- #1

sith

- 14

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[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