The cocycle property of stochastic differential equations driven by G-Brownian motion

Abstract

In this paper, solutions of the following non-Lipschitz stochastic differential equations driven by G-Brownian motion:σ(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.

Path independence of the additive functionals for stochastic differential equations driven by G-Lévy processes

“…So, X is a d-dimensional G-Lévy process. Although the Itô integrals with respect to G-Brownian motions have been introduced in [8], we need to introduce two related spaces used in the sequel. Take 0 = t…”

“…Recently, development of mathematical finance forces the appearance of a type of processes-G-Brownian motions( [5]). And then the related theory, such as stochastic calculus and stochastic differential equations (SDEs in short) driven by G-Brownian motions, are widely studied( [1,5,6,8]). However, in some financial models, volatility uncertainty makes G-Brownian motions insufficient for simulating these models.…”

Exponential Stability of Solutions to Stochastic Differential Equations Driven by G-Lévy Process

“…In [4], the author proved the BDG inequality for G-stochastic calculus with respect to G-Brownian motion. In this article, we will prove the BDG-type inequality for G-stochastic calculus with respect to G-Lévy Process, which will be used in section 3. In [20] and [1] the authors considered the stochastic differential equations driven by G-Brownian motion, where the coefficients do not satisfy the Lipschitz condition. Motivated by the aforementioned works, in section 3, the following stochastic differential equations driven by G-Lévy process (GSDEs) is studied:…”

“…In [20] and [1] the authors considered the stochastic differential equations driven by G-Brownian motion, where the coefficients do not satisfy the Lipschitz condition. Motivated by the aforementioned works, in section 3, the following stochastic differential equations driven by G-Lévy process (GSDEs) is studied:…”