A Girsanov Result Through Birkhoff Integral

Abstract

A vector-valued version of the Girsanov theorem is presented, for a scalar process with respect to a Banach-valued measure. Previously, a short discussion about the Birkhoff-type integration is outlined, as for example integration by substitution, in order to fix the measure-theoretic tools needed for the main result, Theorem 6, where a martingale equivalent to the underlying vector probability has been obtained in order to represent the modified process as a martingale with the same marginals as the original one.

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

“…Theorems 3.5 and 3.6 were announced in [11,Theorems 5,6]; the last one with also a brief sketch of the proof.…”

“…Other results on the Brownian motion subject are given also in [10,26,29]. This paper is inspired by [11,39], in particular some of the results were announced at the ICSSA 2018 Conference. Now, we give a plan of the paper.…”

“…For the second one we need to recall the definition of Fréchet derivative, for other definitions see for example [2,27,28]. Now we are ready to define the stochastic integral of the second summand in (11). Definition 4.4.…”

The Riemann-Lebesgue Integral of Interval-Valued Multifunctions

“…In the literature several methods of integration for functions and multifunctions have been studied extending the Riemann and Lebesgue integrals. In this framework a generalization of Riemann sums was given in [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37] while another generalization is due to Kadets and Tseytlin [38], who introduced the absolute Riemann-Lebesgue |RL| and unconditional Riemann-Lebesgue RL integrability, for Banach valued functions with respect to countably additive measures. They proved that in finite measure space, the Bochner integrability implies |RL| integrability which is stronger than RL integrability that implies Pettis integrability.…”