# Meeting Details

Title: Utility maximation and (F)BSDE of quadratic growth Seminar on Probability and its Application Peter Imkeller, Humboldt University Berlin A financial market model is considered on which agents (e.g. insurers) are subject to an exogenous financial risk, which they trade by issuing a risk bond. They are able to invest in a market asset correlated with the exogenous risk. We investigate their utility maximization problem, and calculate bond prices using utility indifference. In the case of exponential utility, this hedging concept is interpreted by means of martingale optimality, and solved with BSDE with drivers of quadratic growth in the control variable. For more general utility functions defined on the whole real line we show that if an optimal strategy exists then it is given in terms of the solution $(X,Y,Z)$ of a fully coupled FBSDE. Conversely if the FBSDE admits a solution $(X,Y,Z)$ then an optimal strategy can be obtained. In the complete market case, an assumption on the risk aversion guarantees that the FBSDE admits a solution for any finite time horizon. As a particular example of our approach we recover the BSDE for exponential utility, and are able to treat non-classical utility functions like the sum of exponential ones. For utility functions defined on the half line we also reduce the maximization problem to the solution of FBSDE connected with the ones obtained by Peng (1993). In complete markets, once again we provide conditions for solvability that are applicable to the power, the logarithmic and some non-classical utilities. In our approach we propose an alternative form of the maximum principle for which the Hamiltonian is reflected in a remarkable martingale. This is joint work with U. Horst, Y. Hu, A. R\'eveillac, and J. Zhang.