Abstract
In the previous chapter, we have considered SDEs where the integral is with respect to a general semimartingale. In this chapter, we focus our attention on a much more specialized setting, where the integral is taken with respect to time (i.e. Lebesgue measure), a Brownian motion and a compensated Poisson random measure. Working in this setting allows the Markovian properties of the Brownian motion and the Poisson process to be inherited by the SDE solution. A full treatment of this topic would require consideration of general Markov processes. For this, see Ethier and Kurtz [77], or the more specialized treatments in Karatzas and Shreve [117] or Revuz and Yor [155]. We shall instead present only a selection of these issues.
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Notes
- 1.
When thinking about these equations, the key issues are typically present for the simple case N = 1. Conceptually, this is also a little easier to work with, so the reader should feel free to assume this for the sake of simplicity. However, the general case of \(N \in \mathbb{N} \cup \{\infty \}\), \(m \in \mathbb{N}\) follows with no notational changes. For \(N = \infty \), we identify \(\mathbb{R}^{N}\) with ℓ 2, so \(\mathcal{B}(\mathbb{R}^{N})\) refers to the Borel topology inherited from the ℓ 2 metric.
- 2.
Throughout this chapter, we simply write \(\mathcal{H}^{2}\) for the space of \(\mathbb{R}^{m}\)-valued processes with components in \(\mathcal{H}^{2}\), and similarly for \(\mathcal{H}_{\text{loc}}^{2}\).
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Cohen, S.N., Elliott, R.J. (2015). Markov Properties of SDEs. In: Stochastic Calculus and Applications. Probability and Its Applications. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-2867-5_17
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DOI: https://doi.org/10.1007/978-1-4939-2867-5_17
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