A regularized Kellerer theorem in arbitrary dimension

2022/10/25 by Gudmund Pammer, Pammer, Gudmund, Benjamin A. Robinson +3 · 1 citation
Mathematics · #60G44 #60H10 #60J25 (Primary) 60F99 (Secondary) #60J60 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2210.13847

openalex publication_date 2022/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Itô diffusion. A novel compactness result for martingale diffusions is a key tool in our proof. Moreover, we provide counterexamples to show, in dimension d ≥ 2, that uniqueness may not hold, and that some regularization is necessary to guarantee existence of a mimicking Markov martingale.

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