Abstract
In this chapter we extend our discussion of martingales to allow a continuous-time processes. Throughout we take \(\mathbb{T} = [0,\infty [\) or \([0,\infty ]\).
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Notes
- 1.
The Dirac delta function at y, denoted δ x = y , is the ‘function’ with the defining property that \(\int _{\mathbb{R}^{d}}f(x)\delta _{x=y}dx = f(y)\) for any bounded measurable function y. Hence, δ x = y can be thought of as an infinitely tall spike at y. While this is not a function, but a linear map \(L^{\infty }\rightarrow \mathbb{R}\) (so an element of the dual of \(L^{\infty }\)), its integral is still well defined, and this serves as convenient notation.
References
D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, 3rd edn (Springer, Berlin, 1999)
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Cohen, S.N., Elliott, R.J. (2015). Martingales in Continuous Time. In: Stochastic Calculus and Applications. Probability and Its Applications. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-2867-5_5
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DOI: https://doi.org/10.1007/978-1-4939-2867-5_5
Publisher Name: Birkhäuser, New York, NY
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