There are lots of differences between SDE and ODE. From the theoretical point of view an also from the numerical algorithms used for simulations. But I am interested in knowing if there is a point when you can use a ODE with a random force to model a SDE.


For example, in which conditions the SDE: $$ d X\_t = -\theta X\_t dt + \sigma d W\_t $$


could be represented by an ODE like: $$ \frac{dX(t)}{dt} = -\theta X(t) + \sigma \xi(t) $$


Where $W\_t$ is a Weiner process and $\xi(t)$ is an equivalent random force.


Thanks



What are the difference between modeling with stochastic differential equations (SDE) and ordinary differential equations (ODE) with a random force?
You might want to look into the Wong-Zakai theorem. Essentially, it states that if $\xi\_\epsilon$ is a sequence of smooth approximations to white noise, then the solutions to the random ODE
$$
{dx \over dt} = f(x) + g(x) \xi\_\epsilon\;,
$$
converge to the solutions to the SDE (in Stratonovich form)
$$
dx = f(x)\,dt + g(x)\circ dW\;.
$$
There are of course technical assumptions, but hitting white noise with a smooth enough mollifier satisfies these assumptions, as do piecewise constant approximations (i.e. piecewise linear approximations of the Wiener process).


