I am looking at a process with increments:[$]dX/X = \alpha \sqrt{dt} - \beta \max(0,dW_t)[$]where W is a standard BMHas anyone studied this kind of process? I would assume it to be rather explosive except when alpha = beta/sqrt(2 *Pi )Any pointers appreciatedThank youe.

It's obscure, non-standard notation IMO, but I will guess that it: (i) indeed can be made sensible (only) in the special case you mention, and (ii) is equivalent in that special case to [$]dX_t/X_t = \sigma \, dB_t[$], where [$]B_t[$] is another BM, and where(iii) [$]\sigma^2 = \frac{\beta^2}{2 \pi} \, E [(1 - \sqrt{2 \pi} Z^+)^2][$], using [$]Z^+ = \max(0,Z)[$], and [$]Z[$] a standard normal variate. This SDE in (ii) of course has an easy and well-known solution. In other words, my guess is it's just a GBM process obscurely disguised. Vaguely, this kind of thing comes up in Nelson's construction of a diffusion limit of GARCH-type models.(`ARCH models as diffusion approximations') There, the "dt's" would appear as [$]\Delta t[$]'s which are tending to 0, and the "dW_t's" are really [$]Z_t \sqrt{\Delta t}[$]'s,all coming from the discrete-time GARCH model. In that theory, you get a conventional SDE in the continuum limit by moment matching,and you would never write [$]\sqrt{dt}[$], etc.

Hi AlanThank you for your response and the connection with GARCHDiscrete time analysis of this type of process indeed seems the way to go (except in the special case mentioned)e.