Let (d X_{t}=h_{t} d t+sigma_{t} d W_{t}+varphi_{t} d M_{t}) be a mixed process and (S_{t}=e^{X_{t}}). Prove that

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Let $d X_{t}=h_{t} d t+\sigma_{t} d W_{t}+\varphi_{t} d M_{t}$ be a mixed process and $S_{t}=e^{X_{t}}$. Prove that the dynamics of $S$ are

\[d S_{t}=S_{t^{-}}\left(\left(h_{t}+\frac{1}{2} \sigma_{t}^{2}+\left(e^{\varphi_{t}}-1-\varphi_{t}\right) \lambda(t)\right) d t+\sigma_{t} d W_{t}+\left(e^{\varphi_{t}}-1\right) d M_{t}\right)\]

Conversely, if

\[d S_{t}=S_{t^{-}}\left(\mu_{t} d t+\sigma_{t} d W_{t}+\psi_{t} d M_{t}\right)\]

with $\psi_{t}>-1$, prove that $S_{t}=e^{Y_{t}}$, where $Y$ is a mixed process.

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Mathematical Methods For Financial Markets

ISBN: 9781447125242

1st Edition

Authors: Monique Jeanblanc, Marc Yor, Marc Chesney

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Question Posted: Apr 03, 2024 01:01 AM

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Let (d X_{t}=h_{t} d t+sigma_{t} d W_{t}+varphi_{t} d M_{t}) be a mixed process and (S_{t}=e^{X_{t}}). Prove that

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To prove the dynamics of St in terms of Xt well use Itos lemma Then well show the converse proving t...View the full answer

Mathematical Methods For Financial Markets

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