Martingale Representation Theorem for the G-Expectation

By H. M. Soner, Nizar Touzi, Jianfeng Zhang. 2010

214 citations recorded by the source corpus.

This paper considers the nonlinear theory of G-martingales as introduced by Peng in [16, 17]. A martingale representation theorem for this theory is proved by using the techniques and the results established in [20] for the second order stochastic target problems and the second order backward stochastic differential equations. In particular, this representation provides a hedging strategy in a market with an uncertain volatility.

Subfields: Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/ca480619e42cd3954bcb42e975268e85680123aa

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Citations in the corpus

UTILITY MAXIMIZATION UNDER MODEL UNCERTAINTY IN DISCRETE TIME (2013, background): “The robust superhedging problem has been studied in several settings; see [1, 4, 6, 11, 13, 17, 23, 26, 32, 33], among others.”
Wellposedness of second order backward SDEs (2010, methodology): “(ii) In our paper [16], we used the norm ‖ · ‖ L p,1 H .”
Some properties on G-evaluation and its applications to G-martingale decomposition (2010, methodology): “Soner et al. [ 8 ] made a progress in this direction.”
Stochastic control for a class of nonlinear kernels and applications (2015, unknown): “Since the early works [85, 86], the space L t,ω(FT ) has been recognised to be particularly suited for 2BSDE theory.”
ROBUST FUNDAMENTAL THEOREM FOR CONTINUOUS PROCESSES (2014, background): “…Fernholz and Karatzas (2011), Neufeld and Nutz (2013), Nutz and Soner (2012), Nutz and Zhang (2015), Peng (2010), Possamai, Royer, and Touzi (2013), Soner, Touzi, and Zhang (2011), Soner, Touzi, and Zhang (2013), among others, or when all paths are considered possible for the stock and options are…”
Dual Formulation of Second Order Target Problems (2010, methodology): “More connections between this paper and G−martingales are established in our accompanying paper [24].”
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.”
Financial markets with volatility uncertainty (2010, methodology): “The following example (see also Soner, Touzi, and Zhang (2010b)) illustrates some issues that arise when we deal with uncertain volatilities.”
Pathwise construction of stochastic integrals (2011, background): “[15], the second order backward stochastic di erential equations and target problems of Soner et al.”
Properties of hitting times for G-martingales and their applications (2011, background): “In [6], they proved that L p G(ΩT ) ⊂ L 2 E (ΩT ) for any p > 2, which is a particular case of Theorem A.”
Superreplication under Volatility Uncertainty for Measurable Claims (2012, methodology): “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 discussed below.”