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Consider the following stochastic differential equation: $$\ddot{X_t}+\omega_0^2X_t=dW_t.$$ I thought the way to do this would be to let: $$dX_t=\dot{X}_tdt,$$ $$d\dot{X}_t=-\omega_0^2X_tdt+dW_t,$$ however, this post says that the system of equations should be this: $$dX_t=\dot{X}_tdt+dW_t,$$ $$d\dot{X}_t=-\omega_0^2X_tdt.$$ The way it is done in that post seems to be correct as they are able to reproduce theoretical results numerically.

Edit:

The code in the linked post does not work if I switch the force vector around? Do you know why it doesn't work?

Peanutlex
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  • It seems incorrect to me, the regularity is wrong. $X_t$ should be classically differentiable (once, not twice). A simpler scenario in which you can see that this is wrong would be $\ddot{X}_t=\dot{W}_t$, which is to be interpreted as $X_t=x_0+tv_0+\int_0^t W_s ds$ (where $x_0$ is the initial condition for position and $v_0$ is the initial condition for velocity). – Ian Jan 08 '19 at 23:59
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    In the notation of the answer you referred to the issue is simply that $dW_t$ should be in the second component of the forcing vector. – Ian Jan 09 '19 at 00:01

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