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Change of variables in a second order SDE

  1. Sep 1, 2012 #1
    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

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

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

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

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

    [itex]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)[/itex]

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

    Thanks in advance. /Simon
     
  2. jcsd
  3. Sep 3, 2012 #2
    I have made some progress in the work. Treating [itex]H[/itex] as constant [itex]\sigma_v[/itex] can be found to be

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

    by using Itô's lemma on the more simple form

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

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

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

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

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

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