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A compactness principle for bounded sequences of martingales with applications. (English) Zbl 0939.60024

Dalang, Robert C. (ed.) et al., Seminar on Stochastic analysis, random fields and applications. Centro Stefano Franscini, Ascona, Italy, September 1996. Basel: Birkhäuser. Prog. Probab. 45, 137-173 (1999).
The authors provide a compactness principle for \(H^1\)-bounded sequences of martingales which is related to the Kadeč-Pełczyński decomposition for bounded sequences in \(L^1\) [cf. M. Kadeč and A. Pełczyński, Stud. Math. 25, 297-323 (1965; Zbl 0135.34504)]. The idea is to decompose a given sequence in an \(H^1\)-weakly compact part on the one hand and a manageable singular part on the other hand. This principle allows the authors to prove that, for any \(H^1\)-bounded sequence of continuous local martingales \((M^n)\), there is an increasing sequence of stopping times \((T_n)\) with \(P[T_n < \infty] \to 0\) and such that the sequence of stopped processes \(((M^n)^{T_n})\) is relatively weakly compact in \(H^1\). For the discontinuous case, it is shown that increasing stopping times can be found such that the sequence of martingales \(N^n = (M^n)^{T_n}-\Delta M_{T_n}1_{[[T_n,\infty]]}+C^n\) is weakly compact in \(H^1\), where \(C^n\) denotes the compensator of \(\Delta M_{T_n}1_{[[T_n,\infty]]}\). Moreover, by passing to convex combinations, strong convergence in \(H^1\) of \((N^n)\) can be obtained in such a way that the corresponding sequence of convex combinations of \((C^n)\) converges a.s. to a càdlàg optional process of finite variation. As a first application of their results, the authors give an alternative proof of Kramkov’s optional decomposition theorem [cf., e. g., D. O. Kramkov, Probab. Theory Relat. Fields 105, No. 4, 459-479 (1996; Zbl 0853.60041)]. A second application is a new proof of Yor’s theorem on the closedness of the space of stochastic integrals in \(H^1\) and \(L^1\), respectively [see M. Yor, in: Sémin. Probab. XII. Lect. Notes Math. 649, 265-309 (1978; Zbl 0391.60046)].
For the entire collection see [Zbl 0914.00071].

MSC:

60G44 Martingales with continuous parameter
60H05 Stochastic integrals
46N30 Applications of functional analysis in probability theory and statistics
91B28 Finance etc. (MSC2000)
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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