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.

**Definition 35.1** (*C*^{2}-Function). A function *f* on [*a, b*] is called a *C*^{2}-*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

where *X*_{a} is -measurable, , and .

For convenience, ...

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