Gaofeng Zong

In this paper, we investigate the second Borel–Cantelli lemma for capacity without the assumption of independence for events. We obtain a sufficient condition under which the second Borel–Cantelli lemma for capacity holds. Our results are natural extensions of the classical Borel–Cantelli lemma. However, the proof is different from the existing literature.

Physical models that simulate viscoelastic phenomena with random effects often use stochastic wave equations, but to simulate viscoelastic phenomena depends on past random effects, stochastic wave equation with time-fractional derivative is the best choice. In this paper, we introduce a class of stochastic semi-linear wave equations with time-fractional derivatives, and study the existence, uniqueness, and the existence of continuous modification of mild solutions to this time-fractional stochastic semi-linear wave equation by establishing a prior estimation.

This paper deals with a stochastic optimal feedback control problem for the controlled stochastic partial differential equations. More precisely, we establish the existence of stochastic optimal feedback control for the controlled stochastic partial differential equations with fully monotone coefficients by a minimizing sequence for the control problem. Using the Faedo-Galerkin approximations, the uniform estimates and the tightness in some appropriate space for the Faedo-Galerkin approximating solution can be obtain to prove the well-posedness of the controlled stochastic partial differential equations with fully monotone coefficients. The results obtained in the present paper may be applied to various types of controlled stochastic partial differential equations, such as the controlled stochastic convection diffusion equation.

We study the mean-field type stochastic control problem where the dynamics is governed by a general L\’{e}vy process with moments of all orders. For this, we introduce the power jump processes and the related Teugels martingales and give the Malliavin derivative with respect to Teugels martingales. We derive necessary and sufficient conditions for optimality of our control problem in the form of a mean-field stochastic maximum principle.

We consider the Cox–Ingersoll–Ross (CIR) model in time-dependent domains, that is, the CIR process in time-dependent domains reflected at the time-dependent boundary. This is a very meaningful question, as the CIR model is commonly used to describe interest rate models, and interest rates are often artificially set within a time-dependent domain by policy makers. We consider the most fundamental question of recurrence versus transience for normally reflected CIR process with time-dependent domains, and we examine some precise conditions for recurrence versus transience in terms of the growth rates of the boundary. The drift terms and the diffusion terms of the CIR processes in time-dependent domains are carefully provided. In the transience case, we also investigate the last passage time, while in the case of recurrence, we also consider the positive recurrence of the CIR processes in time-dependent domains.

In this article, we consider the Nash equilibrium of stochastic differential game where the state process is governed by a controlled stochastic partial differential equation and the information available to the controllers is possibly less than the general information. All the system coefficients and the objective performance functionals are assumed to be random. We find an explicit strong solution of the linear stochastic partial differential equation with a generalized probabilistic representation for this solution with the benefit of Kunita’s stochastic flow theory. We use Malliavin calculus to derive a stochastic maximum principle for the optimal control and obtain the Nash equilibrium of this type of stochastic differential game problem.