A Vector Girsanov Result and its Applications to Conditional Measures via the Birkhoff Integrability

Abstract

Some integration techniques for real-valued functions with respect to vector measures with values in Banach spaces (and viceversa) are investigated in order to establish abstract versions of classical theorems of Probability and Stochastic Processes. In particular the Girsanov Theorem is extended and used with the treated methods.2010 Mathematics Subject Classification. 28B20, 58C05,28B05, 46B42, 46G10, 18B15.

A Girsanov Result Through Birkhoff Integral

“…Proof. Taking into account Proposition 2.11, the proof is analogous to that of [8,Theorem 2.13] where the Birkhoff integrability of first type of Φ(•)φ(•) is substituted by its Pettis integrability and φ is Bartle-Dunford-Schwartz integrable.…”

“…We want to point out that this topic is interesting also from the point of view of measure and integration theory, as showed in the papers [1, 2, 4-6, 9, 10, 12, 13, 15-18, 20, 24, 25, 28-30, 33, 35, 37]. In a previous research we have obtained a Girsanov result for the Birkhoff integral of a vector-valued function [7,8] and a Radon Nikodym result in [3]. In this paper we want to weaken the hypothesis of integrability and examine the case of the Pettis integrability when the space X is a Banach space not necessarily separable, the use of non-absolute integrals is also motivated by applications, as shown in [11,14,18].…”

“…8), then thanks to[32, Theorem 10.5], the measure Q = y T dP defined using Y T as Radon-Nikodym derivative, is an equivalent martingale measure with respect to P and the process wt is a Brownian motion under the new probability Q. This allows us to define the stochastic Pettis integral (P e) )dw s + (P e) t 0 r(s)Ψ(s)ds as a vector equivalence of Pettis stochastic integral and Pettis integral.…”