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?
Hey. Why do people upvote a post that says "Taleb is a jerk" and makes a reputation argument in a forceful manner (reminiscent of Taleb's own style) -- and then downvote a post with technical detail?
Probably the fact that you flatly ignored the remark about stuffing your post with opaque technical terms--I know a lot of maths, but I can't make heads or tails of what you're saying. Who are you even talking to? I assume it means something to you, but it doesn't even look like you're trying to communicate your point in a clear manner. Just that you enjoy using words. When trying to read around those technical terms, what is left has a rather nasty and arrogant tone. Therefore, doesn't add much to the discussion = downvote.
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?