The Fubini Theorem for Stochastic Integrals with Respect to $L^0 $-Valued Random Measures Depending on a Parameter

1996 ◽  
Vol 40 (2) ◽  
pp. 285-293 ◽  
Author(s):  
V. A. Lebedev
Keyword(s):  
Fubini Theorem ◽  
Stochastic Integrals ◽  
Random Measures

scholarly journals A Fubini theorem for iterated stochastic integrals

1978 ◽  
Vol 84 (1) ◽  
pp. 159-161 ◽  
Author(s):  
Marc A. Berger ◽  
Victor J. Mizel
Keyword(s):  
Fubini Theorem ◽  
Stochastic Integrals

scholarly journals On representation of integer-valued random measures by means of stochastic integrals with respect to the Poisson measure

1971 ◽  
Vol 11 (1) ◽  
pp. 93-108
Author(s):  
B. Grigelionis
Keyword(s):  
Stochastic Integrals ◽  
Poisson Measure ◽  
Random Measures

The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: Б. Григелионис. О представлении целочисленных случайных мер как стохастических интегралов по пуассоновской мере B. Grigelionis. Apie atsitiktinių matų su sveikomis reikšmėmis išreiškimų stochastiniais integralais Puasono mato atžvilgiu

scholarly journals Burkholder–Davis–Gundy Inequalities in UMD Banach Spaces

2020 ◽  
Vol 379 (2) ◽  
pp. 417-459
Author(s):  
Ivan Yaroslavtsev
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Stochastic Integrals ◽  
Reflexive Banach Spaces ◽  
Random Measures ◽  
Umd Spaces ◽  
Independent Increments ◽  
The Right ◽  
Umd Banach Spaces ◽  
Vector Valued

Abstract In this paper we prove Burkholder–Davis–Gundy inequalities for a general martingale M with values in a UMD Banach space X. Assuming that $$M_0=0$$ M 0 = 0 , we show that the following two-sided inequality holds for all $$1\le p<\infty $$ 1 ≤ p < ∞ : Here $$ \gamma ([\![M]\!]_t) $$ γ ( [ [ M ] ] t ) is the $$L^2$$ L 2 -norm of the unique Gaussian measure on X having $$[\![M]\!]_t(x^*,y^*):= [\langle M,x^*\rangle , \langle M,y^*\rangle ]_t$$ [ [ M ] ] t ( x ∗ , y ∗ ) : = [ ⟨ M , x ∗ ⟩ , ⟨ M , y ∗ ⟩ ] t as its covariance bilinear form. This extends to general UMD spaces a recent result by Veraar and the author, where a pointwise version of ($$\star $$ ⋆ ) was proved for UMD Banach functions spaces X. We show that for continuous martingales, ($$\star $$ ⋆ ) holds for all $$0<p<\infty $$ 0 < p < ∞ , and that for purely discontinuous martingales the right-hand side of ($$\star $$ ⋆ ) can be expressed more explicitly in terms of the jumps of M. For martingales with independent increments, ($$\star $$ ⋆ ) is shown to hold more generally in reflexive Banach spaces X with finite cotype. In the converse direction, we show that the validity of ($$\star $$ ⋆ ) for arbitrary martingales implies the UMD property for X. As an application we prove various Itô isomorphisms for vector-valued stochastic integrals with respect to general martingales, which extends earlier results by van Neerven, Veraar, and Weis for vector-valued stochastic integrals with respect to a Brownian motion. We also provide Itô isomorphisms for vector-valued stochastic integrals with respect to compensated Poisson and general random measures.

Stochastic Fubini Theorem for Semimartingales in Hilbert Space

1990 ◽  
Vol 42 (5) ◽  
pp. 890-901 ◽  
Author(s):  
Jorge A. León
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Measure Space ◽  
Sufficient Conditions ◽  
Linear Operators ◽  
Fubini Theorem ◽  
Stochastic Integrals ◽  
Bounded Linear ◽  
Image Position ◽  
Stochastic Fubini Theorem

In this paper we will study the Fubini theorem for stochastic integrals with respect to semimartingales in Hilbert space.Let (Ω, , P) he a probability space, (X, , μ) a measure space, H and G two Hilbert spaces, L(H, G) the space of bounded linear operators from H into G, Z an H-valued semimartingale relative to a given filtration, and φ: X × R+ × Ω → L(H, G) a function such that for each t ∈ R+ the iterated integrals are well-defined (the integrals with respect to μ are Bochner integrals). It is often necessary to have sufficient conditions for the process Y1 to be a version of the process Y2 (e.g. [1], proof of Theorem 2.11).

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