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Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach by Hong Xie, Chaoqun Ma, Guojun Gan

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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 {Xt : atb} is called an Itô process if it has the form

(35.1) equation

where Xa is -measurable, , and .

For convenience, ...

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