Portfolio optimization under partial observation: theoretical and numerical aspects∗

We give a survey of the methods involved in portfolio selection with partial ob-servation. We describe both the theoretical and numerical aspects related to these optimization problems. The presentation is divided in two parts. In the first one, we focus on continuous-time problem: here, the mean rates of return of the asset prices are not directly observable. Investors observe only asset prices. By the method of change of probability and innovation process in filtering theory, the partial observation portfolio selection problem is transformed into a full observation one with the additional filter state variable, for which one may apply the martingale or PDE approach. We investi-gate different cases for the modelling of the unobervable mean rate of return: Bayesian, linear-Gaussian and finite state Markov chain. In the second part, we consider discrete-time optimization problems: this context includes the case of unobservable volatility. We are then concerned with the numerical approximation of the optimization problem under partial observation. This is achieved by performing a quantization of the pair process filter-observation, and using backward dynamic programming formulas. Sev-eral numerical experiments illustrate the resuts for hedging problems in the context of partially observed stochastic volatility models. Key words: Partial observation, portfolio optimization, filtering, martingale approach, dynamic programming, quantization