Some properties on G-evaluation and its applications to G-martingale decomposition

By Yongsheng Song. 2010

169 citations recorded by the source corpus.

In this article, a sublinear expectation induced by G-expectation is introduced, which is called G-evaluation for convenience. As an application, we prove that for any ζ ∈ LGβ (ΩT) with some β > 1 the martingale decomposition theorem under G-expectaion holds, and that any β > 1 integrable symmetric G-martingale can be represented as an Itô integral w.r.t. G-Brownian motion. As a byproduct, we prove a regularity property for G-martingales: Any G-martingale {Mt} has a quasi-continuous version.

Subfields: Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/d1bf8e4c6dc6ea31f5bccd14ae5dba4dfbe7ac0a

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Citations in the corpus

On the existence and uniqueness of solutions to stochastic differential equations driven by G-Brownian motion with integral-Lipschitz coefficients (2010, background): “volves Z, i.e., the integrand of the Ito type G-stochastic integral with respect to G-Brownian motion. Based on the great eﬀorts of many authors, such as Xu and Zhang [17], Soner et al. [12] and Song [13, 14], Peng et al. [10] have given a complete theory for G-martingale representation. Subsequentl”
Superreplication under Volatility Uncertainty for Measurable Claims (2012, methodology): “DMS-1208985 is gratefully acknowledged. 1 The duality formula under volatility uncertainty has been established for several cases and through diﬀerent approaches: [6] used ideas from capacity theory, [16, 20, 22] used an approximation by Markovian control problems, and [19, 12] used a method discuss”
Nonlinear Expectations and Stochastic Calculus under Uncertainty (2010, background): “Some very interesting developments to this important problem can be found in Soner, Touzi and Zhang (2009) [114] and Song (2009) [116].”
Martingale Representation Theorem for the G-Expectation (2010, methodology): “After the completion and the submission of this manuscript, we became aware of the manuscript of Song [23] which proves a decomposition result for random variables in LpG with p > 1.”
Backward stochastic differential equations driven by G-Brownian motion (2012, background): “Thanks to an original new norm introduced in Song [28, 2012] for decreasing G-martingales, a representation theorem of G-martingales in a complete subspace of LαG(ΩT ) has been obtained by Peng, Song and Zhang [24, 2012].”
A complete representation theorem for G-martingales (2012, background): “Moreover, combining the arguments in [22] and [9], or following the arguments in [25], one can easily prove”
Properties of hitting times for G-martingales and their applications (2011, methodology): “Moreover we improve the results of [7] on the G-martingale representation by a stopping time technique.”
Arbitrage and duality in nondominated discrete-time models (2013, background): “In the setting where no options are traded and P is supposed to consist of martingale measures in the rst place, there is a substantial literature on superhedging under volatility uncertainty, where S is a (continuous-time) process with continuous trajectories; see, among others, [8, 27, 40, 41, 42,”
Uniqueness of the representation for $G$-martingales with finite variation (2010, unknown): “Let hs = ∑m−1 i=0 ξi1] iT m , (i+1)T m ] (s), where ξi = ϕi(W iT m −W (i−1)T m , · · ·,W T m −W0), ϕi ∈ Cb,lip(Ri), c ≤ |ϕi| ≤ C.”
Quasi-continuous random variables and processes under the G-expectation framework (2014, background): “5 This definition of quasi-continuous of a process is different from the one in [15, 16].”