Stochastic control problems with performance fees and incomplete markets
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This dissertation applies stochastic control theory to two problems: i) portfolio choice of hedge fund managers compensated by performance fees, and ii) consumption and investment in an incomplete market. Part I. Optimal portfolios are derived in closed form for a fund manager, who is paid performance fees with a high-water mark provision, and invests both the fund's assets and private wealth in separate and potentially correlated, constant investment opportunities. The manager's goal is to maximize expected utility from private wealth in the long run, with constant relative risk aversion. At the optimum, the private portfolio depends only on the private investment opportunity, and the fund's portfolio only on fund's opportunity. The manager invests earned fees in the safe asset, allocating remaining private wealth in a constant-proportion portfolio with his or her own risk aversion. The fund is managed as a constant-proportion portfolio with risk aversion shifted towards one. The optimal welfare is the maximum between the optimal welfare of each investment opportunity alone. Part II. An agent maximizes isoelastic utility from consumption with infinite horizon in an incomplete market, in which state variables are driven by diffusions. First, a general verification theorem is provided, which links the solution of the Hamilton-Jacobi-Bellman equation to the optimal consumption and investment policies. To tackle the analytical intractability of such problems, approximate policies are proposed, which admit an upper bound, in closed-form for their utility loss. These policies are optimal for the same agent in an artificial and complete market, in which the safe rate and the state variable follow different dynamics, but excess returns remain the same. The approximate policies have closed form solutions in common models, and become optimal if the market is complete, or utility is logarithmic.