The Ambit Framework

Abstract

This chapter contains an extensive introduction to fundamental concepts leading to the definition of ambit fields and processes. The theory of Levy bases is followed by concepts of stochastic integration in time and space necessary in order to state a mathematically rigorous definition of an ambit field. The integration theory with respect to Levy bases is not as well-developed as the temporal counterpart (i.e. the integration with respect to Levy process); however, the integration theories introduced by Rajput and Rosinski (Probab Theory Related Fields 82(3):451–487, 1989), Walsh (An introduction to stochastic partial differential equations. In Carmona R, Kesten H, Walsh J (eds) Lecture notes in mathematics 1180, Ecole d’Ete de Probabilites de Saint-Flour XIV (1984), Springer, 1986) and Bichteler and Jacod (Random measures and stochastic integration. In Kallianpur G (ed) Theory and application of random fields, vol 49. Lecture notes in control and information sciences. Springer, Berlin, pp 1–18, 1983) are available and fit our purposes. Following the mathematical definition of ambit fields, we give a series of basic properties focussing on the characteristic function and the second order properties of the fields. Moreover, we explore the link between stochastic partial differential equations and ambit fields. The final part of Chap. 5 is devoted to subordination of Levy bases, where we define the concepts of metatimes and chronometers. Chapter 5 contains a series of particular examples of ambit fields, highlighting the theory as well as serving for later analysis.