Operator Methods for Continuous-Time Markov Processes

This chapter surveys a set of mathematical and statistical tools that are valuable in understanding and characterizing nonlinear Markov processes. Such processes are used extensively as building blocks in economics and finance. Operator methods begin with a local characterization of the Markov process dynamics. This local specification takes the form of an infinitesimal generator. The infinitesimal generator is itself an operator mapping test functions into other functions. From the infinitesimal generator, one constructs a family (semigroup) of conditional expectation operators. The operators exploit the time-invariant Markov structure. Each operator in this family is indexed by the forecast horizon; the interval of time between the information set used for prediction and the object that is being predicted. Operator methods allow ascertaining global and, in particular, long-running implications from the local or infinitesimal evolution. These global implications are reflected in the implied stationary distribution; the analysis of the eigenfunctions of the generator that dominate in the long run; and the construction of likelihood expansions and other estimating equations.