Utility maximization and quadratic BSDEs under exponential moments

In the present thesis we consider the problem of maximizing the power utility from terminal wealth where the stocks have continuous semimartingale dynamics and there are investment and information constraints on the agent’s strategies. The main focus is on the backward stochastic differential equation (BSDE) that encodes the dynamic value process and on transferring new results on quadratic semimartingale BSDEs to the portfolio choice problem, in particular to its stability properties. This is accomplished under the assumption of finite exponential moments of the meanvariance tradeoff, generalizing previous results which require boundedness. We first recall the precise relationship between the duality and BSDE approaches to solving the above problem and then study the quadratic semimartingale BSDE which arises in such a problem when the market price of risk is of BMO type. We show that even for a bounded mean-variance tradeoff there is always a continuum of distinct solutions to the BSDE with square-integrable martingale part (with only one of them being a bounded solution). For this we prove that in contrast to the classical Itô decomposition theorem an L2-representation of random variables in terms of stochastic exponentials is not unique. We then provide a new sharp condition on the dynamic exponential moments of the mean-variance tradeoff which guarantees the boundedness of BSDE solutions in a general filtration. The main results are complemented by several interesting examples which illustrate their sharpness as well as important properties of the utility maximization BSDE. In a subsequent step we establish existence, uniqueness and stability results for general quadratic continuous BSDEs under an exponential moments condition. An important additional result is that the martingale part of a solution does provide a true measure change even though the first component of the solution triple might not be bounded, equivalently, even though the martingale part might not be of BMO type. As a consequence, the verification argument for the utility maximization problem becomes feasible for unbounded mean-variance tradeoff variables which satisfy an appropriate exponential moments assumption. We use these results to study the portfolio selection problem when there are conic investment constraints. Building on a decomposition result for the elements of the so-called dual domain we have the associated BSDE satisfied by the dynamic value process (the opportunity process) and show that by our moments assumptions this value process is contained in a specific space in which BSDE solutions are unique. This provides an argument for verification. A consequence of the stability result for BSDEs is then the continuity of the optimizers with respect to the input parameters of the model, i.e. the investor’s relative risk aversion, the market price of risk process, the statistical probability measure and the constraints sets, in the semimartingale topology. This result integrates previous research into a unified BSDE framework. Finally, we study the existence, uniqueness and stability of the optimal investment problem under exponential moments, compact constraints and restricted information. This is done by referring to BSDE results only and performing the verification argument using the previous measure change result, which is beyond the boundedness assumptions of recent literature.