## Filtering, Approximation and Portfolio Optimization for Shot-Noise Models and the Heston Model

• We consider a continuous time market model in which stock returns satisfy a stochastic differential equation with stochastic drift, e.g. following an Ornstein-Uhlenbeck process. The driving noise of the stock returns consists not only of Brownian motion but also of a jump part (shot noise or compound Poisson process). The investor's objective is to maximize expected utility of terminal wealth under partial information which means that the investor only observes stock prices but does not observe the drift process. Since the drift of the stock prices is unobservable, it has to be estimated using filtering techniques. E.g., if the drift follows an Ornstein-Uhlenbeck process and without jump part, Kalman filtering can be applied and optimal strategies can be computed explicitly. Also in other cases, like for an underlying Markov chain, finite-dimensional filters exist. But for certain jump processes (e.g. shot noise) or certain nonlinear drift dynamics explicit computations, based on discrete observations, are no longer possible or existence of finite dimensional filters is no longer valid. The same computational difficulties apply to the optimal strategy since it depends on the filter. In this case the model may be approximated by a model where the filter is known and can be computed. E.g., we use statistical linearization for non-linear drift processes, finite-state-Markov chain approximations for the drift process and/or diffusion approximations for small jumps in the noise term. In the approximating models, filters and optimal strategies can often be computed explicitly. We analyze and compare different approximation methods, in particular in view of performance of the corresponding utility maximizing strategies.

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