A generator approach to stochastic monotonicity and propagation of order

2018/04/26 by Kraaij, Richard C., Schauer, Moritz
#06A06 (secondary) #60J35 (primary) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1804.10222

Abstract

We study stochastic monotonicity and propagation of order for Markov processes with respect to stochastic integral orders characterized by cones of functions satisfying Φf ≥ 0 for some linear operator Φ. We introduce a new functional analytic technique based on the generator A of a Markov process and its resolvent. We show that the existence of an operator B with positive resolvent such that ΦA - B Φ is a positive operator for a large enough class of functions implies stochastic monotonicity. This establishes a technique for proving stochastic monotonicity and propagation of order that can be applied in a wide range of settings including various orders for diffusion processes with or without boundary conditions and orders for discrete interacting particle systems.

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