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

By Laurent Denis, Mingshang Hu, Shige Peng. 2008

654 citations recorded by the source corpus.

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.

Subfields: Demography, Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/919d56dbb0d0eaeadf860de743896c938eca4943

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Citations in the corpus

The p-th moment stability of solutions to impulsive stochastic differential equations driven by G-Brownian motion (2017, background): “[2], we know that P is tight and C is a Choquet capacity.”
Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II (2012, background): “As in Denis, Hu and Peng [8], the set Pt L induces the following capacity: C t [A] := sup P∈Pt L P[A], for all A ∈ F t T .”
Dual Formulation of Second Order Target Problems (2010, background): “Indeed a similar extension of G-martingales is given in [8] showing that they cover the same space as in the quasi-sure analysis of [7].”
Financial markets with volatility uncertainty (2010, methodology): “Denis and Martini (2006) also obtained the identity hup = EG(HTγ −1 T ) for the case when HT ∈ Cb(Ω) by using quasi–sure analysis.(11) We formulate the financial market in the presence of volatility uncertainty using the G–framework of Peng (2007). Utilizing the set of multiple priors P induced by E”
Exponential inequalities under the sub-linear expectations with applications to laws of the iterated logarithm (2014, methodology): “ 9 the monotonicity and sub-additivity, and the elementary inequality |xy| 6 1 p|x|p + 1 q|y|q, where p,q>1 are two real numbers satisfying 1 p + 1 q = 1 (c.f, Proposition 16 of Denis, Hu and Peng [3]). Lemma 4.1. (Ho¨lder’s inequality)Letp,q>1 betworealnumberssatisfying 1 p + 1 q = 1. Then fo”
Normal approximation by Stein’s method under sublinear expectations (2017, background): “A functional N is a sublinear expectation on Cb,Lip(R) if and only if it can be represented as the supremum expectation of a weakly compact subset Θ of probability measures on (R,B(R)) (see [2]), N [φ] = sup μ∈Θ μ[φ], for all φ ∈ Cb,Lip(R).”
Lyapunov-type conditions and stochastic differential equations driven by $G$-Brownian motion (2014, methodology): “We can extend the domain of G-expectation E[·] from Lip(Ω) to L (Ω) by the procedure introduced in [4], i.”
The Signature of a Rough Path: Uniqueness (2014, background): “See again [20] for a detailed discussion and other related properties.”
Quasi-continuous random variables and processes under the G-expectation framework (2014, methodology): “Noting that the completion of Mc(0, T ) under the norm || · ||Mp is M G(0, T ), then, by the same analysis as in Propositions 18 and 24 in [1], we can get M p G(0, T ) ⊂ Jp.”