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Financial Math Seminar

Mean-Variance Portfolio Selection for Partially-Observed Point Processes

We study the mean-variance portfolio selection problem for a class of models well fit 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, mildly 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.