I'm trying to benchmark SDE numerical solver, which is easy for scalar noise, since sufficient number of examples of exactly solvable scalar SDEs are avaliable (cf Kloden-Platen Numerical Solution of Stochastic Differential Equations).
I want to benchmark some methods in the vector case, the problem here is that the only examples of exactly integrable vector SDE's I've found are ones with either linear noise or additive noise. These won't work for me because coefficients by higher stochastic integrals are all zero (cf Kloden-Platen 3.14)
I need at least minimally nonlinear vector SDE with at least two driving processes that can be solved exactly, where can I find one? Is it easy to construct such equation?
Simplest case I've found so far is two dimensional simple random walk recast in polar coordinates. The equations are as follows:
$$ \mathrm{d}r = \frac{1}{2r} \mathrm{d}t + \cos \phi \,\mathrm{d}W_x + \sin \phi \, \mathrm{d}W_y\\ \mathrm{d}\phi = -\frac{\sin\phi}{r} \,\mathrm{d}W_x + \frac{\cos\phi}{r} \, \mathrm{d}W_y $$
with solution
$$ r = \sqrt{W_x^2 + W_y^2}\\ \phi = \arctan(W_x,W_y) $$
Equations are coupled and all coefficients have all (even mixed) derivatives nonzero, which is exactly what I needed for benchmarking.
More generally any curved coordinate chart on $\mathbb{R}^2$ would lead to a similar pair of equations (easily obtained using vector variant of Ito's lemma).