G-Brownian Motion and Dynamic Risk Measure under Volatility Uncertainty

By Shige Peng. 2007

228 citations recorded by the source corpus.

We introduce a new notion of G-normal distributions. This will bring us to a new framework of stochastic calculus of Ito's type (Ito's integral, Ito's formula, Ito's equation) through the corresponding G-Brownian motion. We will also present analytical calculations and some new statistical methods with application to risk analysis in finance under volatility uncertainty. Our basic point of view is: sublinear expectation theory is very like its special situation of linear expectation in the classical probability theory. Under a sublinear expectation space we still can introduce the notion of distributions, of random variables, as well as the notions of joint distributions, marginal distributions, etc. A particularly interesting phenomenon in sublinear situations is that a random variable Y is independent to X does not automatically implies that X is independent to Y. Two important theorems have been proved: The law of large number and the central limit theorem.

Subfields: Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/7c33d06c087b1f134f615bdbae1efa5d5b6fb2cb

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Citations in the corpus

Nonlinear Expectations and Stochastic Calculus under Uncertainty (2010, background): “This chapter is mainly from Peng (2007) [102].”
The maximum maximum of a martingale with given $\mathbf{n}$ marginals (2012, background): “…the non-dominated setup when P α may be mutually singular posed new challenges and was investigated starting with Avellaneda et al. [1] and Lyons [26], through Denis and Martini [16] to several recent works e.g. Peng [34], Soner, Touzi and Zhang [37], see also [19] and the references therein.”
Martingale Representation Theorem for the G-Expectation (2010, methodology): “1 Introduction The notion of a G-expectation as recently introduced by Peng [16, 17] has several motivations and applications.”
Dual Formulation of Second Order Target Problems (2010, background): “Indeed the standard BSDEs, as introduced by Pardoux and Peng [16], are known to provide a stochastic representation for the solutions of semi-linear PDEs in the Markov case.”
Normal approximation by Stein’s method under sublinear expectations (2017, methodology): “Motivated by problems of model uncertainty in statistics, measures of risk and superhedging in finance, Peng (2007) introduced the notion of sublinear expectations.”
On viscosity solutions of path dependent PDEs (2011, background): “We shall prove the existence, uniqueness, stability and comparison principle for the viscosity solutions.”
Pathwise properties and homeomorphic flows for stochastic differential equations driven by G-Brownian motion (2009, background): “19 in [13], 〈B〉t ∈ tΣ = {t × γ γ τ ; γ ∈ Γ }.”
Optimal Stopping under Nonlinear Expectation (2012, background): “[12] Peng, S. (2004) Filtration consistent nonlinear expectations and evaluations of contin- gent claims, Acta Mathematicae Applicatae Sinica (English Series), 20, 191-214.”
Function Spaces and Capacity Related to a Sublinear Expectation: Application to G-Brownian Motion Paths (2008, background): “Motivated by the problem of coherent risk measures under the volatility uncertainty, [21, 22] introduced a sublinear expectation on a well-defined space L 1 (�) under which the increments of the canonical process (Bt)t≥0 are zero-mean, independent and stationary and can be proved to be ‘G-normally d”