The differentiability of quadratic BSDEs driven by martingales and an application to utility maximization

In a continuous martingale setting we discuss exponential utility indifference prices and delta hedges for derivatives or liabilities written on non-tradable underlyings in incomplete financial markets. We use the fact that the optimal investment strategy can be described in terms of solutions of Backward Stochastic Differential Equations (BSDEs) with quadratic driver. After establishing some regularity properties for BSDEs, we show that the control process of the BSDE can be described in terms of a differential operator of the solution process and a correlation coefficient. This formula generalizes the results obtained by several authors in the Brownian setting, designed to represent the optimal delta hedge in the context of cross hedging insurance derivatives that generalizes the derivative hedge in the Black-Scholes model. It involves Malliavin's calculus which is not available in the general martingale setting. Consequently, we replace it by new tools based on stochastic calculus techniques.