On representation theorem of G-expectations and paths of G-Brownian motion

By Mingshang Hu, Shige Peng. 2009

192 citations recorded by the source corpus.

AbstractWe give a very simple and elementary proof of the existence of a weakly compact family of probability measures {Pθ: θ ∈ gJ} representing an important sublinear expectation— G-expectation $$ \mathbb{E} $$[·]. We also give a concrete approximation of a bounded continuous function X(ω) by an increasing sequence of cylinder functions Lip(Ω) in order to prove that Cb(Ω) belongs to the completion of Lip(Ω) under the natural norm $$ \mathbb{E} $$[| · |].

Subfields: Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/3283cd457688c41d84054e26706ea8cf210edec2

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Citations in the corpus

Note on Viscosity Solution of Path-Dependent PDE and G-Martingales (2011, unknown): “|u(ωt)− u(υs)| ∨ |v(ωt)− v(υs)| ≤ ρ(d(ωt, υs)), (15)”
Stochastic functional differential equations with infinite delay driven by G‐Brownian motion (2013, background): “Peng S. G-Brownian motion and dynamic risk measures under volatility uncertainty, 2007. arXiv: 0711.2834v1.”
G-Lévy processes under sublinear expectations (2009, background): “zed G-Brownian motion, where G(p,A) = lim t↓0 Eˆ[ 1 2 hAXt,Xti +hp,Xti]t−1 for (p,A) ∈ Rd×S(d). Remark 18 In fact, paths of (Xt)t≥0 in the above theorem are continuous due to the condition (iii) (see [12, 16]). In the following sections, we consider L´evy processes without the condition (iii) which ”
Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations (2009, background): “The following result was established in [18]. See also [ 20 ].”
Quadratic BSDEs with convex generators and unbounded terminal conditions (2007, background): “1 ([7, 16]) There exists a weakly compact set P of probability measures on (Ω,B(Ω)), such that Ê[X ] = sup P∈P E [X ] for any X ∈ L(1)G(Ω).”
Some properties on G-evaluation and its applications to G-martingale decomposition (2010, background): “3) Noting the discussion in Remark 2.7, we can prove the desired representation by just the same arguments as those in [ 2 ].”
Backward stochastic differential equations driven by G-Brownian motion (2012, unknown): “7 ([5, 7]) There exists a tight subset P ⊂ M1(ΩT ), the set of probability measures on (ΩT ,B(ΩT )), such that Ê[ξ] = sup P∈P EP [ξ] for all ξ ∈ H T .”
Nonlinear Expectations and Stochastic Calculus under Uncertainty (2010, background): “Hu and Peng (2009) [58] then have introduced an intrinsic and simple approach.”
Quasi-continuous random variables and processes under the G-expectation framework (2014, methodology): “5 ([1, 5]) Let Xk ∈ L(1)G(Ω), k ≥ 1, be such that Xk ↓ X q.”
Comparison Theorem, Feynman-Kac Formula and Girsanov Transformation for BSDEs Driven by G-Brownian Motion (2012, unknown): “7 ([5, 8]) There exists a weakly compact set P ⊂ M1(ΩT ), the set of probability measures on (ΩT ,B(ΩT )), such that Ê[ξ] = sup P∈P EP [ξ] for all ξ ∈ H T .”
Backward Stochastic Differential Equation, Nonlinear Expectation and Their Applications (2011, methodology): “Without this condition, B may become a G-Lévy process (see Hu and Peng (2009b)).”
Risk measuring under model uncertainty (2010, background): “It is proved in [15] and [23] that every G-expectation E has a representation with respect to a weakly relatively compact set of probability measures P : E(f ) = supP∈P EP (f ) for all f in H.”