I am wondering if the following integral of stochastic Brownian motions has an analytical solution?
$$ \int_{0}^{t}e^{\nu \tilde{V}_{\tau} - \frac{1}{2}\nu^{2}\tau}d\tilde{W}_{\tau} $$
where $\tilde{W}_{\tau},\tilde{V}_{\tau}$ are two independent Brownian motions, $N(0,\tau)$. I did some search on the subject and found this reference based on Bougerol work (in french). However, I cannot find a satisfactory answer.
In the papers they give 2 results:
$$ \sinh(\tilde{V}_{t}) \stackrel{law}{=} \int_{0}^{t}e^{\tilde{V}_{s}}d\tilde{W}_{s} $$
or
$$ \sinh(\tilde{V}_{t}+\tilde{\varepsilon}t) \stackrel{law}{=} \int_{0}^{t}e^{\tilde{V}_{s}+s}d\tilde{W}_{s} $$
where $\tilde{\varepsilon}$ is a symmetric Bernoulli variable between {-1,1}.
I am looking for an answer like that but for the integral:
$$ \int_{0}^{t}e^{\nu \tilde{V}_{\tau} - \frac{1}{2}\nu^{2}\tau}d\tilde{W}_{\tau} \stackrel{law}{=} \dots $$
Thanks in advance for your replies.
I previously posted this question on Mathoverflow but Mathematics seems to be more adapted.