ABSTRACT
The financial market is modeled as a nonlinear stochastic system with continuous Wiener and discontinuous Poisson random fluctuations. In the financial market, most managers or investors hope their investment policies to be with the not only high profit but also low risk. In general, managers and investors involved pursue their own interests but are partly conflicting with others. Stochastic game theory has been widely applied to multi-person noncooperative decision-making problem of financial market. However, for the nonlinear stochastic financial system with random fluctuations, it still lacks an analytical or computational scheme to effectively solve the complex multi-player noncooperative game strategy design problem. In this chapter, the stochastic multi-person noncooperative https://www.w3.org/1998/Math/MathML"> H ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429432941/63df2486-ef31-4a70-9449-3488247ff9a8/content/inline15_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> game strategy in nonlinear stochastic financial systems is transformed to a multi-tuple Hamilton–Jacobi–Isaac inequalities (HJIIs)-constrained multiobjective optimization problem (MOP). This HJIIs-constrained MOP solution is also found to be the Nash equilibrium solution of multi-person noncooperative https://www.w3.org/1998/Math/MathML"> H ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429432941/63df2486-ef31-4a70-9449-3488247ff9a8/content/inline15_2.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> game strategy in nonlinear stochastic financial systems. In order to simplify design procedure by the global linearization theory, a set of local linear stochastic financial systems are interpolated to approximate the nonlinear stochastic financial system so that the m-tuple HJIIs-constrained MOP for multi-person noncooperative https://www.w3.org/1998/Math/MathML"> H ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429432941/63df2486-ef31-4a70-9449-3488247ff9a8/content/inline15_3.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> game strategy of nonlinear stochastic financial system could be converted to a linear matrix inequalities (LMIs)-constrained MOP. Finally, an LMIs-constrained multiobjective evolutionary algorithm (MOEA) is explored for effectively solving the multi-person noncooperative https://www.w3.org/1998/Math/MathML"> H ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429432941/63df2486-ef31-4a70-9449-3488247ff9a8/content/inline15_4.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> game strategy in nonlinear stochastic financial systems. Two design examples are also given for the illustration of the design procedure and the performance validation of the proposed https://www.w3.org/1998/Math/MathML"> H ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429432941/63df2486-ef31-4a70-9449-3488247ff9a8/content/inline15_5.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> stochastic noncooperative investment strategy in the nonlinear stochastic financial systems.
