# Change of variables in a second order SDE

## Main Question or Discussion Point

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 $v_r$.

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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.