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Stochastic Differential Equations

  • Vincenzo Capasso
  • David Bakstein
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)

Abstract

Let \((W_{t})_{t\in \mathbb{R}_{+}}\) be a Wiener process on the probability space \((\varOmega,\mathcal{F},P)\), equipped with its natural filtration \((\mathcal{F}_{t})_{t\in \mathbb{R}_{+}}\), \(\mathcal{F}_{t} =\sigma (W_{s},0 \leq s \leq t)\). Furthermore, let a(t, x), b(t, x) be deterministic measurable functions in \([t_{0},T] \times \mathbb{R}\) for some \(t_{0} \in \mathbb{R}_{+}.\) Finally, consider a real-valued random variable u 0; we will denote by \(\mathcal{F}_{u^{0}}\) the σ-algebra generated by u 0, and we assume that \(\mathcal{F}_{u^{0}}\) is independent of \((\mathcal{F}_{t})\) for t ∈ (t 0, +). We will denote by \(\mathcal{F}_{u^{0},t}\) the σ-algebra generated by the union of \(\mathcal{F}_{u^{0}}\) and \(\mathcal{F}_{t}\) for t ∈ (t 0, +). 

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Vincenzo Capasso
    • 1
  • David Bakstein
    • 1
  1. 1.ADAMSS (Interdisciplinary Centre for Advanced Applied Mathematical and Statistical Sciences)Università degli Studi di MilanoMilanItaly

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