Continuous-Time Mean-Variance Portfolio Selection In International Security Markets

This paper is concerned with a continuous-time mean-variance portfolio selection model in international securities markets that is formulated as a bi-criteria optimization problem.In this model, the investor invests his money in two different security markets and buys two securities ,one of which is bond in the domestic and another of which is stock in a foreign country. the price of stock is influenced by uncertainty factors(for example,the regulations,the volatilities),moreover,influenced by exchange rate.In the optimization problem,the control system includes two one-dimensional mutually dependent Brownian motions and the cost functionals includes two parts,namely, maximize the expected terminal return and minimize the variance of the terminal wealth,therefore,this is a multi-objective optimization problem. First, the multi-objective optimization problem is transformed to a single-objective optimization problem P(μ),Stochastic linear-quadratic(LQ)problem(auxiliary problem A(μ, λ) with parameters (μ, λ). Then the analytical solution of optimal control and cost functional of auxiliary problem are obtained by using dynamic programming principle,the economical analysis to the optimal choice of the investor is given through the investment theory.In the last of this paper, an analytical optimal portfolio policy and an explicit expression of efficient frontier for the original problem are derived according to the former conclusions; Also, by discussing an example,the price of risk diagram on some parameters is given and the influence of the parameters on the optimal choice of the investor and price of risk is also considered.Chapter two:Given the model of international security markets , suppose the investor wants to invest the bonds in his home country and another kind of stock in a foreign country,the price of dom(?)stic bonds ,the stock in the foreign country and exchange rate satisfydP₀(t) = r(t)P₀(t)dt,de{t) = e{t)b_e(t)dt + e(t)σe(t)dB₂(t),respectively.The terms B₁(t)and B₂(t) are two one-dimensional mutually dependent Brownian motion.The process of the investor's wealth satisfiesdx(t) = {r(t)x(t) + [b₁(t) - r(t)]u(t)}dt + σ_tu(t)dB₁(t) + σ_e(t)u(t)dB₂(t),chapter three:the original problem is transformed to a Stochastic linear-quadratic(LQ)problem(auxiliary problem A(μ, λ))to obtain the optimal portfolio and the efficient frontier of the mean-variance problem.chapter four:Applying the dynamic programming principle,we obtained the optimal feedback control and cost function of the auxiliary problem.namely Theorem 4.1.The stochastic LQ problem( auxiliary problem)has an optimalfeedback controlmoreover,the optimal cost value ischapter five:The optimal portfolio and the efficient frontier of the mean-variance problem are given,namely...