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I have to derive several things for my thesis, however, I have the following expression:

$$ \int^{t}_{0} \exp\{\sigma W_{t}\}.dW_{t} $$

Does anyone know what the solution for this is?

Kind regards.

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1 Answer 1

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Unlike for many ordinary integrals in calculus there is not always a solution in Ito calculus.

Partial Answer and Hints

The process $$ M_t=\exp\{\sigma\,W_t-\sigma^2t/2\} $$ is a martingale that satisfies $$ dM_t=\sigma\,M_t\,dW_t\,,\quad\text{ or in integral form }M_t=1+\int_0^t\sigma\,M_s\,dW_s\,. $$ Therefore, $$ \int_0^t\sigma\,M_s\,dW_s=\sigma\int_0^t\exp\{\sigma\,W_s-\sigma^2s/2\}\,dW_s=M_t-1=\exp\{\sigma\,W_t-\sigma^2t/2\}-1\,. $$ If you want to get rid of the $-\sigma^2s/2$ term in the exponential you could apply Girsanov's theorem which will introduce a drift in the integrating BM $dW_t$.

I think some integral will always remain "unsolved".

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