G -Expectation, G -Brownian Motion and Related Stochastic Calculus of Itô Type

By Shige Peng. 2006

774 citations recorded by the source corpus.

We introduce a notion of nonlinear expectation --G--expectation-- generated by a nonlinear heat equation with infinitesimal generator G. We first discuss the notion of G-standard normal distribution. With this nonlinear distribution we can introduce our G-expectation under which the canonical process is a G--Brownian motion. We then establish the related stochastic calculus, especially stochastic integrals of Ito's type with respect to our G--Brownian motion and derive the related Ito's formula. We have also give the existence and uniqueness of stochastic differential equation under our G-expectation. As compared with our previous framework of g-expectations, the theory of G-expectation is intrinsic in the sense that it is not based on a given (linear) probability space.

Subfields: Finance, Modeling and Simulation, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/4b90847369ff38f24340ca506650424dfb832e4e

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Citations in the corpus

Rosenthal’s inequalities for independent and negatively dependent random variables under sub-linear expectations with applications (2014, methodology): “a sub-linear expectation space. Give a sub-linear expectation Eb, let us denote the conjugate expectation Ebof bEby Eb[X] := −Eb[−X], ∀X ∈ H . Obviously, for all X ∈ H , Eb[X] ≤ Eb[X]. Deriniton 1.2 (Peng (2006,2008b)) (i) (Independence) Let X1 and X2 be two n-dimensional random vectors deﬁned respe”
Properties of hitting times for G-martingales and their applications (2011, background): “Recently, [2,4] introduced the notion of sublinear expectation space, which is a generalization of probability space.”
Risk measuring under model uncertainty (2010, background): “Peng introduced the notion of sublinear expectation and of G-expectations [26] [27] defined on a vector lattice H of real functions containing 1 and included in Cb(Ω).”
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.”
Backward Stochastic Differential Equation, Nonlinear Expectation and Their Applications (2011, methodology): “From the above theorem, we have ob- tained a family of mappings E g s,t : L 2 P (Ft) 7→ L 2 P (Fs), 0 ≤ s ≤ t ∞, (1.8) with “backward semigroup property” (see Peng (2007a)): E g s,t[E g t,T [ξ]] = E g s,T [ξ], E g T,T [ξ] = ξ, for s ≤ t ≤ T ∞, ∀ξ ∈ L2(FT ).”
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.”
Robust Markowitz mean‐variance portfolio selection under ambiguous covariance matrix (2016, neutral): “We model the uncertainty about the volatility matrix of the risky assets by using the probabilistic setup as in [10], [28] or [35].”
Exponential inequalities under the sub-linear expectations with applications to laws of the iterated logarithm (2014, methodology): “Non-additive probabilities and non-additive expectations are useful tools for studying uncertainties in statistics, measures of risk, superhedging in finance and non-linear stochastic calculus, c.f. Denis and Martini (2006), Gilboa (1987), Marinacci (1999), Peng (1997, 1999, 2006, 2008a) etc.”