by the forward component.