A kind of optimal investment problem under non-Markovian regime-switching model with random horizon

In this paper, we study a kind of optimal portfolio problem with regime-switching and random horizon in the international market. Unlike previous works, the dynamic of assets are described by non-Markovian regime-switching models in the sense that all the market parameters are predictable with respect to the filtration generated jointly by Markov chain and Brownian motion. The investor can invest the wealth in a foreign bonds or in an international real project with production. We consider the influence of tax rate and exchange rate. In particular for the CRRA case, we give the explicit form of optimal portfolio by the method of completion of squares and the maximum utility by the solution of a regime-switching stochastic Riccati equation. Moreover, we present several numerical examples to show the application of the results.