Mean-Variance Portfolio Selection for Partially Observed Point Processes

Abstract

We study the mean-variance portfolio selection problem for a class of price models, which well fit the two features of time-stamped transactions data. The price process of each stock is described by a collection of partially observed point processes. They are the noisy observation of an intrinsic value process, assumed to be Markovian. However, the control problem with partial information is non-Markovian and depends on an infinite-dimensional measure-valued input. To solve the challenging problem, we first establish a separation principle, which divides the filtering and the control problems and reduces the infinite-dimensional input to finite-dimensional ones. Building upon the result of nonlinear filtering with counting process observations, we solve the control problem by employing the stochastic maximum principle for control with forward-backward SDEs developed in [SIAM J. Control Optim., 48 (2009), pp. 2945--2976]. We explicitly obtain the efficient frontier and derive the optimal strategy, which is based on the filtering estimators.