I thought of a simple example on my way to my commute. (It's a short walk.) The Tanaka equation is an example (admittedly: not a diffusion, not a case o plug-a

I thought of a simple example on my way to my commute. (It's a short walk.) The Tanaka equation is an example (admittedly: not a diffusion, not a case o plug-and-chug the Ito lemma) of a process driven by a Brownian with a discontinuous probability distribution. How the hell? From memory, dTNK = dB if dB>0 else -dB Now imagine a model with three equations. X1 is a bog-standard geometric diffusion (we could have picked something that's chi-squared distributed driven by a Brownian from standard interest rate models) as in the 1970s Black-Scholes models. But instead of having an exogenous volatility, it has its Brownian term dB1 multiplied by a second equation, X2. dX2 could be a standard mean-reversion equation, but again its dB2 is multiplied by X3. dX3 is something like abs(dB3 - dB4* X3). Voilà, an equation (X1) with a sudden break driven by the level of a mean-reverting equation (X2, which tells us volatility should come down in finite time even if it grows by a lot at times) that's set to blow up at a X2-dependent but stochastic level. Don't get me wrong, Poisson-like jumps are very common (they're precisely the limiting process for sudden jumps) but people overstate (perhaps because they didn't really read the conditions for Ito isometry) how much a Brownian motion forces a system into normality or smoothness. But hey, people get away with being hipsters about programming languages, why shouldn't they do that for stochastic calculus too, you know?