Existence and Stability of Solutions to Highly Nonlinear Stochastic Differential Delay Equations Driven by G-Brownian Motion

Under linear expectation (or classical probability), the stability for stochastic differential delay equations (SDDEs), where their coefficients are either linear or nonlinear but bounded by linear functions, has been investigated intensively. Recently, the stability of highly nonlinear hybrid stochastic differential equations is studied by some researchers. In this paper, by using Peng’s G-expectation theory, we first prove the existence and uniqueness of solutions to SDDEs driven by G-Brownian motion (G-SDDEs) under local Lipschitz and linear growth conditions. Then the second kind of stability and the dependence of the solutions to G-SDDEs are studied. Finally, we explore the stability and boundedness of highly nonlinear G-SDDEs.

Multiple G-Stratonovich Integral Driven by G-Brownian Motion

In this paper, we propose the multiple Stratonovich integral driven by G-Brownian motion under the G-expectation framework. Then based on G-Itöformula, we obtain the relationship between Hermite polynomials and multiple G-Stratonovich integrals by using mathematical induction method.

On Representation Theorem of G-Expectations and Paths of G-Brownian Motion

We give a very simple and elementary proof of the existence of a weakly compact family of probability measures {Pθ ： θ∈θ} representing an important sublinear expectation- G-expectation E[·]. We also give a concrete approximation of a bounded continuous function X（ω） by an increasing sequence of cylinder functions Lip（Ω） in order to prove that Cb（Ω） belongs to the completion of Lip（Ω） under the natural norm E[｜·｜].

On the Existence and Uniqueness of Solutions to Stochastic Differential Equations Driven by G-Brownian Motion with Integral-Lipschitz Coefficients

In this paper, we study the existence and uniqueness of solutions to stochastic differential equations driven by G-Brownian motion （GSDEs） with integral-Lipschitz coefficients.

Some Properties of Stochastic Differential Equations Driven by the G-Brownian Motion

In this paper, we study the property of continuous dependence on the parameters of stochastic integrals and solutions of stochastic differential equations driven by the G-Brownian motion. In addition, the uniqueness and comparison theorems for those stochastic differential equations with non-Lipschitz coefficients are obtained.

G-布朗运动驱动的随机微分方程的全局渐近稳定性

利用一致渐近稳定函数,对G-布朗运动驱动的随机微分方程给出了其平凡解在拟必然意义下全局渐近稳定的一个充分条件,通过例子展示了结果的有效性。

Numerical simulations for G-Brownian motion

This paper is concerned with numerical simulations for the G- Brownian motion （defined by S. Peng in Stochastic Analysis and Applications, 2007, 541-567）. By the definition of the G-normal distribution, we first show that the G-Brownian motions can be simulated by solving a certain kind of Hamilton-Jacobi-Bellman （HJB） equations. Then, some finite difference methods are designed for the corresponding HJB equations. Numerical simulation results of the G-normal distribution, the G-Brownian motion, and the corresponding quadratic variation process are provided, which characterize basic properties of the G-Brownian motion. We believe that the algorithms in this work serve as a fundamental tool for future studies, e.g., for solving stochastic differential equations （SDEs）/stochastic partial differential equations （SPDEs） driven by the G-Brownian motions.

How big are the increments of G-Brownian motion?

In this paper,we investigate the problem:How big are the increments of G-Brownian motion.We obtain the Csrg and R′ev′esz’s type theorem for the increments of G-Brownian motion.As applications of this result,we get the law of iterated logarithm and the Erds and R′enyi law of large numbers for G-Brownian motion.Furthermore,it turns out that our theorems are natural extensions of the classical results obtained by Csrg and R′ev′esz(1979).

Differentiability of stochastic differential equations driven by the G-Brownian motion

In this paper,we study the differentiability of the solutions of stochastic differential equations driven by the G-Brownian motion with respect to the initial data and the parameter.

Harnack Inequality and Applications for SDEs Driven by G-Brownian Motion

In this paper,Wang's Harnack and shift Harnack inequality for a class of stochastic differential equations driven by G-Brownian motion are established.The results generalize the ones in the linear expectation setting.Moreover,some applications are also given.

Reflected stochastic differential equations driven by G-Brownian motion with nonlinear resistance

We study the uniqueness and existence of solutions of reflected G-stochastic differential equations （RGSDEs） with nonlinear resistance under an integral-Lipschitz condition of coefficients. Moreover, we obtain the comparison theorem for RGSDEs with nonlinear resistance.

The Cocycle Property of Stochastic Differential Equations Driven by G-Brownian Motion

In this paper,solutions of the following non-Lipschitz stochastic differential equations driven by G-Brownian motion：Xt=x＋∫^t0b（s,w,Xs）ds＋∫^t0h（s,ω,Xs）ds＋∫^t0σ（s,ω,Xs）dBs are constructed.It is shown that they have the cocycle property.Moreover,under some special non-Lipschitz conditions,they are bi-continuous with respect to t,x.

Convergence to a self-normalized G-Brownian motion

G-Brownian motion has a very rich and interesting new structure that nontrivially generalizes the classical Brownian motion.Its quadratic variation process is also a continuous process with independent and stationary increments.We prove a self-normalized functional central limit theorem for independent and identically distributed random variables under the sub-linear expectation with the limit process being a G-Brownian motion self-normalized by its quadratic variation.To prove the self-normalized central limit theorem,we also establish a new Donsker’s invariance principle with the limit process being a generalized G-Brownian motion.

Stochastic calculus with respect to G-Brownian motion viewed through rough paths

We study rough path properties of stochastic integrals of Ito's type and Stratonovich's type with respect to G-Brownian motion. The roughness of G-Brownian motion is estimated and then the pathwise Norris lemma in G-framework is obtained.

Optimal control with delayed information flow of systems driven by G-Brownian motion

In this paper,we study strongly robust optimal control problems under volatility uncertainty.In the G-framework,we adapt the stochastic maximum principle to find necessary and sufficient conditions for the existence of a strongly robust optimal control.

The Support of the Solution for Stochastic Differential Equations Driven by G-Brownian Motion

By a linear interpolation approximation method, we obtain a characterization of the support of the solution for stochastic differential equations driven by G-Brownian motion.

Reflected Quadratic BSDEs Driven by G-Brownian Motions

In this paper,the authors consider a reflected backward stochastic differential equation driven by a G-Brownian motion(G-BSDE for short),with the generator growing quadratically in the second unknown.The authors obtain the existence by the penalty method,and some a priori estimates which imply the uniqueness,for solutions of the G-BSDE.Moreover,focusing their discussion at the Markovian setting,the authors give a nonlinear Feynman-Kac formula for solutions of a fully nonlinear partial differential equation.

由G-布朗运动驱动的具有一致连续性生成元的BSDE的解的极限定理

研究了由G-布朗运动驱动的具有一致连续性生成元的倒向随机微分方程的解的极限定理,并由此得到了该方程的逆比较定理.

局部非利普希茨条件下G-随机微分方程的解的逼近

考虑了一类由G布朗运动驱动的随机微分方程,在其参数满足局部非利普希茨条件下,采用逐步逼近的方法,得到了方程的局部解的存在性和唯一性.

由G-布朗运动驱动的某些欧式期权的定价

主要介绍G期望,并利用G-几何布朗运动描述标的资产的价格变动,进而得到带有常数红利率的欧式看涨期权和欧式看涨幂期权的动态定价公式。