Two behavioural pseudometrics for continuous-time Markov processes

TL;DR

This paper introduces and compares two new behavioural pseudometrics for continuous-time Markov processes, extending the concept of bisimulation metrics from discrete to continuous time, with applications to diffusions and jump diffusions.

Contribution

It proposes a second pseudometric based on trajectories for continuous-time Markov processes and compares it with an existing pseudometric, advancing the understanding of behavioral equivalences in continuous time.

Findings

Two pseudometrics for continuous-time Markov processes are developed.

The trajectory-based pseudometric is compared with the kernel-based pseudometric.

The paper demonstrates the applicability of these pseudometrics to diffusions and jump diffusions.

Abstract

Bisimulation is a concept that captures behavioural equivalence of states in a variety of types of transition systems. It has been widely studied in discrete-time settings where a key notion is the bisimulation metric which quantifies "how similar two states are". In [ 11], we generalized the concept of bisimulation metric in order to metrize the behaviour of continuous-time Markov processes. Similarly to the discrete-time case, we constructed a pseudometric following two iterative approaches - through a functional and through a real-valued logic, and showed that the outcomes coincide: the pseudometric obtained from the logic is a specific fixpoint of the functional which yields our first pseudometric. However, different from the discrete-time setting, in which the process has a step-by-step dynamics, the behavioural pseudometric we constructed applies to Markov processes that evolve continuously through time, such as diffusions and jump diffusions. While our treatment of the pseudometric in [11] relied on the time-indexed Markov kernels, in [ 8 , 9, 10 ], we showed the importance of trajectories in the consideration of behavioural equivalences for true continuous-time Markov processes. In this paper, we take the work from [11 ] further and propose a second behavioural pseudometric for diffusions based on trajectories. We conduct a similar study of this pseudometric from both the perspective of a functional and the viewpoint of a real-valued logic. We also compare this pseudometric with the first pseudometric obtained in [11].