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 ≤ s ≤ T 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 t ↦ B 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.
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Baldi, P. (2017). The Stochastic Integral. In: Stochastic Calculus. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-62226-2_7
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DOI: https://doi.org/10.1007/978-3-319-62226-2_7
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