Quasi-continuous random variables and processes under the G-expectation framework

Abstract

In this paper, we first use PDE techniques and probabilistic methods to identify a kind of quasi-continuous random variables. Then we give a characterization of the G-integrable processes and get a kind of quasicontinuous processes by Krylov's estimates. This result is useful for the development of G-stochastic analysis theory. Moreover, it also provides a tool for the study of the non-Markovian Itô processes.

Lévy's martingale characterization and reflection principle of G-Brownian motion

“…Similar to Theorems 4.15 and 4.16 in [4], we have the following proposition which contains the case σ 2 = 0. The proof is omitted.…”

Backward Stochastic Differential Equations Driven by G-Brownian Motion with Uniformly Continuous Generators

“…which, together with Theorem 4.16 in [10] and Lemma 4.1, implies that both ϕ n (., X t,x . , y, z) and ϕ n (., X t,x .…”

“…Note that |f 0 (s)| + 2L|Y s | + L|Z n s | + L|Z s | ∈ M 2 G (0, T ). Thus with the help of Theorem 4.7 in [10], we get that…”

Theory, methods and meaning of nonlinear expectation theory

“…事实上, 在非线性期望框架下控制收敛定 理不再自动成立, 从而产生一些极具挑战性的且有实际背景的研究问题. 为此, 文献 [64] 通过偏微分 方程技术给出了 Ω 上的一大类拟连续的简单函数, 这说明 G-期望空间具有充分多的元素和丰富的 结构.…”