A regularized Kellerer theorem in arbitrary dimension

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 $d\ge 2$, that uniqueness may not hold, and that some regularization is necessary to guarantee existence of a mimicking Markov martingale.

25 pages, 1 figure

References in corpus (6)

• Fitting Martingales To Given Marginals
• Limits Of One Dimensional Diffusions
• Properties of Expectations of Functions of Martingale Diffusions
• A Generalized Backward Equation For One Dimensional Processes
• Optimal control of martingales in a radially symmetric environment
• SDEs with no strong solution arising from a problem of stochastic control

Cited by in corpus (1)

• Calibration of the Bass Local Volatility model