A monotone scheme for nonlinear partial integro-differential equations with the convergence rate of $alpha$-stable limit theorem under sublinear expectation

In this paper, we propose a monotone approximation scheme for a class of fully nonlinear partial integro-differential equations (PIDEs) which characterize the nonlinear $alpha$-stable L{e}vy processes under sublinear expectation space with $alpha in(1,2)$. Two main results are obtained: (i) the error bounds for the monotone approximation scheme of nonlinear PIDEs, and (ii) the convergence rates of a generalized central limit theorem of Bayraktar-Munk for $alpha$-stable random variables under sublinear expectation. Our proofs use and extend techniques introduced by Krylov and Barles-Jakobsen.

A note on the convergence rate of Pengs law of large numbers under sublinear expectations

This short note provides a new and simple proof of the convergence rate for Pengs law of large numbers under sublinear expectations, which improves the corresponding results in Song [15] and Fang et al. [3].

Dynamic programming principle and Hamilton-Jacobi-Bellman equation under nonlinear expectation

In this paper, we study a stochastic recursive optimal control problem in which the value functional is defined by the solution of a backward stochastic differential equation (BSDE) under $tilde{G}$-expectation. Under standard assumptions, we establish the comparison theorem for this kind of BSDE and give a novel and simple method to obtain the dynamic programming principle. Finally, we prove that the value function is the unique viscosity solution of a type of fully nonlinear HJB equation.

L{e}vys martingale characterization and reflection principle of $G$-Brownian motion

In this paper, we obtain L{e}vys martingale characterization of $G$-Brownian motion without the nondegenerate condition. Base on this characterization, we prove the reflection principle of $G$-Brownian motion. Furthermore, we use Krylovs estimate to get the reflection principle of $tilde{G}$-Brownian motion.

On the strong Markov property for stochastic differential equations driven by $G$-Brownian motion

In this paper we study the stochastic differential equations driven by $G$-Brownian motion ($G$-SDEs for short). We extend the notion of conditional $G$-expectation from deterministic time to the more general optional time situation. Then, via this conditional expectation, we develop the strong Markov property for $G$-SDEs. In particular, we obtain the strong Markov property for $G$-Brownian motion. Some applications including the reflection principle for $G$-Brownian motion are also provided.

Product space for two processes with independent increments under nonlinear expectations

In this paper, we consider the product space for two processes with independent increments under nonlinear expectations. By introducing a discretization method, we construct a nonlinear expectation under which the given two processes can be seen as a new process with independent increments.