Product random measures and double stochastic integrals

1982 ◽  
pp. 181-199 ◽  
Author(s):  
Jan Rosiński ◽  
Jerzy Szulga
Keyword(s):  
Stochastic Integrals ◽  
Random Measures

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.

scholarly journals On Itô formulas for jump processes

2021 ◽  
Author(s):  
István Gyöngy ◽  
Sizhou Wu
Keyword(s):  
Jump Processes ◽  
Stochastic Pdes ◽  
Stochastic Integrals ◽  
Itô Formula ◽  
Random Measures ◽  
Ito Formula ◽  
Poisson Random Measures ◽  
Infinite Dimensional ◽  
Wiener Processes ◽  
Finite Dimensional

AbstractA well-known Itô formula for finite-dimensional processes, given in terms of stochastic integrals with respect to Wiener processes and Poisson random measures, is revisited and is revised. The revised formula, which corresponds to the classical Itô formula for semimartingales with jumps, is then used to obtain a generalisation of an important infinite-dimensional Itô formula for continuous semimartingales from Krylov (Probab Theory Relat Fields 147:583–605, 2010) to a class of $$L_p$$ L p -valued jump processes. This generalisation is motivated by applications in the theory of stochastic PDEs.

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