G-Brownian Motion and Itô’s Calculus

Abstract

The aim of this chapter is to introduce the concept of G-Brownian motion, study its properties and construct Itô’s integral with respect to G-Brownian motion. We emphasize here that this G-Brownian motion $B_t$, $t\ge 0$ is consistent with the classical one.

Notes and Comments

Bachelier [7]  proposed to use the Brownian motion as a model of the fluctuations of stock markets. Independently, Einstein [56]  used the Brownian motion to give experimental confirmation of the atomic theory, and Wiener [173] gave a mathematically rigorous construction of the Brownian motion. Here we follow Kolmogorov’s idea [103] to construct G-Brownian motions by introducing finite dimensional cylinder function space and the corresponding family of infinite dimensional sublinear distributions, instead of (linear) probability distributions used in [103].

The notions of G-Brownian motions and the related stochastic calculus of Itô’s type were firstly introduced by Peng [138]  for the 1-dimensional case and then in (2008) [141] for the multi-dimensional situation. It is very interesting that Denis and Martini [48] studied super-pricing of contingent claims under model uncertainty of volatility. They have introduced a norm in the space of continuous paths $\Omega =C([0,T])$ which corresponds to the $L^2_G$-norm and developed a stochastic integral. In that paper there are no notions such as nonlinear expectation and the related nonlinear distribution, G-expectation, conditional G-expectation, the related G-normal distribution and independence. On the other hand, by using powerful tools from capacity theory these authors obtained pathwise results for random variables and stochastic processes through the language of “quasi-surely” (see e.g. Dellacherie [42], Dellacherie and Meyer [43], Feyel  and de La Pradelle [65]) in place of “almost surely” in classical probability theory.

One of the main motivations to introduce the notion of G-Brownian motions was the necessity to deal with pricing and risk measures under volatility uncertainty in financial markets (see Avellaneda, Lévy and Paras [6]  and Lyons [114]). It was well-known that under volatility uncertainty the corresponding uncertain probability measures are singular with respect to each other. This causes a serious problem in the related path analysis to treat, e.g., when dealing with path-dependent derivatives, under a classical probability space. The notion of G-Brownian motions provides a powerful tool to study such a type of problems. Indeed, Biagini, Mancin and Meyer Brandis studied mean-variance hedging under the G-expectation framework in [18]. Fouque, Pun and Wong investigated the asset allocation problem among a risk-free asset and two risky assets with an ambiguous correlation through the theory of G-Brownian motions in [67]. We also remark that Beissner and Riedel [15] studied equilibria under Knightian price uncertainty through sublinear expectation theory, see also [14, 16].

The new Itô’s calculus with respect to G-Brownian motion was inspired by Itô’s groundbreaking work of [92] on stochastic integration, stochastic differential equations followed by a huge progress in stochastic calculus. We refer to interesting books cited in Chap. 4. Itô’s formula given by Theorem 3.6.5 is from [138, 141]. Gao [72]  proved a more general Itô’s formula for G-Brownian motion. On this occasion an interesting problem appeared: can we establish an Itô’s formula under conditions which correspond to the classical one? This problem will be solved in Chap. 8 with quasi surely analysis approach.

Using nonlinear Markovian semigroups known as Nisio’s semigroups (see Nisio [119]), Peng [136] studied the processes with Markovian properties under a nonlinear expectation. Denk, Kupper and Nendel studied the relation between Lévy processes under nonlinear expectations, nonlinear semigroups and fully nonlinear PDEs, see [50].