Isometric Approach to Stochastic Integral of Certain Random Measures

Conventions and Notations (1) The measure space (M;M; ) is interpreted as having M as the underly set, M the sigma algebra on M , and the mesaure on M: If M is a metric space then M is taken as the sigma algebra generated by all its metric-open sets. (2) For two measure spaces (M;M; ) ; (M1;M1; 1) the measure space (M M1;M M1; 1) is their product measure space. All measurs in the paper are considered complete. (3) Let the triple ( ;F ; P ) be a proability space andR = (Rn;B (Rn) ; ) the Euclidean-metric mesure space where is the Lebesgue measure on Rn. Let S=(K;U ; ) be their product space, that is, K = [0;1) ; U = B ([0;1)) F ; = P . (4) A …ltration of a measure space (M;M; ) is a non-decreasing family of sub-sigma algebras fMt : t 2 Tg ofM where T is some index set. Filtrations in this paper are augmented with -mull sets of M . (5) As a basic assumption, all stochastic process are Rn-valued, measurable random functions fX (t; !)g from S to R and are adapted to the …ltration fFt : t 2 [0;1)g of ( ;F ; P ) : (6) Conventions and Notations (7) "Return" means "log-Return" 1. Introduction When modelling asset returns by continuous time models, the key is to model the ‡uctuations (or volatility) of the assert price for small durations of time. To allow for uncertainty in assert price, a stochastic part is intergated in each such model by appropriately de…ning the notion of "di¤erentiatial of a random process" or in other words the notion of "stochastic integral of a random process". Highly dependent on both on the path properties of the trajectry of process of integration and the regularity of the integrand, there are usually four ways to de…ne stochastic integral with respect to a random process, namely, parth-wise integration, isometric integration, multiple Ito Date : Feburary 26, 2009. 1991 Mathematics Subject Classi…cation. Primary 05C38, 15A15; Secondary 05A15, 15A18.