Continuous-time clock representing trajectories for stochastic Moore machines

Establish that, within a systems theory for continuous-time stochastic Moore machines (interpreted as open stochastic differential equations), the behavior functor is representable by a clock system whose state space is the disjoint union \bar{Ω} = ⨿_{t ≥ 0} C([0,t], Ω) of continuous paths up to time t (for an appropriate choice of continuity structure on Ω), where the update operation extends a path by appending a Brownian-motion segment, and where behaviors correspond to maps that perform stochastic integration along such paths to yield the system state at time t.

Background

The paper develops clock systems that represent behaviors for discrete-time deterministic, stochastic (Giry monad on Meas), and nondeterministic (powerset monad on Set) Moore machines within a double-categorical systems-theoretic framework. In the stochastic case, behaviors are represented by morphisms from a stochastic clock built from a filtered probability space, recovering classical notions via conditional probabilities.

The authors seek to extend these results to continuous time, aiming at a systems theory for continuous-time stochastic Moore machines akin to open stochastic differential equations. They conjecture a specific form of the clock object that accumulates continuous sample paths, updated by Brownian increments, and a corresponding notion of behavior via stochastic integration along those paths. However, they note that foundational aspects of the continuous-time systems theory and the precise continuity structure on Ω remain to be fixed.