def. Ito Integral. Let the following:

  • Brownian Motion , an Ito process
  • and thus
  • Then the Ito integral of with respect to is defined as such:
  • Visualization. Think of the 3D visualization of the Riemann–Stieltjes integral, but both and are zigzags.
  • The solution to the Ito integral is also a stochastic process . It also has the following properties because the integrator is
    • is a martingale
    • let
      • Ito Isometry:

Remark. We often use shorthand notation (abuse of notation) to write an integral term like this:

  • by definition of the Reimann Integral
  • Now, these also exist but they are trivial (Ito Isometry):

Reference Table of Common Ito Integrals

Stochastic IntegralResultVariance

Motivation. Unfortunately we cannot calculate ito intergrals directly. For example, as we saw in the above example finding using just the definition was a laborious process. In order to make this easier, we use the following lemma.

thm. Ito’s Lemma. (Ito’s Chain Rule) The stochastic calculus chain rule. Stated in two ways.

  • is an Ito process
  • is a normal function
  • Then:
  • ,

Example. Suppose we want to find the integral . Since it’s nearly impossible to find it from the definition of the Ito integral, let us use Ito’s lemma. Arbitrarily set an ordinary function .

  • ! Now, it is important to note that we normally don’t know what function to use, but in this case assume we are given a nice that will lead to the answer. Given this function, we then have:
  • , , and
  • Because is a standard brownian motion it conforms to the form which means for Ito’s lemma . Then, we use Ito’s lemma:


Motion: Differential:

thm. Ito’s Product Rule. for two stochastic processes

Ito Processes

def. Ito Process. An Ito process is a type of Stochastic Process that is defined by a certain :

  • must be integrable in the ordinary sense
  • must be integrable in the stochastic sense
    • It actually needs to be integrable twice As an abuse of notation we can write:
  • This is called a Stochastic Differential Equation (even if it’s not actually a differential equation). Essentially, a function of an Ito process is also an Ito process.