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In this paper, we introduce the concept of the p-mean almost periodicity for stochastic processes in non linear expectation spaces. The existence and uniqueness of squaremean almost periodic solutions to some non linear stochastic differential equations driven by G-Brownian motion are established under some assumptions for the coefficients. The asymptotic stability of the unique square-mean almost periodic solution in the square-mean sense is also discussed.

This paper introduces a class of backward stochastic differential equations (BS-DEs), whose coefficients not only depend on the value of its solutions of the present but also the past and the future. For a sufficiently small time delay or a sufficiently small Lipschitz constant, the existence and uniqueness of such BSDEs is obtained. As an adjoint process, a class of stochastic differential equations (SDEs) is introduced, whose coefficients also depend on the present, the past and the future of its solutions. The existence and uniqueness of such SDEs is proved for a sufficiently small time advance or a sufficiently small Lipschitz constant. A duality between such BSDEs and SDEs is established.

In this paper we study the weak laws of large numbers for sublinear expectation. We prove that, without any moment condition, the weak laws of large numbers hold in the sense of convergence in capacity induced by some general sublinear expectations. For some specific sublinear expectation, for instance, mean deviation functional and one-side moment coherent risk measure, we also give weak laws of large numbers for corresponding capacity.

In this paper, we establish an exact asymptotic formula for the finite-time ruin probability of a nonstandard compound renewal risk model with constant force of interest in which claims arrive in groups, their sizes in one group are identically distributed but negatively dependent, and the inter-arrival times between groups are negatively dependent too.