Refine
Is part of the Bibliography
- yes (2)
Keywords
- Compound Poisson processes (1)
- Jump processes (1)
- Markov processes (1)
- Reciprocal processes (1)
- Stochastic bridges (1)
- coupling (1)
- dynamical system representation (1)
- monotone random (1)
- monotonicity conditions (1)
- partial ordering (1)
Institute
- Institut für Mathematik (2)
- Extern (1)
We formalize and analyze the notions of monotonicity and complete monotonicity for Markov Chains in continuous-time, taking values in a finite partially ordered set. Similarly to what happens in discrete-time, the two notions are not equivalent. However, we show that there are partially ordered sets for which monotonicity and complete monotonicity coincide in continuoustime but not in discrete-time.
Processes having the same bridges as a given reference Markov process constitute its reciprocal class. In this paper we study the reciprocal class of compound Poisson processes whose jumps belong to a finite set . We propose a characterization of the reciprocal class as the unique set of probability measures on which a family of time and space transformations induces the same density, expressed in terms of the reciprocal invariants. The geometry of plays a crucial role in the design of the transformations, and we use tools from discrete geometry to obtain an optimal characterization. We deduce explicit conditions for two Markov jump processes to belong to the same class. Finally, we provide a natural interpretation of the invariants as short-time asymptotics for the probability that the reference process makes a cycle around its current state.