Remarks on factorization property of some stochastic integrals

TL;DR

This paper simplifies the proof of a factorization property between two families of improper stochastic integrals introduced by Sato (2006), extending the results to measures on Banach spaces and establishing new relations.

Contribution

The paper provides a simpler proof of the factorization property and extends the results to measures on Banach spaces using random integral mappings.

Findings

Simplified proof of the factorization property.

Extension of the property to Banach space measures.

New relations between the two families of improper stochastic integrals.

Abstract

In the paper Sato (2006) there are introduced two families of improper random integrals and the corresponding two convolution semigroups of infinitely divisible laws on $\Rset^d$. Theorem 3.1 gives a relation (a factorization property) between those two integrals. Here, using \emph{the random integral mappings} $I^{h,r}_{(a,b]}$ (cf. the survey article Jurek (2011)), we give a simpler proof that is also valid for measures on Banach spaces. Furthermore, using our technique we establish yet other relations between those two families of improper stochastic integrals.