Show that in Example 21.7 [X_{t}^{circ}=exp left(-int_{0}^{t}left(beta(s)-delta^{2}(s) / 2 ight) d s ight) exp left(-int_{0}^{t} delta(s) d
Question:
Show that in Example 21.7
\[X_{t}^{\circ}=\exp \left(-\int_{0}^{t}\left(\beta(s)-\delta^{2}(s) / 2\right) d s\right) \exp \left(-\int_{0}^{t} \delta(s) d B_{s}\right)\]
and verify the expression given for \(d X_{t}^{\circ}\). Moreover, show that
\[X_{t}=\frac{1}{X_{t}^{\circ}}\left(X_{0}+\int_{0}^{t}(\alpha(s)-\gamma(s) \delta(s)) X_{s}^{\circ} d s+\int_{0}^{t} \gamma(s) X_{s}^{\circ} d B_{s}\right)\]
Data From 21.7 Example
Data from 21.6 Example
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Related Book For
Brownian Motion A Guide To Random Processes And Stochastic Calculus De Gruyter Textbook
ISBN: 9783110741254
3rd Edition
Authors: René L. Schilling, Björn Böttcher
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