We use a minimisation schemeto define a discrete-time Markov chain with the martingale property on a Euclidean space R . As we send the step size of the chain to zero, we obtain an Ito diffusion by a theorem by Stroock and Varadhan. Since this variational approach only uses the metric property of Rd, we conjecture that this approach can be generalised to arbitrary metric spaces. We prove that the method works for diffusions on Rd and sketch the idea for diffusions on general metric spaces, and in particular for the case when the metric space is a Wasserstein space. The resulting measure-valued diffusion is known as a measure-valued martingale.
|Date of Award||1 Apr 2020|
|Supervisor||Johannes Zimmer (Supervisor) & Alex Cox (Supervisor)|