Let (B_{t}=left(b_{t}, beta_{t}ight), t geqslant 0) be a (mathrm{BM}^{2}) and set (r_{t}:=left|B_{t}ight|=sqrt{b_{t}^{2}+beta_{t}^{2}}). a) Show that the stochastic

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Let $B_{t}=\left(b_{t}, \beta_{t}ight), t \geqslant 0$ be a $\mathrm{BM}^{2}$ and set $r_{t}:=\left|B_{t}ight|=\sqrt{b_{t}^{2}+\beta_{t}^{2}}$.

a) Show that the stochastic integrals $\int_{0}^{t} b_{s} / r_{s} d b_{s}$ and $\int_{0}^{t} \beta_{s} / r_{s} d \beta_{s}$ exist.

b) Show that $W_{t}:=\int_{0}^{t} b_{s} / r_{s} d b_{s}+\int_{0}^{t} \beta_{s} / r_{s} d \beta_{s}$ is a $\mathrm{BM}^{1}$.

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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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Question Posted: Mar 17, 2024 01:00 AM

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Let (B_{t}=left(b_{t}, beta_{t}ight), t geqslant 0) be a (mathrm{BM}^{2}) and set (r_{t}:=left|B_{t}ight|=sqrt{b_{t}^{2}+beta_{t}^{2}}). a) Show that the stochastic

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Brownian Motion A Guide To Random Processes And Stochastic Calculus De Gruyter Textbook

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