Abstract
For any two processes \(\xi ,\eta \), the stochastic integral of \(\eta \) with respect to ξ is the process \(\int \eta \mathrm{d}\xi \) defined by \(\int_{0}^{s}\eta \mathrm{d}\xi =\int_{0}^{s}\eta (t)\mathrm{d}\xi (t) =\sum\limits_{t<s}\eta (t)\mathrm{d}\xi (t)\) for all \(s\,\in \,\mathbf{T}\).
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Notes
- 1.
We denote this random variable by \(R\left (t +\mathrm{ d}t\right )\) rather than \(R\left (t\right )\) because it is \({\mathcal{F}}_{t+\mathrm{d}t}\) -measurable, but in general not \({\mathcal{F}}_{t}\) -measurable.
- 2.
For more on Lévy processes—from the perspective of radically elementary probability theory—see Chap. 9.
References
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Nelson, E.: Radically elementary probability theory. Annals of Mathematics Studies, vol. 117. Princeton University Press, Princeton, NJ (1987)
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Herzberg, F.S. (2013). Radically Elementary Stochastic Integrals. In: Stochastic Calculus with Infinitesimals. Lecture Notes in Mathematics, vol 2067. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33149-7_3
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