SDEs with uniform distributions: Peacocks, Conic martingales and mean reverting uniform diffusions.

Summary We introduce a way to design Stochastic Differential Equations of diffusion type admitting a unique strong solution distributed as a uniform law with conic time-boundaries. We connect this general result to some special cases that where previously found in the peacock processes literature, and with the square root of time boundary case in particular. We introduce a special case with linear time boundary. We further introduce general mean-reverting diffusion processes having a constant uniform law at all times. This may be used to model random probabilities, random recovery rates or random correlations. We study local time and activity of such processes and verify via an Euler scheme simulation that they have the desired uniform behaviour.