Abstracts

A basic problem in mathematical finance is the problem of an agent wants to maximize his expected utility of terminal wealth. In this paper, we study this problem in a financial market where the risky asset evolves according to the following stochastic volatility model:
\begin{align*}
dS_{t}&=S_{t}[\mu_{t}dt +g(V_{t}) dB_{t}]\\
dV_{t}&=f(\beta_{t},V_{t}) dt + h(V_{t}) dW_{t}
\end{align*}
The special feature of this paper is that the only information available to the investor is the one generated by the asset prices. Especially, the trend $\mu_{t}$ cannot be observed directly but can be modelled by a stochastic process, for example, an Ornstein-Uhlenbeck. $\beta_{t}$ is either a stochastic process or a constant. $B$ and $W$ are two correlated Brownian motions. $g$,$f$,$h$ are Borel mesurable functions such that a unique solution of the above dynamics exist.
We are in the context of a portfolio optimization problem under partial information. In order to solve it, we need first to reduce this problem to one with complete information. This step can be done with the change of probability method and filtering theory. The idea is to exploit all the information coming from the market, in order to update the knowledge of the not fully know quantities, precisely, we need to replace the unknown risk premia processes of the models by their filter estimates, which are given by the conditional expectations of these variables with respect to the filtration generated by the price process. Secondly, we aim to find an explicit computations of these filters. We show that in the uncorrelated case ($B$ and $W$ are independent), we can use the Bayesian framework to deduce an explicit computations of these filters. But in the correlated case, we use the non-linear filtering theory to show that these processes satisfy a Kushner-Stratonovich equations.
Finally, according to the aboves steps, we reduced our problem to a full observation optimization problem. We solve it with the martingale duality approach in the decorrelated case, and with the dynamic programming approach in the correlated case. We show that the dynamic approach leads to a characterization of the value function as viscosity solution of a nonlinear PDE (Hamilton-Jaccobi-Bellman equation). But, by logarithm transformation, we show that the value function and the optimal portfolio can be expressed in terms of a smooth solution of a semilinear parabolic equation.