Differentiability of quadratic BSDEs generated by continuous martingales

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. The representation formula of the main result describes this delta hedge in terms of the derivative of the BSDEs solution process on the one hand and the correlation structure of the internal uncertainty captured by the forward process and the external uncertainty responsible for the market incompleteness on the other hand. The formula extends the scope of validity of the results obtained by several authors in the Brownian setting. It is designed to extend a genuinely stochastic representation of the optimal replication in cross hedging insurance derivatives from the classical Black–Scholes model to incomplete markets on general stochastic bases. In this setting, Malliavin’s calculus which is required in the Brownian framework, is replaced by new tools based on techniques related to a calculus of quadratic covariations of basis martingales.