Let (f) be a non-constant holomorphic function on (mathbb{C}) and (Z=X+i Y) a complex Brownian motion. Prove

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Let \(f\) be a non-constant holomorphic function on \(\mathbb{C}\) and \(Z=X+i Y\) a complex Brownian motion. Prove that there exists another complex Brownian motion \(B\) such that \(f\left(Z_{t}\right)=f\left(Z_{0}\right)+B\left(\int_{0}^{t}\left|f^{\prime}\left(Z_{s}\right)\right|^{2} d\langle Xangle_{s}\right)\). As an example, \(\exp \left(Z_{t}\right)=1+B_{\int_{0}^{t} d s \exp \left(2 X_{s}\right)}\).

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Mathematical Methods For Financial Markets

ISBN: 9781447125242

1st Edition

Authors: Monique Jeanblanc, Marc Yor, Marc Chesney

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