Quadratic and superquadratic backward stochastic differential equations and applications

We begin by the study of the problem of the exponential utility maximization. As opposed to most of the papers dealing with this subject, the investors’ trading strategies we allow underly constraints described by closed, but not necessarily convex, sets. Instead of the well-known convex duality approach, we apply a backward stochastic differential equation (BSDE) approach. This leads to the study of quadratic BSDEs. The second part gives the recent result on the existence and uniqueness of solution to quadratic BSDEs. We give also the connection between these BSDEs and quadratic PDEs. The last part will show that quadratic BSDE is critic. That is, if the BSDE is superquadratic, there exists always some BSDE without solution; and there is infinite many solutions when there is one solution. This phenomenon does not exist for quadratic and superquadratic PDEs.