We prove the existence of the unique solution of a general Backward Stochastic Differential Equation with quadratic growth driven by martingales. Some kind of comparison theorem is also proved.

We consider a problem of minimization of a hedging error, measured by a positive convex random function, in an incomplete financial market model, where the dynamics of asset prices is given by an Rd-valued continuous semimartingale. Under some regularity assumptions we derive a backward stochastic PDE for the value function of the problem and show that the strategy is optimal if and only if the corresponding wealth process satisfies a certain forward-SDE. As an example the case of mean-variance hedging is considered.

The paper studies the robust maximization of utility from terminal wealth in a diffusion financial market model. The underlying model consists of a tradable risky asset whose price is described by a diffusion process with misspecified trend and volatility coefficients, and a non-tradable asset with a known parameter. The robust functional is defined in terms of a utility function. An explicit characterization of the solution is given via the solution of the Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation.