Statistical Differential Equation

[fuchs]

Stochastic Differential Equation

Hi there,
I am trying to solve (analytically) a stochastic differential equation of the form:
$\frac{d^2}{dt^2}x +\left(k(t)+\delta k \ t\right)x)=0$

Here \delta k is a random (gaussian) white noise. Note, that in the differential equation it is multiplied by t which makes this equation hard to solve.

I would like to calculate <x(t)> and <x(t)^2>. Any suggestion what formulas I could use? Unfortunately I never had a lecture in SDE.

 

[gtoguy97]

Thanks a lot for your help!This type of differential equation can be solved using the method of stochastic averaging. This method is based on the idea that we can represent the solution to the SDE as a collection of random variables that have a mean value that converges to the true solution as the number of random variables approaches infinity. To do this, we first introduce the new variable:z(t) = x(t) + \delta k \int_0^t x(s) dsNow, we can rewrite the differential equation as:\frac{d^2}{dt^2}z +k(t)z = 0 This equation can be solved analytically, and the solution is:z(t) = A e^{-\int_0^t k(s)ds} + B e^{\int_0^t k(s)ds}Where A and B are constants determined by the initial conditions.From this solution we can calculate the mean value of x(t):<x(t)> = \frac{A e^{-\int_0^t k(s)ds} + B e^{\int_0^t k(s)ds}}{1 + \delta k \int_0^t \frac{A e^{-\int_0^s k(u)du} + B e^{\int_0^s k(u)du}}{1 + \delta k \int_0^s \frac{A e^{-\int_0^u k(v)dv} + B e^{\int_0^u k(v)dv}}ds}And the mean value of x(t)^2:<x(t)^2> = \frac{A^2 e^{-2\int_0^t k(s)ds} + 2AB e^{-\int_0^t k(s)ds}e^{\int_0^t k(s)ds} + B^2 e^{2\int_0^t k(s)ds}}{1 + 2\delta k \int_0^t \frac{A e^{-\int_