Abstract
Given our understanding of general stochastic processes, we now set our sights on establishing a theory of stochastic integration. We do this in stages, beginning with the simple case where we take the integral with respect to a process which does not vary ‘too much’, that is, where its paths are of finite variation for almost all ω. This first step is deceptively simple, as we can establish our integral pathwise, simply by appealing to the Lebesgue–Stieltjes integral considered in Chapter 1 We then use this theory to establish the stochastic integral for more general processes, over the coming chapters.
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Notes
- 1.
We formally allow s = “0 −″, so as to include the case X t = I B without additional notation. For consistency, A t∧s : = A 0− = 0 whenever s = 0−.
- 2.
Neveu proves the result with x p replaced by a general convex function.
References
J. Neveu, Martingales à temps discret (Masson & Cie, Paris, 1972)
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Cohen, S.N., Elliott, R.J. (2015). Processes of Finite Variation. In: Stochastic Calculus and Applications. Probability and Its Applications. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-2867-5_8
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DOI: https://doi.org/10.1007/978-1-4939-2867-5_8
Publisher Name: Birkhäuser, New York, NY
Print ISBN: 978-1-4939-2866-8
Online ISBN: 978-1-4939-2867-5
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