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

Summary It is known since Kellerer (1972) that for any peacock process there exist mar-tingales with the same marginal laws. Nevertheless, there is no general method for finding such martingales that yields diffusions. Indeed, Kellerer's proof is not constructive: finding the dynamics of processes associated to a given peacock is not trivial in general. In this paper we are interested in the uniform peacock that is, the peacock with uniform law at all times on a generic time-varying support [a(t), b(t)]. We derive explicitly the corresponding Stochastic Differential Equations (SDEs) and prove that, under certain conditions on the boundaries a(t) and b(t), they admit a unique strong solution yielding the relevant diffusion process. We discuss the relationship between our result and the previous derivation of diffusion processes associated to square-root and linear time-boundaries, emphasizing the cases where our approach adds strong uniqueness, and study the local time and activity of the solution processes. We then study the peacock with uniform law at all times on a constant support [−1, 1] and derive the SDE of an associated mean-reverting diffusion process with uniform margins that is not a martingale. For the related SDE we prove existence of a solution in [0, T ]. Finally, we provide a numerical case study showing that these processes have the desired uniform behaviour. These results may be used to model random probabilities, random recovery rates or random correlations.