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.

**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 *X*_{t} 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 {*X*_{t} : *a* ≤ *t* ≤ *b*} is said to be predictable if the function

is -measurable. ...

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