×

Optimizing the terminal wealth under partial information: the drift process as a continuous time Markov chain. (English) Zbl 1063.91040

The authors consider a financial market consisting of one bank account with stochastic interest rate \(r\) and \(n\) stocks driven by a geometric Brownian motion with a constant volatility matrix \(\sigma\) and rates of return \(\mu\) modelled as a continuous time finite state Markov chain. Since \(\mu\) is a part of a drift process of the observable excess return process \[ \widetilde{R}=\int_0^t (\mu_s-r_s\mathbf1_n)\,ds+\sigma W_t,\quad t\in[0,T], \] one has a hidden Markov model (HMM) for \(\widetilde{R}\), since an investor can only observe the prices (partial information). The Brownian motion and the drift process are not observable. The model represents \(\mu\) as \(\mu_t=B Y_t\), where \(Y\) is a stationary continuous time Markov chain whose state space consists of the standard unit vector in \(\mathbb R^d\), and the matrix \(B\) contains the states of \(\mu\). Conditions on \(r\) are quite restrictive, but they allow the consideration of historical rates in the numerical part and they show how the optimal strategies depend on \(r\). The main result provides an explicit representation of the optimal trading strategy in terms of an unnormalized filter, the interest rates, the excess return process and the parameters of the model which are observable r can be estimated. For logarithmic and power utilities, optimal trading strategies are computed in terms of the current unnormalized filter, the current interest rates and the scalar current wealth rather than the multivariable excess return process.
Reviewer: Yuliya Mishura

MSC:

91G10 Portfolio theory
60G44 Martingales with continuous parameter
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
91G30 Interest rates, asset pricing, etc. (stochastic models)
PDFBibTeX XMLCite
Full Text: DOI