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Radically Elementary Stochastic Integrals

Part of the Lecture Notes in Mathematics book series (LNM,volume 2067)

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 \(s\,\in \,\mathbf{T}\).

Keywords

  • Wiener Process
  • Harnack Inequality
  • Stochastic Integral
  • Mathematical Finance
  • Elementary Analogue

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

  1. 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. 2.

    For more on Lévy processes—from the perspective of radically elementary probability theory—see Chap. 9.

References

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  2. Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–654 (1973)

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  3. 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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