Lu Yang

Aiming at the objective uncertainty, subjective uncertainty, and extreme events may be in a dynamic system simultaneously. This paper focuses on the differential game problem of a linear quadratic jump uncertain stochastic system. The system is described by both a jump uncertain differential equation and a stochastic differential equation. The principle of optimality and equation of optimality are established. Then, a differential game model based on linear quadratic jump uncertain stochastic system is constructed. Furthermore, Nash equilibrium are discussed by using the obtained equation. Finally, its application in the dynamic investment decision of enterprises is given, and numerical simulations are performed. This approach offers a new method for quantitative analysis in future studies.

Due to the conjunction of huge expenditure and latent inefficiencies, competitive advertising has always been occupying a front‐and‐center place in marketing, among which a bilinear system is one of the most important. Because much real business is two giants playing among other emerging challengers and the players can only inspect the market through a related noisy process by the random sampling survey, the sales‐advertising dynamics is a partially observed nonzero‐sum stochastic differential game. However, by the intrinsic difficulties of mathematics, there is no extant research to augment the analysis on the infinite‐time horizon, which is of particular significance to implementing the online sales campaign and evaluating the consequence of endless competition. Motivated by this, controllability and stability analysis are explored first for the iterative approximation to bilinearity. Then, the simulation provides a rapid prototyping and automatic approach for the most efficient strategies. Moreover, four implications have been discovered by comparison with eight examples as the application of four encounters on finite‐ and infinite‐time horizons. Several new insights have been revealed for those struggling to develop successful advertising strategies.

This paper investigates the guaranteed cost control problem and guaranteesthe cost Nash equilibrium for discrete-time Markovian jump Linearsystem (MJLS) with incomplete transition rates. By employing matrixinequalities and the free-connection weighting matrix approach, severalsufficient conditions are proposed to ensure the applicability of theguaranteed cost control strategy. Moreover, we provide sufficient conditionsto highlight further the correctness and lower conservativeness ofthe utilized theorems. Ensuring these conditions for the existence of aguaranteed cost Nash equilibrium affords to transform them into an optimizationproblem that simultaneously satisfies a set of BMI and matrixinequalities. Furthermore, to make the results more general, this paperdesigns guaranteed cost Nash equilibrium strategies for multiple players.Finally, two detailed numerical examples validate the paper’s theorems.

In this study, a partially observed linear quadratic (LQ) non‐cooperative stochastic differential game of Markov jump linear system (MJLS) is considered. First, a new proof of filtering theory is given, using the separation principle and filtering technology, partial observations can be transformed into complete observation. Second, nonzero‐sum stochastic differential game problem for MJLS are studied using the dynamic programming principle and completion square method. Then, the sufficient and necessary conditions for the Nash equilibrium are obtained. Finally, a numerical simulation is performed.