Risk Minimizing Strategies for Tracking a Stochastic Target

Part 3: Stochastic Optimization and ControlInternational audienceWe consider a stochastic control problem of beating a stochastic benchmark. The problem is considered in an incomplete market setting with external economic factors. The investor preferences are modelled in terms of HARA-type utility functions and trading takes place in a finite time horizon. The objective of the investor is to minimize his expected loss from the outperformance of the benchmark compared to the portfolio terminal wealth, and to specify the optimal investment strategy. We prove that for considered loss functions the corresponding Bellman equation possesses a unique solution. This solution guaranties the existence of a well defined investment strategy. We prove also under which conditions the verification theorem for the obtained solution of the Bellman equation holds