G-Brownian Motion Research Articles

The Cox-Ingersoll-Ross process under volatility uncertainty

Due to the significance of the Cox-Ingersoll-Ross process in various areas of finance, a wide range of studies and investigations on this model have been carried out. In cases of ambiguity, we characterize it by applying the theory of G-expectation and the associated G-Brownian motion. In this paper, we establish the existence and uniqueness of the solution for the Cox-Ingersoll-Ross process in the presence of volatility uncertainty. In addition, certain properties of the solution are indicated, such as regularity and the strong Markov property. Furthermore, we compute some moments of the Cox-Ingersoll-Ross process by employing an extension of the nonlinear Feynman-Kac theorem.

Reflected Backward Stochastic Differential Equations Driven by G-Brownian Motion Under Monotonicity Condition

Reflected Backward Stochastic Differential Equations Driven by G-Brownian Motion Under Monotonicity Condition

Quadratic BSDEs with mean reflection driven by G-brownian motion

In this paper, the well-posedness of a kind of backward stochastic differential equation driven by [Formula: see text]-Brownian motions ([Formula: see text]-BSDEs for short) with mean reflection is investigated. In particular, the generator is allowed having quadratic growth in [Formula: see text]. Combining a representation of the solution with [Formula: see text]-BMO martingale techniques, fixed point argument, and [Formula: see text]-method, existence and uniqueness results for such [Formula: see text]-BSDEs are provided under bounded and unbounded terminal condition.

Mean square exponential stability of stochastic function differential equations in the G-framework

Abstract This research focuses on the stochastic functional differential equations driven by G-Brownian motion (G-SFDEs) with infinite delay. It is proved that the trivial solution of a G-SFDE with infinite delay is exponentially stable in mean square. An example is also presented to illustrate the effectiveness of the obtained theory.

Nonparametric Estimation of Trend for Stochastic Processes Driven by G-Brownian Motion with Small Noise

Nonparametric Estimation of Trend for Stochastic Processes Driven by G-Brownian Motion with Small Noise

Quasi-Sure Consensus of Multiagent Systems: G-Noise Communication Protocols Under DoS Attacks

This paper addresses the quasi sure consensus problem for multi-agent systems (MASs) in the context of Denial-of-Service (DoS) attacks. Instead of adopting the conventional noise-based protocols driven by Brownian motion, a novel G-noise communication protocol is first proposed to realize the consensus. Moreover, the consensus protocol not only contains time-varying and nonlinear characteristics, but also focuses on DoS attacks. The protocol driven by G-Brownian motion is characterized in the sublinear expectation framework, which leads to the infeasibility of the existing theory. To overcome the difficulty, a new theoretical framework is established to develop consensus criterion. More specifically, first, the non-zero property of MASs is introduced to ensure the rationality of the construction for the Lyapunov function. Then a sufficient criterion for guaranteeing the consensus of MASs is demonstrated by utilizing Lyapunov method, graph theory, stochastic analysis theory, which depends on DoS duration, noise intensity and G-Brownian motion parameters. Ultimately, theoretical results are associated with islanded microgrid systems with a numerical simulation being exploited.

Mean-Square Stability of Uncertain Delayed Stochastic Systems Driven by G-Brownian Motion

This paper investigates the mean-square stability of uncertain time-delay stochastic systems driven by G-Brownian motion, which are commonly referred to as G-SDDEs. To derive a new set of sufficient stability conditions, we employ the linear matrix inequality (LMI) method and construct a Lyapunov–Krasovskii function under the constraint of uncertainty bounds. The resulting sufficient condition does not require any specific assumptions on the G-function, making it more practical. Additionally, we provide numerical examples to demonstrate the validity and effectiveness of the proposed approach.

Time-averaging principle for G-SDEs based on Lyapunov condition

In this paper, we tame the uncertainty about the volatility in time-averaging principle for stochastic differential equations driven by G-Brownian motion (G-SDEs) based on the Lyapunov condition. That means we treat the time-averaging principle for stochastic differential equations based on the Lyapunov condition in the presence of a family of probability measures, each corresponding to a different scenario for the volatility. The main tool for the mathematical analysis is the G-stochastic calculus, which is introduced in the book by Peng (Nonlinear Expectations and Stochastic Calculus Under Uncertainty. Springer, Berlin, 2019). We show that the solution of a standard equation converges to the solution of the corresponding averaging equation in the sense of sublinear expectation with the help of some properties of G-stochastic calculus. Numerical results obtained using PYTHON illustrate the efficiency of the averaging method.

Practical stability analysis of stochastic functional differential systems with G-Brownian motion and impulsive effects

This article is concerned with the practical stability performance of nonlinear impulsive stochastic functional differential systems driven by G-Brownian motion (G-ISFDSs). Comparing with traditional Lyapunov stability theory, practical stability can portray qualitative behavior and quantitative properties of suggested systems. By employing G-Itô formula, Lyapunov-Razumikhin approach and stochastic analysis theory, some novel conditions for pth moment practical exponential stability and quasi sure global practical uniform exponential stability of G-ISFDSs are established. The obtained results show that impulses may influence dynamic behavior of the addressed system. Two numerical examples are given to verify the validity of our developed results.

A deviation inequality for increment of a [formula omitted]-Brownian motion under [formula omitted]-expectation and applications

In this paper, we prove a deviation inequality for increments of a G-Brownian motion under G-expectation. As an application, we obtain a functional modulus of continuity for a G-Brownian motion.

Forward-backward stochastic differential equations driven by G-Brownian motion under weakly coupling condition

In this paper, we obtain the existence and uniqueness theorem of Lp-solution for coupled forward-backward stochastic differential equations driven by G-Brownian motion (G-FBSDEs) with arbitrary T under weakly coupling condition. Specially, the result for p∈(1,2) is completely different from the one for p≥2. Furthermore, by considering the dual linear FBSDE under a suitable reference probability, we establish the comparison theorem for G-FBSDEs under weakly coupling condition.

Asymptotic Stability in Distribution of Highly Nonlinear Stochastic Differential Equations with G-Brownian Motion

Asymptotic Stability in Distribution of Highly Nonlinear Stochastic Differential Equations with G-Brownian Motion

A Generalized Stochastic Process: Fractional G-Brownian Motion

A Generalized Stochastic Process: Fractional G-Brownian Motion

Fractional Stochastic Differential Equations Driven By G-Brownian Motion with Delays

Fractional Stochastic Differential Equations Driven By G-Brownian Motion with Delays

On the capacity for degenerated G-Brownian motion and its application

On the capacity for degenerated G-Brownian motion and its application

Stability of stochastic Hopfield neural networks driven by [formula omitted]-Brownian motion with time-varying and distributed delays

In this paper, pth exponential stability and quasi-surely exponential stability of stochastic Hopfield neural networks driven by G-Brownian motion are investigated. Under a sublinear expectation framework, we give several lemmas related to the Halanay inequality and the Burkholder-Davis-Gundy ineqaulity. Our stability results is based on the class of the multidimensional Halanay inequalities, as well as the Burkholder-Davis-Gundy inequalities. The main originality lies in the fact that we consider Knightian uncertainty of the theory model of stochastic Hopfield neural networks. Finally, two numerical examples are presented to illustrate our new theory.

Boundedness analysis of stochastic distributed delay-coupled systems on networks driven by G-Brownian motion

The present article is devoted to discuss the pth moment exponential ultimate boundedness for a class of stochastic distributed delay-coupled systems on networks driven by G-Brownian motion. By combining the method of proof by contradiction with Lyapunov method and graph theory, some sufficient condition for both the Lyapunov-type theorem and the coefficients-type theorem are proposed to ensure the pth moment exponential ultimate boundedness of the considered systems. As applications, the boundedness criteria are applied to stochastic distributed delay-coupled neural networks and stochastic distributed delay-coupled control systems in the framework of G-Brownian motion.

Large deviation principle for Reflected Stochastic Differential Equations driven by G-Brownian motion in non-convex domains

In this paper, we establish a large deviation principle for the solution of Reflected Stochastic Differential Equations driven by G-Brownian motion in non-convex domains. Moreover, we prove that the solution converges to the solution of a deterministic Skorohod equation.

Almost sure exponential stability of nonlinear stochastic delay hybrid systems driven by G-Brownian motion

G-Brownian motion has potential applications in uncertainty problems and risk measures, which has attracted the attention of many scholars. This study investigates the almost sure exponential stability of nonlinear stochastic delay hybrid systems driven by G-Brownian motion. Due to the non-linearity of G-expectation and distribution uncertainty of G-Brownian motion, it is difficult to study this issue. Firstly, the existence of the global unique solution is derived under the linear growth condition and local Lipschitz condition. Secondly, the almost sure exponential stability of the system is analyzed by applying the G-Lyapunov function and G-Itô formula. Finally, an example is introduced to illustrate the stability. The conclusions of this paper can be applied to the stability and risk management of uncertain financial markets.

Reflected Forward-Backward Stochastic Differential Equations Driven by G-Brownian Motion with Continuous Monotone Coefficients

This paper is devoted to investigating the existence of solution to a class of reflected forward-backward stochastic differential equations driven by G-Brownian motion (G-RFBSDEs). We construct a solution to the equations by monotone convergence argument when the generator of backward equation and the drift of forward equation satisfy the monotonicity and uniformly continuous condition.