From Diffusions to Semimartingales

This chapter is a quick review of the theory of semimartingales, these processes being those for which statistical methods are considered in this book. A process is a collection X = (Xt) of random variables with values in the Euclidean space R for some integer d ≥ 1, and indexed on the half line R+ = [0,∞), or a subinterval of R+, typically [0, T ] for some real T > 0. The distinctive feature however is that all these variables are defined on the same probability space (Ω,F ,P). Therefore, for any outcome ω ∈ Ω one can consider the path (or “trajectory”), which is the function t 7→ Xt(ω), and X can also be considered as a single random variable taking its values in a suitable space of R-valued functions on R+ or on [0, T ]. In many applications, including the modeling of financial data, the index t can be interpreted as time, and an important feature is the way the process evolves through time. Typically an observer knows what happens up to the current time t, that is, (s)he observes the path s 7→ Xs(ω) for all s ∈ [0, t], and wants to infer what will happen later, after time t. In a mathematical framework, this amounts to associating the “history” of the process, usually called the filtration. This is the increasing family (Ft)t≥0 of σ-fields associated with X in the following way: for each t, Ft is the σ-field generated by the variables Xs for s ∈ [0, t] (more precise specifications will be given later). Therefore, of particular interest is the law of the “future” after time t, that is of the family (Xs : s > t), conditional on the σ-field Ft.