## Continuous-time mean-variance portfolio selection with liability and regime switching.(English)Zbl 1231.91417

Summary: A continuous-time mean-variance model for individual investors with stochastic liability in a Markovian regime switching financial market, is investigated as a generalization of the model of X.Y. Zhou and G. Yin [SIAM J. Control Optim. 42, 1466–1482 (2003; Zbl 1175.91169)]. We assume that the risky stock’s price is governed by a Markovian regime-switching geometric Brownian motion, and the liability follows a Markovian regime-switching Brownian motion with drift, respectively. The evolution of appreciation rates, volatility rates and the interest rates are modulated by the Markov chain, and the Markov switching diffusion is assumed to be independent of the underlying Brownian motion. The correlation between the risky asset and the liability is considered. The objective is to minimize the risk (measured by variance) of the terminal wealth subject to a given expected terminal wealth level. Using the Lagrange multiplier technique and the linear-quadratic control technique, we get the expressions of the optimal portfolio and the mean-variance efficient frontier in closed forms. Further, the results of our special case without liability is consistent with those results of Zhou and Yin [loc. cit.].

### MSC:

 91G10 Portfolio theory 91G80 Financial applications of other theories 60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) 60J60 Diffusion processes

Zbl 1175.91169
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