I want to solve a scalar differential equation with a random forcing drawn from a power law distribution. It looks like $$ \frac{dx}{dt} = f(x,t) + \sigma(x,t) \eta_t $$ where $\eta$ represents a power-law noise term that is bounded and always positive.

How can I solve the equation numerically?

  • 1
    $\begingroup$ No differently than you would solve the same SDE using white noise (Wiener process). There is a book by Kloeden and Platen that describes various discretization schemes. $\endgroup$
    – Kurt G.
    Jan 19, 2022 at 12:36
  • $\begingroup$ I'm having trouble understanding how to do this when the noise term has a nonzero mean. Doing the simple Euler-Maruyama recursion, $x_{i+1} = x_i + f(x_i,t) \Delta t + \sqrt{\Delta t} \sigma(x_i,t) \eta_t$, gives results that depend on the step size. Any pointers? $\endgroup$
    – qsfzy
    Jan 25, 2022 at 21:59
  • $\begingroup$ I know the mean of the noise term, so I could just subtract it out $\endgroup$
    – qsfzy
    Jan 25, 2022 at 22:17


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