Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation

By Shige Peng. 2006

638 citations recorded by the source corpus.

We develop a notion of nonlinear expectation-G-expectation-generated by a nonlinear heat equation with infinitesimal generator G. We first study multi-dimensional G-normal distributions. With this nonlinear distribution we can introduce our G-expectation under which the canonical process is a multi-dimensional G-Brownian motion. We then establish the related stochastic calculus, especially stochastic integrals of Ito's type with respect to our G-Brownian motion, and derive the related Ito's formula. We have also obtained the existence and uniqueness of stochastic differential equations under our G-expectation.

Subfields: Economics and Econometrics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/23ff74f99812616d96df5abb2bf3675ade10b67e

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Citations in the corpus

Comparison Theorem, Feynman-Kac Formula and Girsanov Transformation for BSDEs Driven by G-Brownian Motion (2012, background): “The readers may refer to [7], [19], [20], [21], [22], [24] for more details.”
Pathwise properties and homeomorphic flows for stochastic differential equations driven by G-Brownian motion (2009, methodology): “The G-Itô’s formula in [11] is extended by the localization method.”
Risk measuring under model uncertainty (2010, background): “Peng introduced the notion of sublinear expectation and of G-expectations [26] [27] defined on a vector lattice H of real functions containing 1 and included in Cb(Ω).”
Random G-expectations. (2010, background): “The author thanks Shige Peng, Mete Soner and Nizar Touzi for stimulating discussions as well as Laurent Denis, Sebastian Herrmann and the anonymous referees for helpful comments.”
Nonlinear Lévy Processes and their Characteristics (2016, background): “Keywords Nonlinear Lévy process; Sublinear expectation; Partial integro-differential equation; Semimartingale characteristics; Knightian uncertainty AMS 2010 Subject Classification 60G51; 60G44; 93E20”
The p-th moment stability of solutions to impulsive stochastic differential equations driven by G-Brownian motion (2017, background): “Introduction Since Peng [1] introduced the theory of G-expectation and its related stochastic calculus, they have attracted researchers’ great interest for their potential applications in uncertainty problems, risk measures, superhedging in finance and so on.”
A New Central Limit Theorem under Sublinear Expectations (2008, background): “Recently problems of model uncertainties in statistics, measures of risk and superhedging in finance motivated us to introduce, in [13] and [14] (see also [11], [12] and references herein), a new notion of sublinear expectation, called “G-expectation”, and the related “G-normal distribution” (see De”
Normal approximation by Stein’s method under sublinear expectations (2017, background): “05 38 4v 1 [ m at h. PR ] 1 5 N ov Peng (2008)([12]) proved the Central Limit Theorem under a sublinear expectation: Let (Xi)i≥1 be a sequence of i.i.d random variables under a sublinear expectation Ê with Ê[X1] = Ê[−X1] = 0 and Ê[|X1”
Strong limit theorems for extended independent and extended negatively dependent random variables under non-linear expectations (2016, methodology): “…of this paper is to study the strong law of large numbers and the law of the iterated logarithm for a sequence of random variables in a sub-linear expectation space under a concept of extended independence which is much weaker and easier to verify than the independence proposed by Peng (2008b).”
Donsker’s Invariance Principle Under the Sub-linear Expectation with an Application to Chung’s Law of the Iterated Logarithm (2015, methodology): “Roughly speaking, a sub-linear expectation is related to a group of unknown linear expectations and the distribution under a sub-linear expectation is related to a group of probabilities (c.f. Lemma 2.4 of Peng (2008b)).”
Uniqueness of the representation for $G$-martingales with finite variation (2010, background): “More details of this section can be found in [4, 5, 6, 7].”
Backward Stochastic Differential Equation, Nonlinear Expectation and Their Applications (2011, methodology): “Multidimensional case can be found in Peng (2008a, 2010a).”
Stopping times and related Itô's calculus with G-Brownian motion (2009, methodology): “If ξ ∈ Cb(Ωt), then by [Peng], we have the following Itô’s formula: ξ2[(BT −Bt)2 − (〈BT 〉 − 〈Bt〉)] = 2 ∫ T t ξ2BsdBs.”
G-Lévy processes under sublinear expectations (2009, background): “This new type of sublinear distribution was first introduced in [16] (see also [17−20]) for a new type of G-Brownian motion and the related calculus of Itô's type.”
Quasi-continuous random variables and processes under the G-expectation framework (2014, methodology): “The readers may refer to [11], [12], [13] [14] for more details.”
Rosenthal’s inequalities for independent and negatively dependent random variables under sub-linear expectations with applications (2014, methodology): “Deriniton 1.2 (Peng (2006,2008b)) (i) (Identical distribution) Let X1 and X2 be two n-dimensional random vectors defined respectively in sub-linear expectation spaces (Ω1,H1, Ê1) and (Ω2,H2, Ê2).”