ABSTRACT
Decision making in all spheres of activity involves uncertainty. If rational decisions have to be made, they have to be based on the past observations on the phenomenon. Data collection, model building and inference from the data collected, validation of the model, and refinement of the model are the key steps involved in any rational decision-making process. Stochastic processes are widely used for model building, and the subject of inference for stochastic processes is of importance both from the theoretical point of view as well as from the application aspect. Several books have been published during the last twenty years, starting with Basawa and Prakasa Rao [5], Prakasa Rao [52, 53], Grenander [18], Kutoyants [38], and more recently by Karr [35] and Fleming and Harrington [15] on inference for stochastic processes. Prakasa Rao [54], Prakasa Rao and Bhat [56], Prabhu [50], Prabhu and Basawa [51], and Basawa and Prabhu [4] give comprehensive accounts of recent developments in the area of inference for stochastic processes. In a recent book, Statistical Inference for Diffusion Type Processes (cf. [55]), we have discussed various methods of estimation of the parameters involved in the process when the process is continuously observed over time or when a sampled data on the process is available as is generally feasible. The class of diffusion-type processes is a subclass of the class of semimartingales. Our discussion here concerns the general asymptotic theory of semimartingales and their statistical inference. The notion of a semimartingale has been found to be of major interest in stochastic modeling, as it includes several types of processes such as point processes, diffusion processes, diffusion processes withjumps, etc., that are widely used for model building. Asernimartingale is essentially the sum of alocal martingale and a process that is of bounded variation. A brief introduction for such 2processes is given in [53]. As the concept of a semimartingale and its properties are not widely discussed in the books on statistical inference and are not widely known to the statisticians and modelers, we give a review of some results from the theory of semimartingales in this chapter, giving proofs occasionally. The books by Elliott [12], Bremaud [6], Liptser and Shiryayev [44, 45], and Kallianpur [33], for instance, deal extensively with this topic, and our review in this chapter is based on these as well as a few other related books.
