John Noble: Time Homogeneous Diffusions with Given Marginal at a Deterministic Time

The subject of the seminar is a proof of the result that for any probability law μ with compact support in ℝ and a given deterministic time t > 0, there exists a gap diffusion martingale with this law at a time t > 0.

The method starts by constructing a discrete time process X on a finite state space, where X_τ has law μ, for a geometric time τ, independent of the diffusion. This argument is developed, using a fixed point theorem, for τ an independent time with negative binomial distribution, using the fact that a negative binomial variable is the sum of independent geometrically distributed variables. Reducing the time mesh gives a continuous time diffusion with prescribed law for a random time τ with Gamma distribution. Keeping E[τ] = t fixed, the parameters of the Gamma distribution are altered, giving the prescribed law for deterministic time. An approximating sequence, using some theory from Krein's strings establishes the result for arbitrary state space with compact support in ℝ.