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Properties of G-martingales with finite variation and the application to G-Sobolev spaces

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  • Song, Yongsheng

Abstract

As is known, if B=(Bt)t∈[0,T] is a G-Brownian motion, a process of form ∫0tηsd〈B〉s−∫0t2G(ηs)ds, η∈MG1(0,T), is a non-increasing G-martingale. In this paper, we shall show that a non-increasing G-martingale cannot be form of ∫0tηsds or ∫0tγsd〈B〉s, η,γ∈MG1(0,T), which implies that the decomposition for generalized G-Itô processes is unique: For arbitrary ζ∈HG1(0,T), η∈MG1(0,T) and non-increasing G-martingales K,L, if ∫0tζsdBs+∫0tηsds+Kt=Lt,t∈[0,T],then we have η≡0, ζ≡0 andKt=Lt. As an application, we give a characterization to the G-Sobolev spaces introduced in Peng and Song (2015).

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  • Song, Yongsheng, 2019. "Properties of G-martingales with finite variation and the application to G-Sobolev spaces," Stochastic Processes and their Applications, Elsevier, vol. 129(6), pages 2066-2085.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:6:p:2066-2085
    DOI: 10.1016/j.spa.2018.07.002
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    References listed on IDEAS

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    1. Soner, H. Mete & Touzi, Nizar & Zhang, Jianfeng, 2011. "Martingale representation theorem for the G-expectation," Stochastic Processes and their Applications, Elsevier, vol. 121(2), pages 265-287, February.
    2. Hu, Mingshang & Ji, Shaolin & Peng, Shige & Song, Yongsheng, 2014. "Comparison theorem, Feynman–Kac formula and Girsanov transformation for BSDEs driven by G-Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 124(2), pages 1170-1195.
    3. Hu, Mingshang & Ji, Shaolin & Peng, Shige & Song, Yongsheng, 2014. "Backward stochastic differential equations driven by G-Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 759-784.
    4. Peng, Shige, 2008. "Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation," Stochastic Processes and their Applications, Elsevier, vol. 118(12), pages 2223-2253, December.
    5. Song, Yongsheng, 2011. "Properties of hitting times for G-martingales and their applications," Stochastic Processes and their Applications, Elsevier, vol. 121(8), pages 1770-1784, August.
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    Cited by:

    1. Xiao, Guanli & Wang, JinRong & O’Regan, Donal, 2020. "Existence, uniqueness and continuous dependence of solutions to conformable stochastic differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
    2. Hu, Ying & Tang, Shanjian & Wang, Falei, 2022. "Quadratic G-BSDEs with convex generators and unbounded terminal conditions," Stochastic Processes and their Applications, Elsevier, vol. 153(C), pages 363-390.
    3. Hu, Mingshang & Wang, Falei, 2021. "Probabilistic approach to singular perturbations of viscosity solutions to nonlinear parabolic PDEs," Stochastic Processes and their Applications, Elsevier, vol. 141(C), pages 139-171.
    4. Hanwu Li & Yongsheng Song, 2021. "Backward Stochastic Differential Equations Driven by G-Brownian Motion with Double Reflections," Journal of Theoretical Probability, Springer, vol. 34(4), pages 2285-2314, December.

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