I have a stochastic process that describes a particle moving through a field of randomly distributed particles and undergoing scattering collisions (modeled simplistically) off of them. In its simplest form, the equation is
$$ \Delta \vec{v} = \frac{1}{N}\sum_i^N\left[k_1\frac{\hat{x}\,\rm{cos}\,\theta_i + \hat{y}\,\rm{sin}\,\theta_i}{\sqrt{b_i}}+k_2\frac{\hat z}{b_i}\right] $$
with randomly chosen variables
$$ b_i = \rm{rand}\left[0,b_\rm{max}\right]\, ,\,\, \theta_i = \rm{rand}\left[0,2\pi\right] $$
I am looking to find a function that describes the probability distribution of this process. I don't know how to do a stochastic integral analytically but I did a numerical calculation with $k_1=k_2=b_\rm{max}=1, N = 10^4$ and the distribution that I found is shown in the image below. The numerical solution is a good start, but I would like to have something that is a function of the input variables.
Numerical solution: numerical solution plot http://www.rearviewminor.com/logdistribution3.png.
Is it possible to find an analytic solution that describes the 2-D probability distribution of this process? And if so, how is this accomplished? I've tried reading some literature on stochastic calculus but as it's not typically a part of my field I have a difficult time understanding the syntax and lingo.
Thank you!
Edit: Since I received a few upvotes for this question, but no answer, I've asked another question here that is maybe a simplified version of this one.