A regularized Kellerer theorem in arbitrary dimension

TL;DR

This paper extends Kellerer's theorem to multiple dimensions, demonstrating the existence of Markov martingales for peacocks after Gaussian regularization, and discusses non-uniqueness issues in higher dimensions.

Contribution

It provides a multidimensional extension of Kellerer's theorem, introduces a new compactness result for martingale diffusions, and highlights the necessity of regularization for existence.

Findings

Existence of Markov martingales for multidimensional peacocks after Gaussian regularization.

Counterexamples showing non-uniqueness in dimensions d ≥ 2.

A new compactness result for martingale diffusions.

Abstract

We present a multidimensional extension of Kellerer's theorem on the existence of mimicking Markov martingales for peacocks, a term derived from the French for stochastic processes increasing in convex order. For a continuous-time peacock in arbitrary dimension, after Gaussian regularization, we show that there exists a strongly Markovian mimicking martingale It\^o diffusion. A novel compactness result for martingale diffusions is a key tool in our proof. Moreover, we provide counterexamples to show, in dimension $d \geq 2$, that uniqueness may not hold, and that some regularization is necessary to guarantee existence of a mimicking Markov martingale.