This paper studies the question of filtering and maximizing terminal wealth from expected utility in a  stochastic volatility models. The special feature is that the only information available to the investor is the one generated by the asset prices and, in particular, the return processes cannot be observed directly and assumed to be modelled by a stochastic differential equation. Using stochastic non-linear filtering and change of measure techniques, the partial observation context can be transformed into a full information context such that coefficients depend only on past history of observed prices (filters processes). The main difficulty is that these filters are valued in infinite-dimensional space: it satisfy a stochastic partial differential equations named "Kushner-Stratonovich equations". We also show that we need to introduce an a priori models for the trend and the stochastic volatility in order to evaluate the filters processes. The dynamic programming or maximum principle are still applicable and the associated Bellman equationor Hamiltonian system are now in infinite dimension .
This paper studies the question of filtering and maximizing terminal wealth from expected utility in a partially information stochastic volatility models. The special features is that the only information available to the investor is the one generated by the asset prices , and the unobservable processes will be modeled by a stochastic differential equations. Using the change of measure techniques, the partial observation context can be transformed into a full information context such that coefficients depend only on past history of observed prices (filters processes). Adapting the stochastic non-linear filtering, we show that under some assumptions on the model coefficients, the estimation of the filters depend on a priorimodels for the trend and the stochastic volatility. Moreover, these filters satisfy a stochastic partial differential equations named "Kushner-Stratonovich equations". Using the martingale duality approach in this partially observed incomplete model, we can characterize the value function and the optimal portfolio. The main result here is that the dual value function associated to the martingale approach can be expressed, via the dynamic programmingapproach, in terms of the solution to a semilinear partial differential equation. We illustrate our results with some examples of stochastic volatility models popular in the financial literature .
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and, in particular, the return processes cannot be observed directly and assumed to be modelled
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equation. Using stochastic non-linear filtering and
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The main difficulty is that these filters are valued in infinite-dimensional space: it
Adapting the stochastic non-linear filtering, we show that under some assumptions on the model coefficients, the estimation of the filters depend on a priorimodels for the trend and the stochastic volatility. Moreover, these filters
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We also show that we need to introduce an a priori models for the trend and the stochastic volatility in order to evaluate the filters processes. The dynamic programming or maximum principle are still applicable and the associated Bellman equationor Hamiltonian system are now in infinite dimension
Using the martingale duality approach in this partially observed incomplete model, we can characterize the value function and the optimal portfolio. The main result here is that the dual value function associated to the martingale approach can be expressed, via the dynamic programmingapproach, in terms of the solution to a semilinear partial differential equation. We illustrate our results with some examples of stochastic volatility models popular in the financial literature