Abstract
The dynamics of a Markov process are often specified by its infinitesimal generator or, equivalently, its symbol. This paper contains examples of analytic symbols which do not determine the law of the corresponding Markov process uniquely. These examples also show that the law of a polynomial process in the sense of [4, 5, 11] is not necessarily determined by its generator if it has jumps. On the other hand, we show that a combination of smoothness of the symbol and ellipticity warrants uniqueness in law. The proof of this result is based on proving stability of univariate marginals relative to some properly chosen distance.
Highlights
Consider a system whose state at time t is represented by a vector X(t) in Rd
On the other hand, X(t) is random, it may be viewed as a Markov process whose local dynamics can be specified in terms of a stochastic differential equation, its infinitesimal generator, its local semimartingale characteristics, or its symbol
As in the deterministic case, this immediately leads to the question of existence and uniqueness of a stochastic process exhibiting the given local dynamics
Summary
Consider a system whose state at time t is represented by a vector X(t) in Rd. In applications the dynamics of such a system are often described by specifying how X(t) changes as a function of the current state X(t). Ellipticity, on the other hand, requires a continuous martingale part to be present, which often is not the case either This piece of research is motivated by the desire to come up with a general uniqueness result for Markov processes that may not have a continuous martingale part or a natural representation as a SDE. This is the first main result of this paper. Further unexplained notation is used as in [9, 21]
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