First of all, apologies in advance for any imprecision in terms- it's been a while since I worked with these concepts and I'm quite rusty.

Consider a stochastic system $X_t$ that's just a simple thermal Brownian motion:

$$dX_t = \alpha dW_t$$

Let's assume that we subject this system to some kind of auxiliary force proportional to the cube root of its unforced motion, so that we have a new system defined:

$$dX_t = \alpha dW_t + \beta \sqrt[3]{dX_t}$$

If $\beta>0$, this acts as an motive force pushing it further than it would have gone otherwise, if $\beta<0$, this acts as a resistive force. I have no difficulties coding up a discrete approximation for this, where you generate dW from $N(0,\sqrt{\Delta T})$ and then solve implicitly for the $dX_t$ to make your next step, and indeed it "works" more or less like you'd expect, with the damped path tracking the undamped one but with smaller movements:

enter image description here

My question is simple: is there any ways to recast this equation into a more standard form of $dX_t = a(X_t,t)dt + b(X_t,t)dW_t$, eliminating terms of the form $dX_t^n$ using Ito's Lemma or similar, so I can generate the yellow path in the above graph quickly and efficiently? Or does the singularity at 0 in the cube root mean that this is not possible in the limit, even though it appears to work in the discrete case? And can this even work for $\beta >0$, or does that cause some other problem because it isn't a single-valued function to invert?

Thanks in advance!


If such a representation existed, we would have a continuous version. For the sake of sketching an example we could take $\alpha=\beta=1$.

The term under the cube root is very significant so we would have $\mathbb{E}(dX_t)= dt^{1/6}$. Which contradicts the conditions for the SDE of the representation to have a solution.


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