Matrix Game with Payoffs Represented by Triangular Dual Hesitant Fuzzy Numbers

Matrix Game with Payoffs RepresentedDue to the complexity of information or the inaccuracy of decision-makers’ cognition, it is difficult for experts to quantify the information accurately in the decision-making process. However, the integration of the fuzzy set and game theory provides a way to help decision makers solve the problem. This research aims to develop a methodology for solving matrix game with payoffs represented by triangular dual hesitant fuzzy numbers (TDHFNs). First, the definition of TDHFNs with their cut sets are presented. The inequality relations between two TDHFNs are also introduced. Second, the matrix game with payoffs represented by TDHFNs is investigated. Moreover, two TDHFNs programming models are transformed into two linear programming models to obtain the numerical solution of the proposed fuzzy matrix game. Furthermore, a case study is given to to illustrate the efficiency and applicability of the proposed methodology. Our results also demonstrate the advantage of the proposed concept of TDHFNs

Exit Problem and Stochastic Resonance for a Class of Random Perturbations

The asymptotic exit problems for diffusion processes with small parameter were considered in the classic work of Freidlin and Wentzell. In 2000, a mathematical theory of stochastic resonance for systems with random perturbations was established by Freidlin in the frame of the large deviation theory. This dissertation concerns exit problems and stochastic resonance for a class of random perturbations approximating white noise. The tools used in the proofs are the large deviation theory and the Markov property of the processes. The first problem considered is the exit problem and stochastic resonance for random perturbations of random walks. It turns out that a specific random walk can be chosen which approximates the large deviation asymptotics of the Wiener process in the best way. Analogous results concerning exit problems and stochastic resonance for this type of random perturbations were obtained under appropriate assumptions and compared with those of white noise type perturbation. The second problem I consider is the exit problems for random perturbations of a Gaussian process $\eta_{t}^{\mu,\varepsilon}$ which satisfies the equation \mu \dot{\eta}_{t}^{\mu,\varepsilon}=- \eta_{t}^{\mu,\varepsilon}+\sqrt{\varepsilon}\dot{W}_{t}, \,\eta_{0}^{\mu,\varepsilon}=y, \,0<\mu<<1,\,0<\varepsilon<<1 . One can check that $\int_{0}^{t} \eta_{s}^{\mu,\varepsilon}ds$ converges to $\sqrt{\varepsilon}W_{t}$ uniformly on $[0,T]$ in probability as $\mu \downarrow 0$. Results concerning asymptotic exit problems for this type of random perturbation were obtained under appropriate assumptions. Since $\eta_{t}^{\mu,\varepsilon}$ is not a Markov process, this creates some difficulties for the proof. A new Markov process was constructed and the Markov property of the new process was used in the proof