Semi-Martingales and Measure Theory

Abstract

Publisher Summary This chapter introduces semi-martingales as defining measures on the previsible sigma field with values in the space L 0 of measurable functions. A fundamental theorem by Doob states that, if M is a martingale and T a stopping time, the stopped process M T is again a martingale. According to Doob's theorem, a martingale is a local martingale, but, of course, the converse need not be true. In particular, if M is a local martingale, M t need not be integrable. The optional sigman field is the sigma field generated by the real cadlag processes, and the previsible sigma field is the sub sigma field Pre generated by the real continuous processes. The optional sigma field can also be generated by the closed epigraphs of the stopping times and the previsible sigma field by the open epigraphs of the stopping times, a fortiori by the stochastic intervals.