Differentiability of quadratic BSDE generated by continuous martingales and hedging in incomplete markets

In this paper we consider a class of BSDE 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 which from the perspective of a utility optimization interpretation of the underlying control problem on a financial market takes the following form. The control process of the BSDE steers the system into a random liability depending on a market external uncertainty and this way describes the optimal derivative hedge of the liability by investment in a capital market the dynamics of which is described by the forward component. This delta hedge is described in a key formula in terms of a derivative functional of the solution process and the correlation structure of the internal uncertainty captured by the forward process and the external uncertainty responsible for the market incompleteness. The formula largely extends the scope of validity of the results obtained by several authors in the Brownian setting, designed to give a genuinely stochastic representation of the optimal delta hedge in the context of cross hedging insurance derivatives generalizing the derivative hedge in the Black-Scholes model. Of course, Malliavin’s calculus needed in the Brownian setting is not available in the general local martingale framework. We replace it by new tools based on stochastic calculus techniques. AMS subject classifications: Primary 60H10; secondary 60J25.