Capacity and Quasi-surely Analysis for G-Brownian Paths

Abstract

In the last three chapters, we have considered random variables which are elements in a Banach space $L^p_G(\Omega )$. A natural question is whether such elements $\xi$ are still real functions defined on $\Omega$, namely $\xi =\xi (\omega )$, $\omega \in \Omega$.

Notes and Comments

Choquet capacity was first introduced by Choquet [33] , see also Dellacherie [42]  and the references therein for more properties. The capacitability of Choquet capacity was first studied by Choquet [33] under 2-alternating case, see Dellacherie and Meyer [43], Huber and Strassen [89] and the references therein for more general case. It seems that the notion of upper expectations was first discussed by Huber [88] in robust statistics. It was rediscovered in mathematical finance, especially in risk measure, see Delbaen [44, 45], Föllmer and Schied [68] etc.

The fundamental framework of quasi-surely stochastic analysis in this chapter is due to Denis and Martini [48]. The results of Sects. 6.1–6.3 for G-Brownian motions were mainly obtained by Denis, Hu and Peng [47] . The upper probability in Sect. 6.1 was firstly introduced in [47] in the framework of G-expectation. Note that the results established in [47] cannot be obtained by using outer capacity introduced by Denis and Martini [48]. In fact the outer capacity may be strictly bigger than the inner capacity which coincides with the upper probability in Definition 6.1.3, see Exercise 6.5.2. An interesting open problem is to prove, or disprove, whether the outer capacity is equal to the upper probability associated with $\mathcal {P}$.

Hu and Peng [80] have introduced an intrinsic and simple approach. This approach can be regarded as a combination and extension of the construction approach of Brownian motion of Kolmogorov (for more general stochastic processes) and a sort of cylinder Lipschitz functions technique already introduced in Chap. 3. Section 6.1 is from [47] and Theorem 6.2.5 was firstly obtained in the same paper, whereas contents of Sects. 6.2 and 6.3 are mainly from [80].

Section 6.4 is mainly based on Hu et al. [85]. Some related discussions can be found in Song [159].