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The Stochastic Integral

  • Paolo Baldi
Chapter
Part of the Universitext book series (UTX)

Abstract

Let \(B = (\varOmega,\mathcal{F},(\mathcal{F}_{t})_{t},(B_{t})_{t},\mathrm{P})\) be a (continuous) standard Brownian motion fixed once and for all: the aim of this chapter is to give a meaning to expressions of the form \(\displaystyle{ \int _{0}^{T}X_{ s}(\omega )\,dB_{s}(\omega ) }\) where the integrand (X s )0 ≤ sT is a process enjoying certain properties to be specified. As already remarked in Sect.  3.3, this cannot be done path by path as the function tB t (ω) does not have finite variation a.s. The r.v. (7.1) is a stochastic integral and it will be a basic tool for the construction and the investigation of new processes.

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Paolo Baldi
    • 1
  1. 1.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomaItaly

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