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The Itô Integral and Itô's Lemma

This Demonstration illustrates (a discrete version of) the most fundamental concept in stochastic analysis—the Itô integral and its most fundamental property—Itô's lemma. Choose the integrand from the dropdown menu. The graph displays four curves (two of which coincide in the case of the first integrand) that show approximations of a path of Brownian motion (the integrator), the chosen integrand, and the left- and right-hand sides in Itô's formula (see the details). As you decrease the size of the time step the latter two curves come closer together, showing that they coincide in the limit (i.e., actual Brownian motion).
Mouse over a curve to see the stochastic concept for which the curve is an approximation.

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The concept of the Itô stochastic integral is one of the most fundamental concepts of stochastic analysis, with a huge number of applications ranging from quantum mechanics to mathematical finance. Here we consider only the most basic Itô integral with respect to a one-dimensional Brownian motion (Wiener process), where we take as integrand a suitable twice-differentiable function of a Brownian motion . In this situation Itô's lemma can be written as follows:
.
This should be compared with the statement of the fundamental theorem of calculus for the usual Riemann–Stielties integral. The difference between the two is the presence of the time integral term , which denotes the stochastic version of the Riemann–Stieltjes integral. Note that all the expressions in the statement of Itô's lemma above that depend on are in fact stochastic processes. In the Demonstration we graphically construct paths of discrete random walks approximating both sides of the equation, obtained by replacing the Brownian motion with an approximating discrete random walk. As the random walk approximates the Brownian motion more closely, the approximating paths of the processes on both sides of the equation converge to each other. Beside the processes corresponding to both sides of the equation (colored red and green) we also show the path of Brownian motion that drives all the processes and the corresponding path of the integrand process.
Note that the paths of our discrete Itô integral integral are "step functions", which can be seen clearly when the step size is small.
Note that the Itô integral with respect to continuous Brownian motion cannot be defined on a path-by-path basis, as in this Demonstration. However, this discrete version of the Itô lemma can be rigorously proved and the continuous version can be derived from it. This is done in the section entitled "Discrete Version of Itô's Formula", in chapter VII of the reference below:
A. N. Shirayev, "Discrete Version of Itô's Formula," Probability, 2nd ed., New York: Springer, 1995.
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