In the previous two chapters, we introduced stochastic integrals with respect to standard Brownian motions. In this chapter, we expand the definition of stochastic integral such that the integrators are right-continuous and square-integrable martingales.

34.1 Basic Concepts and Facts

Definition 34.1 (Stochastic Process Space be a right-continuous filtration. The stochastic process space ([a, b] × Ω) is the space of all stochastic processes X(t, ω) satisfying the following conditions:

(a) X(t) is adapted to the filtration .
(b) Almost all sample paths of Xt are left-continuous.

Definition 34.2 (Predictable Stochastic Process). Let be the smallest σ-field of subsets of [a, b] × Ω with respect to which all stochastic processes in ([a, b] × Ω) are measurable. A stochastic process {Xt : atb} is said to be predictable if the function


is -measurable. ...

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