Change of variables in a second order SDE

[sith]

Hello everyone! I am fairly new to SDE theory, so I'm sorry if my question may be a bit naive. I have the following coupled set of SDE:s

d\phi = \frac{v - v_r}{R}d t + \frac{\pi}{\sqrt{t_c}}d W
d v = A\cos(n\phi - \phi_w)d t + a_v d t + \sigma_v d W.

W denotes a Wiener process, and the parameters v_r, R, t_c, A and \phi_w are constants. The functions a_v and \sigma_v are the drift and variance in v, respectively, solely due to the stochasticity in \phi. So my question is: how do I derive analytical expressions for a_v and \sigma_v? I don't know if this helps, but when disregarding the stochastic processes in \phi and v it will turn into a second order ODE, and one will have the following constant of motion

H(\phi,v) = \frac{v^2 - 2 v_r v}{2 R} - \frac{A}{n}\sin(n\phi - \phi_w).

My first thought of how to solve the problem was to rewrite the expression as v(\phi, H) and using Itô's lemma

d v = \frac{\partial v}{\partial\phi}d\phi + \frac{\partial v}{\partial H}d H + \frac{1}{2}\left(\frac{\partial^2 v}{\partial\phi^2}d[\phi,\phi] + \frac{\partial^2 v}{\partial\phi\partial H}d[\phi,H] + \frac{\partial^2 v}{\partial H^2}d[H,H]\right)

where d[X,Y] is the quadratic co-/variance. Is this a correct approach? Then how do I calculate the differentials d H, d[\phi,H] and d[H,H] for this particular case? Numerical simulations have indicated that \sigma_v is on the form of a Lorentzian function in v, centered around v_r.

Thanks in advance. /Simon

 

[sith]

I have made some progress in the work. Treating H as constant \sigma_v can be found to be

\sigma_v = \frac{\pi A R}{\sqrt{t_c}(v - v_r)}\cos(n\phi - \phi_w)

by using Itô's lemma on the more simple form

d v = \frac{d v}{d\phi}d\phi + \frac{1}{2}\frac{d^2 v}{d\phi^2}d[\phi,\phi].

I also made a mistake in the previous post. It is not \sigma_v that takes the form of a Lorentzian, but \sigma^2_v. Specifically I have found that the Lorentzian approximation is valid in the limit A \ll \frac{n^3 R}{t_c^2}, and it then takes the form

\sigma_v^2 = \frac{\pi^2 A^2 R^2}{2 t_c[v_B^2 + (v - v_r)^2]}
v_B = \frac{\alpha n R}{t_c},

where \alpha \approx 4.92 is a numerical constant. This expression is consistent with the one derived when assuming H is constant in the limit |v - v_r| \gg v_B, and averaging the expression over phase \phi. The questions I am still left with are:

* How do I derive the Lorentzian form of \sigma_v^2, if it is even valid?
* Is there a phase dependence in \sigma_v? (There are no indications on a phase dependence from numerical simulations)
* Is there a way to derive \sigma_v when the condition A \ll \frac{n^3 R}{t_c^2} is violated?

Of course I am also interested in estimating the drifts due to stochasticity in \phi, which I haven't even begun to look at in numerical simulations.