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.     

Martingale characterization of G-Brownian motion

In this paper, we study the martingale characterization of G-Brownian motion, which was defined by Peng (cf. http://abelsymposium.no/symp2005/preprints/peng.pdf) in 2006. As an application, we present a method for constructing a G-Brownian motion using a Markov chain. Furthermore, we obtain the representation theorem for some special symmetric martingales in the G-framework.     

Stochastic Differential Equations for Eigenvalues and Eigenvectors of a G-Wishart Process with Drift

Ukrainian Mathematical Journal - We propose a system of G-stochastic differential equations for the eigenvalues and eigenvectors of a G-Wishart process defined according to a G-Brownian motion...     

Wong–Zakai Approximation for Stochastic Differential Equations Driven by G-Brownian Motion

Journal of Theoretical Probability - In this paper, we build the Wong–Zakai approximation for Stratonovich-type stochastic differential equations driven by G-Brownian motion and obtain the...     

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

Acta Mathematicae Applicatae Sinica, English Series - In this paper, Wang’s Harnack and shift Harnack inequality for a class of stochastic differential equations driven by G-Brownian motion...     

Large Deviations for Backward Stochastic Differential Equations Driven by G-Brownian Motion

Journal of Theoretical Probability - In this paper, we consider forward–backward stochastic differential equation driven by G-Brownian motion (G-FBSDEs in short) with small parameter...     

Backward Stochastic Differential Equations Driven by G-Brownian Motion with Double Reflections

Journal of Theoretical Probability - In this paper, we study the reflected backward stochastic differential equations driven by G-Brownian motion with two reflecting obstacles, which means that the...     

Backward Stochastic Differential Equations Driven by G-Brownian Motion with Uniformly Continuous Coefficients in (y, z)

Journal of Theoretical Probability - In this paper, we investigate backward stochastic differential equations driven by G-Brownian motion with uniformly continuous coefficients in (y, z)....     

Asymptotic estimates for the solution of stochastic differential equations driven By G-Brownian motion

In this paper, we give four results of asymptotic estimates for the solution of stochastic differential equations driven by G-Brownian motion, which have different convergence rates under different assumptions. One of them can be considered as a Law of the Iterated Logarithm of the solution of G-SDEs under nonlinear conditions.     

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This paper concerns stochastic differential equations driven by G-Brownian motion under non-Lipschitz condition which is a much weaker condition with a wider range of applications. Stochastic averaging is established for such non-Lipschitz SDEs where an averaged system is presented to replace the original one in the sense of mean square. An example is presented to illustrate the averaging principle.     

Invariance principles for the law of the iterated logarithm under G-framework

We obtain a general invariance principle of G-Brownian motion for the law of the iterated logarithm(LIL for short). For continuous bounded independent and identically distributed random variables in G-expectation space, we also give an invariance principle for LIL. In some sense, this result is an extension of the classical Strassen's invariance principle to the case where probability measure is no longer additive. Furthermore,we give some examples as applications.     

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 coeficients 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.     

G-Brown运动驱动的中立型随机时滞微分方程的指数稳定性

研究了一类G-Brown运动驱动的中立型随机时滞微分方程的指数稳定性.在G-框架意义下,运用合适的Lyapunov-Krasovskii泛函,中立型时滞微分方程理论以及随机分析技巧,证明了所研究方程平凡解的p-阶矩指数稳定性,得到了所研究方程平凡解是p-阶矩指数稳定的充分条件.最后通过例子说明所得的结果.     

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.     

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:X_t=x+∫~t_0b(s,w,X_s)ds+∫~t_0h(s,ω,X_s)dBs+∫~t_0σ(s,ω,X_s)dB_s 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.     

Reflected solutions of backward stochastic differential equations driven by G-Brownian motion

In this paper, we study the reflected solutions of one-dimensional backward stochastic differential equations driven by G-Brownian motion. The reflection keeps the solution above a given stochastic process. In order to derive the uniqueness of reflected G-BSDEs, we apply a "martingale condition" instead of the Skorohod condition. Similar to the classical case, we prove the existence by approximation via penalization. We then give some applications including a generalized Feynman-Kac formula of an obstacle problem for fully nonlinear partial differential equation and option pricing of American types under volatility uncertainty.     

Martingale representation theorem for G-Brownian motion

This paper firstly discuss some related properties for a type of G-function by introducing a G-Brownian motion martingale. And a martingale representation theorem is then proved by using Multiple G-Itô integral appeared in [1 Peng, S. (2007). G-expectation G-Brownian motion and related stochastic calculus of Itô type, arXiv:math/0601035v2. [Google Scholar]].     

A Girsanov Type Theorem Under G-Framework

This article establishes a Girsanov type theorem under the G-Framework of Peng [15 Peng , S. 2006 . G-expectation, G-Brownian motion and related calculus of Ito's type. Available at: http://abelsymposium.no/symp2005/preprints/peng.pdf  [Google Scholar]]. Our result generalizes the classical Girsanov theorem for Brownian motion [10 Girsanov , I.V. 1960 . On transforming a certain class of stochastic processes by absolutely continuous substitution of measures . Theory Probability and Its Applications 5 : 285 – 301 .[Crossref] , [Google Scholar]]. As an application, we price the European call option when the underlying asset's price follows the Geometric G-Brownian motion.     

Periodic and chaotic dynamics in an asymmetric elastoplastic oscillator

We consider the dynamics of a harmonically forced oscillator with an asymmetric elastic–perfectly plastic stiffness function. The computed bifurcation diagrams for the oscillator show regions of periodic motion, hysteresis and large regions of chaotic motion. These different regions of dynamical behaviour are plotted in a two-dimensional parameter space consisting of forcing amplitude and forcing frequency. Examples of the chaotic motion encountered are shown using a discontinuity crossing map. Comparisons are made with the symmetric oscillator by computing a typical bifurcation diagram and considering previously published results for the symmetric system. From this we conclude that the asymmetric system is dominated by a large region of chaotic motion whereas in the symmetric oscillator period one motion and coexisting period three motion predominates.     

Evolution of the precessional motion of unbalanced gyrostats of variable structure

The precessional motion of an unbalanced gyrostat of variable structure when acted upon by dissipative and accelerating external and internal moments, which depend on the angular velocities of the bodies (the carrier and the rotor) is considered. A qualitative method of analysing the phase space of non-autonomous dynamical systems is developed, based on the determination of the curvature of the phase trajectory. The motion is analysed and the conditions for obtaining the required modes of nutational-precessional motion of unbalanced gyrostats of variable structure are synthesized using this method. A number of cases of the motion of a gyrostat of variable structure, including free motion, motion when there are constant internal and reactive moments and, also, under the action of the moments of resistance forces, proportional to the angular velocities, is investigated. The possible evolutions in the above-mentioned cases of motion and the causes of these evolutions are determined. The conditions for evolution with a decreasing amplitude of the nutational oscillations are obtained.