Multiobjective Investment Policy for a Nonlinear Stochastic Financial System: A Fuzzy Approach

Abstract

The financial market always suffers from continuous and discontinuous (jump) changes and can be regarded as a nonlinear stochastic jump diffusion system. Most investors expect their investment policies to be not only higher benefits but also lower risk as a multiobjective optimization problem (MOP). In this study, a multiobjective $H_{2}/H_{\infty }$ fuzzy investment is proposed for nonlinear stochastic jump diffusion financial systems to achieve the desired target with minimum investment cost and risk in Pareto optimal sense, simultaneously. The Takagi–Sugeno (T–S) fuzzy model is used to approximate the nonlinear stochastic jump diffusion financial system to simplify the multiobjective $H_{2}/H_{\infty }$ investment policy design procedure. By the help of the T–S fuzzy model, the multiobjective $H_{2}/H_{\infty }$ fuzzy investment policy problem of nonlinear stochastic financial system can be transformed to a linear-matrix-inequality-constrained (LMI-constrained) MOP to avoid solving the annoying Hamilton–Jacobi inequalities. Because the LMI-constrained MOP is not easy to directly calculate its Pareto optimal solutions, an indirect method is proposed to solve this MOP for the multiobjective $H_{2}/H_{\infty }$ fuzzy investment policy design of nonlinear stochastic jump diffusion financial systems. An LMI-constrained multiobjective evolution algorithm (LMI-constrained MOEA) is also developed to efficiently solve the Pareto optimal solutions of the LMI-constrained MOP for the multiobjective $H_{2}/H_{\infty }$ fuzzy investment policy design of nonlinear stochastic jump diffusion financial systems. When the Pareto optimal regulation solutions are solved by the proposed LMI-constrained MOEA, investors can select one investment policy to achieve their desired target with minimum investment cost and risk according to his/her own preference.