Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach

CHAPTER 35

THE ITÔ FORMULA

The Itô formula in stochastic calculus is similar to the chain rule in Leibniz-Newton calculus. In this chapter, we introduce the simple Itô formula and its variations.

35.1 Basic Concepts and Facts

Definition 35.1 (C²-Function). A function f on [a, b] is called a C²-function if and only if f is twice differentiable and the second derivative f″ is continuous on [a, b].

Definition 35.2 (Stochastic Process Space . Let be a filtration under consideration. The space is defined to be the space of all stochastic processes , satisfying the following conditions:

(a) f(t) is adapted to the _t.

(b) a.s.

Definition 35.3 (Itô Process). A stochastic process {X_t : a ≤ t ≤ b} is called an Itô process if it has the form

(35.1)

where X_a is -measurable, , and .

For convenience, ...