Optimal risk transfer and investment policies based upon stochastic differential utilities

This paper addresses the stochastic differential utility (SDU) version of the issue raised by Barrieu and El Karoui (Quantitative Finance, 2:181–188, 2002a) in which optimal risk transfer from a bank to an investor, realized by transacting well-designed derivatives written on relevant illiquid assets, was␣mainly studied in two cases with and without an available financial market. From a stochastic maximum principle as described in Yong and Zhou (Stochastic controls: Hamiltonian systems and HJB equations. Springer-Verlag, New York, 1999) we shall derive necessary and sufficient conditions for optimality in several SDU-based maximization problems. It is also shown that the optimal risk transfer, consumptions, investment policies of both agents are characterized by a forward–backward stochastic differential equation (FBSDE) system.