In this paper, the optimal reinsurance strategy is considered to minimize the ruin probability of a risk model with multiple dependent classes of insurance under variance reinsurance premium principle. Through diffusion approximation of the claim risk process and by applying the dynamic programming approach, explicit expressions of the optimal strategy and the value function are obtained. Moreover, by comparing to the results obtained under the expected value reinsurance premium principle, it is found that the optimal reinsurance form and the retention risk level are both different. By comparing to the results which maximize expected exponential utility, it is found that the optimal reinsurance proportion here depends not only on safety loading, but also on the claim distribution, the counting process and the premium rate of insurance $c$. Finally, combining with numerical example, dynamic impact of the dependence parameter is demonstrated and sensitive correlation between the optimal strategy and $c$ is illustrated.

In this paper, the pricing problems of geometric average Asian options are studied under the nonlinear Black-Scholes model. Firstly, the partial differential equations for the Asian options are transformed into a series of parabolic equations with constant coefficients by the perturbation method of single-parameter. Secondly, the approximate pricing formulae of the geometric average Asian options are given by solving those parabolic equations with constant coefficients. Finally, the error estimates of the approximate solutions are given by using Green function.

Several concepts of recurrence and transience of states for Markov Chains in random environments are introduced under the influence of various environmental factors, and their connections and properties are discussed. These conclusions present that the recurrence and transience of states for Markov Chains in random environments and the recurrence and transience of states for classical Markov chains have obvious differences.

By using the $C_0$-semigroup theory, this paper studies the repairable closed queueing system with server of Erlangian distributed life time. First, by using the Hille-Yosida theorem, Phillips theorem and Fattorini theorem in functional analysis, the existence and uniqueness of nonnegative time dependent solution of system model has been proved. Next, the spectral properties of the operator corresponding to system model are investigated, which show that all points on the imaginary axis except zero belong to the resolvent set of the operator and zero is an eigenvalue of the operator and its adjoint operator with geometric multiplicity one. Thus, the above results give that the time-dependent solution of the system model converges strongly to its steady state solution.

The boundary value problems of a class of quasilinear elliptic equations are considered, which possess periodic variable exponents and concave-convex nonlinearities in $\mathbf{R}^{N}$. Under some weaker assumptions, the multiplicity of solutions for the equations is obtained by applying the Ekeland's variational principle and the Nehari manifold theory.

In this paper, a class of initial boundary problem of the singular perturbed semi-linear delayed parabolic partial differential equation is discussed. The formal asymptotic expansion of the problem is obtained. The maximum principle of the singular perturbed delayed semi-linear parabolic partial differential equations is proved. Then, the maximum-norm estimation and Schauder estimation for this problem are obtained. By the maximum-norm estimation and Schauder estimation for this problem, the existence and uniqueness of the solution of the problem on the columnar zone is proved, and the uniformly valid estimation of the asymptotic expansion is gained.

Double boundary layers of quadratic nonlinear singularly perturbed boundary value problem with infinite boundary values is studied. Using the boundary layer correction function, its asymptotic solution is constructed, and using the theory of differential inequality, the uniformly valid asymptotic estimation is presented. Finally, an example is given to verify the validity of the relevant conclusions.

This paper is devoted to study practical stability of the general impulsive switched systems with time delay. By employing the Razumikhin technique and Lyapunov functions, we establish some sufficient conditions to guarantee the practical stability or uniform practical stability of impulsive switched systems with time delay. Two examples and simulation are also given to illustrate our results.

A split modified method of characteristics mixed finite element (SMMOC-MFE) and a split modified method of characteristics with adjusted advection mixed finite element (SMMOCAA-MFE) are proposed for solving advection-dominated diffusion equations, in which the mixed element systems are symmetric positive definite, and the original variable $u$ and the diffusive flux $\bm\sigma=-\varepsilon\nabla u$ can be solved separately. The optimal-order error estimates in weighted energy norm are derived and some numerical implementations are given to confirm the convergence results.

In port container terminals, a vessel is usually divided longitudinally between head and tail into many holds to store containers, which must be loaded or unloaded by several quay cranes. The scheduling of quay cranes significantly influences the turn-around time of a container vessel. This paper studies a problem of scheduling small number of quay cranes with non-interference constraint. The objective is to minimize the overall time of loading or unloading the containers. New scheduling algorithms are designed and analyzed for three and four quay cranes, which improve previous results on this problem.

In the process of strategy choice problem, players need to estimate and rank the expected return (payoffs), and the selected results are often influenced by risk preferences in reality. So a method for trapezoidal intuitionistic fuzzy bi-matrix game with risk preference is researched in this paper. In this method, a new order relation with risk preference of trapezoidal intuitionistic fuzzy number based on the difference-index of value-index is proposed, and then the parametric bi-matrix game model is solved by bilinear programming. Lastly, the method proposed is demonstrated by a real example of the marketing enterprises’ strategy choice problem, which shows the effective and practical of the method.

The properties of Orlicz spaces $L_{\mathit\Phi}^{*}(0,\infty)$ corresponding to the Young function $\mathit\Phi(x)$ are discussed and the Hardy-Littlewood property of the Orlicz spaces $L_{\mathit\Phi}^{*}(0,\infty)$ is given. Then two kinds of strong converse inequalities of Jacobi weighted simultaneous approximation for Gamma operators are established by modified Jacobi weighted $K$-functional and Jacobi weighted modulus of smoothness in Orlicz spaces $L_{\mathit\Phi}^{*}(0,\infty)$.