Take The Part Of The Debt Information Under The Optimal Portfolio

It is well-known that a financial model with deterministic coefficients are only good for a relative short period of time and cannot respond to changing conditions. The information available to the investor is the filtration generated by the asset price processes only. The investor can in general not directly observe the mean return rate processes and the volatility process of the asset price process. It has very important significance in theory and practical to study the portfolio selection problem under the partial information.The estimation of the mean return rate of the stock price process under the stock price leads to two famous filtering theory. One is the Kalman filter, the mean return rates is modelled by a linear diffusion model. The other is the Wonham filter, the mean return rates is represented by a finite state continuous-time Markov chain.In this paper we study the portfolio selection problem with liability under the partial information. When the liability process is modelled by a diffusion model or jump-diffusion model, we investigate the portfolio selection problem under the Kalman filter. By using the stochastic linear-quadratic control technique, the closed form solutions of the optimal portfolio strategy and the maximal expected exponential utility are obtained. Furthermore, we also study the portfolio selection problem under the Wonham filter when the liability process is modelled by a diffusion model or jump-diffusion model. We also obtain the explicit expression of the optimal portfolio strategy and the maximal expected exponential utility.