G-Brownian Motion Research Articles

A note on sufficient conditions of asymptotic stability in distribution of stochastic differential equations with [formula omitted]-Brownian motion

This paper investigates a sufficient condition of asymptotic stability in distribution of stochastic differential equations driven by G-Brownian motion (G-SDEs). We define the concept of asymptotic stability in distribution under sublinear expectations. Sufficient criteria of the asymptotic stability in distribution based on sublinear expectations are given. Finally, an illustrative example is provided.

Pathwise convergence under Knightian uncertainty

In this note, we will obtain the first quasi-surely convergence rate of approximation of stochastic differential equations driven by G-Brownian motion. We carry it out by making a relation between Lp and quasi-surely convergences by considering this point that in general Lp convergence does not imply quasi-surely one and neither does quasi-surely convergence. This result will show that the rate of quasi-surely convergence can not exceed that of p-th mean.

Stability Analysis of Stochastic Differential Equation Driven by G-Brownian Motion under Non-Lipschitz Condition

This paper is devoted to studying the p-th moment exponential stability for a class of stochastic differential equation (SDE) driven by G-Brownian motion under non-Lipschitz condition. The delays considered in this paper are time-varying delays τ i t 1 ≤ i ≤ 3 . Since the coefficients are non-Lipschitz, the normal enlargement on the coefficients is not available and the Gronwall inequality is not suitable in this case. By Bihari inequality and Itô integral formula, it is pointed out that there exists a constant τ ∗ such that the p-th moment exponential stability holds if the time-varying delays are smaller than τ ∗ .

Pricing interest rate derivatives under volatility uncertainty

In this paper, we study the pricing of contracts in fixed income markets under volatility uncertainty in the sense of Knightian uncertainty or model uncertainty. The starting point is an arbitrage-free bond market under volatility uncertainty. The uncertainty about the volatility is modeled by a G-Brownian motion, which drives the forward rate dynamics. The absence of arbitrage is ensured by a drift condition. Such a setting leads to a sublinear pricing measure for additional contracts, which yields either a single price or a range of prices and provides a connection to hedging prices. Similar to the forward measure approach, we define the forward sublinear expectation to simplify the pricing of cashflows. Under the forward sublinear expectation, we obtain a robust version of the expectations hypothesis, and we show how to price options on forward prices. In addition, we develop pricing methods for contracts consisting of a stream of cashflows, since the nonlinearity of the pricing measure implies that we cannot price a stream of cashflows by pricing each cashflow separately. With these tools, we derive robust pricing formulas for all major interest rate derivatives. The pricing formulas provide a link to the pricing formulas of traditional models without volatility uncertainty and show that volatility uncertainty naturally leads to unspanned stochastic volatility.

Practical Stability with Respect to a Part of the Variables of Stochastic Differential Equations Driven by G-Brownian Motion

Practical Stability with Respect to a Part of the Variables of Stochastic Differential Equations Driven by G-Brownian Motion

Stability with respect to a part of the variables of stochastic nonlinear systems driven by G-Brownian motion

In this paper, we investigate the pth moment exponential stability of stochastic differential equations driven by G-Brownian motion (G-SDEs) with respect to a part of the variables by means of the G-Lyapunov functions and recently developed Itô's calculus for SDEs driven by G-Brownian motion, as well as Gronwall's inequalities. We establish sufficient conditions to ensure the quasi sure exponential stability of stochastic differential equations perturbed by G-Brownian motion with respect to a part of the variables. Some illustrative examples to show the usefulness of the stability with respect to a part of the variables notion are also provided.

Derivation and Application of Some Fractional Black–Scholes Equations Driven by Fractional G-Brownian Motion

Derivation and Application of Some Fractional Black–Scholes Equations Driven by Fractional G-Brownian Motion

Practical exponential stability of stochastic delayed systems with G-Brownian motion via vector G-Lyapunov function

This paper deals with practical stability problem for nonlinear stochastic delayed systems with G-Brownian motion (GSDSs). Practical stability can describe qualitative behavior and quantitative properties of systems in comparison with traditional Lyapunov stability theory. By employing stochastic analysis technique, Razumikhin-type theorem and vector G-Lyapunov function, new sufficient conditions for pth moment practical exponential stability of GSDSs are proposed. Finally, two examples are presented to verify the feasibility of theoretical results.

Delay-dependent Asymptotic Stability of Highly Nonlinear Stochastic Differential Delay Equations Driven by [formula omitted]-Brownian Motion

Based on the classical probability, the stability of stochastic differential delay equations (SDDEs) whose coefficients are growing at most linearly has been investigated intensively. Moreover, the delay-dependent stability of highly nonlinear hybrid stochastic differential equations (SDEs) has also been studied recently. In this paper, using the nonlinear expectation theory, we first explore the delay-dependent criteria on the asymptotic stability for a class of highly nonlinear SDDEs driven by G-Brownian motion (G-SDDEs). Then, the (weak) quasi-sure stability of solutions to G-SDDEs is developed. Finally, an example is analyzed by the φ-max-mean algorithm to illustrate our theoretical results.

Practical stability of impulsive stochastic delayed systems driven by G-Brownian motion

This paper investigates practical stability problem for nonlinear impulsive stochastic delayed systems driven by G-Brownian motion (IGSDSs). Practical stability can describe quantitative properties and qualitative behavior in contrast to traditional Lyapunov stability theory. Based on G-Lyapunov function, Razumikhin-type theorem, G-Itô formula, Burkholder–Davis–Gundy (B-D-G) inequalities I & II and stochastic analysis technique, some new criteria for moment and quasi sure global practical uniform exponential stability of IGSDSs are proposed. Finally, two examples are presented to verify validity of our theoretical results.

Practical stability in relation to a part of variables for stochastic reaction–diffusion systems driven by G-Brownian motion

The main purpose of this paper is to investigate the practical stability in relation to a part of the variables of stochastic reaction–diffusion systems driven by G-Brownian motion (G-SRDSs, in short). With the aid of G-Lyapunov techniques and G-Itô's formula, the criteria of the global practical uniform kth moment exponential stability and the global practical uniform quasi surely exponential stability in relation to a part of variables of G-SRDSs are established. An example is given to illustrate the usefulness and feasibility of the proposed results.

How small are the increments of [formula omitted]-Brownian motion

In this paper we study the problem: How small are the increments of G-Brownian motion? We establish the Csörgő and Révész’s type theorem for the increments of G-Brownian motion.

Comparison theorem for neutral stochastic functional differential equations driven by [formula omitted]-Brownian motion

In this paper, we investigate sufficient and necessary conditions for the comparison theorem of neutral stochastic functional differential equations driven by G-Brownian motion (G-NSFDE). Moreover, the results extend the ones in the linear expectation case (Bai and Jiang, 2016) and nonlinear expectation framework (Huang and Yang, 2021).

Qualitative analysis of A.P.A. solution for fractional order neutral stochastic evolution equations driven by G-Brownian motion

In this paper, we will analyses the square mean almost pseudo automorphic mild solution for fractional order equation, (1) c 0 D α γ [ℵ(γ)−D(γ,ℵ(γ))] = [Aℵ(γ) +φ(γ,ℵ(γ))]dγ + ϕ(γ,ℵ(γ))d<B>(γ) + ψ(γ,ℵ(γ))dB(γ), γ∈R where A(γ) : D(A(γ)) ⊂ L 2 G (F) → L 2 G (F) is densely closed linear operator and the functions D, φ, ϕ and ψ: L 2 G (F) → L 2 G (F) are jointly continuous. We drive square mean almost pseudo automorphic mild solution for fractional order neutral stochastic evolution equations driven by G-Brownian motion is obtain by using evolution operator theorem and fixed point theorem. Moreover, we prove that this mild solution of equation (1) is unique.

Large Deviation Principle for Reflected Stochastic Differential Equations Driven by G-Brownian Motion in Non-Convex Domains

Large Deviation Principle for Reflected Stochastic Differential Equations Driven by G-Brownian Motion in Non-Convex Domains

Two Theorems of Multiple <i>G</i>-Itô Integral under <i>G</i>-Lévy Process

In this paper, according to G-Brownian motion and other related concepts and properties, we define multiple Itô integrals driven by G-Brownian motion and G-Lévy process. By using the G-Itô formula and the properties of G-expectation, two main theorems about Itô integral are obtained and proved. These two theorems provide powerful help for the subsequent research on jump process.

An averaging principle for nonlinear parabolic PDEs via FBSDEs driven by G-Brownian motion

In this paper, we study the averaging problem for a class of forward-backward stochastic differential equations driven by G-Brownian motion (G-FBSDEs) with rapidly oscillating coefficients, which corresponds to the singular perturbation problem of a kind of fully nonlinear partial differential equations (PDEs). With the help of the nonlinear stochastic analysis techniques and viscosity solution methods, we prove that the limit distribution of the solution is the unique viscosity solution to a fully nonlinear PDE.

Term structure modeling under volatility uncertainty

In this paper, we study term structure movements in the spirit of Heath et al. (Econometrica 60(1):77–105, 1992) under volatility uncertainty. We model the instantaneous forward rate as a diffusion process driven by a G-Brownian motion. The G-Brownian motion represents the uncertainty about the volatility. Within this framework, we derive a sufficient condition for the absence of arbitrage, known as the drift condition. In contrast to the traditional model, the drift condition consists of several equations and several market prices, termed market price of risk and market prices of uncertainty, respectively. The drift condition is still consistent with the classical one if there is no volatility uncertainty. Similar to the traditional model, the risk-neutral dynamics of the forward rate are completely determined by its diffusion term. The drift condition allows to construct arbitrage-free term structure models that are completely robust with respect to the volatility. In particular, we obtain robust versions of classical term structure models.

BSDEs driven by G-Brownian motion with non-Lipschitz coefficients

In this paper, we establish the existence and uniqueness theorem of backward stochastic differential equations driven by G-Brownian motion under β-order Mao's condition which is weaker than the Lipschitz one with the help of Picard iteration. Moreover, the corresponding comparison theorem and stability theorem are also obtained.

A stochastic epidemic model with G-Brownian motion

In this paper, we propose a new stochastic epidemic model as follows where The model embodies both the mean and variance uncertainty. It can be seen as an extension of the existing model. It shows a unique positive solution in the sense of quasi surely and will asymptotically tend to zero quasi surely and is pth-moment stable.