Function Spaces and Capacity Related to a Sublinear Expectation: Application to G-Brownian Motion Paths

Abstract

In this paper we give some basic and important properties of several typical Banach spaces of functions of G-Brownian motion paths induced by a sublinear expectation-G-expectation. Many results can be also applied to more general situations. A generalized version of Kolmogorov's criterion for continuous modification of a stochastic process is also obtained. The results can be applied in continuous time dynamic and coherent risk measures in finance, in particular for path-dependence risky positions under situations of volatility model uncertainty.

Martingale representation theorem for theG-expectation

“…Similar to [4], we also provide a dual characterization of G-martingales as an immediate consequence of the results in [4,17]. This observation is one of the key-ingredients of our representation proof.…”

“…Our analysis utilizes the already mentioned duality result of Denis, Hu and Peng [4]. Similar to [4], we also provide a dual characterization of G-martingales as an immediate consequence of the results in [4,17].…”

“…We remark that Denis and Martini [3] uses the space P W [a,a] . But Denis, Hu and Peng [4] and our subsequent work [21] …”

“…A characterization of this space, in particular a Lusin type theorem, is obtained in [4]. However, since these integrability classes are defined through the closure of a rather smooth space L ip , they require substantial "smoothness".…”

“…However, they use a quite different approach which utilizes the set P of all probability measures P so that the canonical map in the Wiener space is a martingale under P and the quadratic variation of this martingale lies between a ≤ a. Although the constructions of the quasi sure analysis and the G-expectations are substantially different, these theories are very closely related as proved recently by Denis, Hu & Peng [4]. The paper [4] also provides a dual representation of the G-expectation as the supremum of expectations over P. This duality and more generally the dynamic programming principle is generalized by Nutz [13] who considers lower and upper bounds a, a that are random processes.…”