Abstract
Let η be a limited, \(\mathcal{F}\) -adapted stochastic process, let ξ be the process defined by ξ(0) = 1 and \(\mathrm{d}\xi (t) = \xi (t)\eta (t)\ \mathrm{d}W(t)\) for all \(t \in \mathbf{T} \setminus \{ 1\}\).
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Notes
- 1.
Bayes’ formula can be proven as follows: For all s > t, every \({\mathcal{F}}_{s}\)-measurable z and every \({\mathcal{F}}_{t}\)-measurable A, one has
$$\begin{array}{rcl} \int_{A}\xi (t){E}^{Q}\left [\left.z\right \vert {\mathcal{F}}_{t}\right ]\ \mathrm{d}Q& =& \int_{A}{E}^{Q}\left [\left.\xi (t)z\right \vert {\mathcal{F}}_{t}\right ]\ \mathrm{d}Q \\ & =& \int_{A}\xi (t)z\ \mathrm{d}Q =\int_{A}\xi (t)z\xi (s)\ \mathrm{d}Q \\ & =& \int_{A}\xi (t)\xi (s)z\ \mathrm{d}P =\int_{A}\xi (s)z\ \mathrm{d}Q \\ & =& \int_{A}E\left [\left.\xi (s)z\right \vert {\mathcal{F}}_{t}\right ]\ \mathrm{d}Q.\end{array}$$
References
Benoît, E.: Random walks and stochastic differential equations. In: Diener, F., Diener, M. (eds.) Nonstandard Analysis In Practice. Universitext, pp. 71–90. Springer, Berlin (1995)
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Herzberg, F.S. (2013). The Radically Elementary Girsanov Theorem and the Diffusion Invariance Principle. In: Stochastic Calculus with Infinitesimals. Lecture Notes in Mathematics, vol 2067. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33149-7_4
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DOI: https://doi.org/10.1007/978-3-642-33149-7_4
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