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.

Thanks in advance for your replies.


In the papers they give 2 results:

$$ \sinh(\tilde{V}_{t}) \stackrel{law}{=} \int_{0}^{t}e^{\tilde{V}_{s}}d\tilde{W}_{s} $$


$$ \sinh(\tilde{V}_{t}+\tilde{\varepsilon}t) \stackrel{law}{=} \int_{0}^{t}e^{\tilde{V}_{s}+s}d\tilde{W}_{s} $$

where $\tilde{\varepsilon}$ is a symmetric Bernoulli variable between {-1,1}.

I was looking for an answer like that but for the integral:

$$ \int_{0}^{t}e^{\nu \tilde{V}_{\tau} - \frac{1}{2}\nu^{2}\tau}d\tilde{W}_{\tau} \stackrel{law}{=} \dots $$


I am not sure how rigorous this is (it has been a while since I touched stochastic calculus) but here goes:

Let $S_t := \exp ( \nu V_t - 0.5 \nu^2 t)$. Trivially, $X_t = \int_{0}^{t} S_z \, dW_z$.

We will solve $X_t = X_0 + \int_{0}^{t} S_z \, dW_z$ for some non zero real number $X_0$.

In differential form we have $d X_t = S_t \, d W_t$. Assuming that $S_t$ is locally Lipschitz continuous (is it?), we have $$X_t = X_0 \, \exp \left( -0.5 \int_{0}^{t} S_z^2 \, dz + W_t \int_{0}^{t} S_z \, dz \right)$$ via Ito's formula.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.