Somebody has an idea on how to tackle this quantity $$ \mathbb{E}\left[ \left. \frac{\int_{0}^{T}{{{e}^{\alpha {{W}_{t}}}}}dt}{\int_{0}^{T}{{{e}^{-\alpha {{W}_{t}}}}}dt+\int_{0}^{T}{{{e}^{\alpha {{W}_{t}}}}}dt}\,\, \right|\,\,{{W}_{T}} \right] $$

For $\alpha \in \mathbb{R}$. The notation $\mathbb{E}[.|W_T]$ means conditional expectation of a stochastic process given $W_t$.

I tried use brownian bridge to build independent quantities but I cannot get a tractable result.



Maybe not as clean as you would hope for but perhaps you could do the following:

$\mathbb{E}_{W_T}\left[ \frac{\int_0^T e^{\alpha W_t} dt}{\int_0^T e^{-\alpha W_t} dt + \int_0^T e^{\alpha W_t} dt} \right] = \mathbb{E}_{W_T}\left[ \frac{1}{\frac{\int_0^T e^{-\alpha W_t} dt}{\int_0^T e^{\alpha W_t} dt} + 1} \right]$

Then at least formally

$ \mathbb{E}_{W_T}\left[ \frac{\int_0^T e^{\alpha W_t} dt}{\int_0^T e^{-\alpha W_t} dt + \int_0^T e^{\alpha W_t} dt} \right] = 1 - \mathbb{E}_{W_T}\left[ \frac{\int_0^T e^{-\alpha W_t} dt}{\int_0^T e^{\alpha W_t} dt}\right] + ...$

Where the ... indicates higher moments of $\frac{\int_0^T e^{-\alpha W_t} dt}{\int_0^T e^{\alpha W_t} dt}$.

Now the expectation here can be expressed as a series in mixed moments of the numerator and denominator. See this http://www.faculty.biol.ttu.edu/rice/ratio-derive.pdf. I think this is more convenient because for moments like that you can pass the expectation through the time integral, and perhaps get something you can evaluate.

If you know that $\alpha$ is large you could probably justify an approximation that only keeps some small number of terms (i.e. hopefully just one correction is okay). You may also have to derive two approximations based on if $W_T$ is positive or negative. I guess for $|W_T| \ll \alpha$ neither approximation would work, but maybe thats okay for your application.

P.S. to evaluate the expectations inside the time integrals you'll probably want to substitute in there a brownian bridge and notice that you're taking moments of multivariate log normal variables https://en.wikipedia.org/wiki/Log-normal_distribution#Multivariate_log-normal. The brownian bridge will involve terms like $W(T)$ and $W(t)$ and you should know the co-variance structure of that.

P.P.S. if instead you are interested in the case $\alpha \ll |W_T|$ you could get a different approximation by just expanding the integrand as a series in $\alpha$. I think it will basically be $\frac{1}{2}\left(1 + \alpha\mathbb{E}_{W_T}\left[\int_0^T W_t \mathrm{d}t\right] \right) + O(\alpha^2) = \frac{1}{2} + \frac{\alpha}{4 T} W_T + O(\alpha^2)$


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.