Convergence in distribution and Skorokhod Convergence for the general theory of processes

Abstract

This paper proves some Skorokhod Convergence Theorems for processes with filtration. Roughly, these are theorems which say that if a family of processes with filtration (X n ,ℱ n ),n∈ℕ, converges in distribution in a suitable sense, then there exists a family of equivalent processes (Y n ,ℊ n ),n∈ℕ, which converges almost surely. The notion of equivalence used is that of adapted distribution, which guarantees that each (X n ,ℊ n ) has the same stochastic properties as (X n ,ℱ n ), with respect to its filtration, such as the martingale property or the Markov property. The appropriate notion of convergence in distribution is convergence in adapted distribution, which is developed in the paper. Fortunately, any tight sequence of processes has a subsequence which converges in adapted distribution. For discrete time processes, (Y n ,ℊ n ),n∈ℕ, and their limit (Y, ℊ) may be taken as all having the same fixed filtrationℊ n =ℊ. In the continuous time case, theY n ,ℊ n may require different filtrationsℊ n , which converge toℊ. To handle this, convergence of filtrations is defined and its theory developed.