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

Yongsheng Song

Stochastic Processes and their Applications, 2019, vol. 129, issue 6, 2066-2085

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).

Keywords: G-martingales with finite variation; Generalized G-itô processes; Unique decomposition; G-Sobolev spaces (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (4)

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DOI: 10.1016/j.spa.2018.07.002

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