An Itô integral is a fundamental concept in stochastic calculus. It allows us to integrate a stochastic process with respect to another stochastic process, most commonly Brownian motion. The standard form is the integral of X sub t with respect to d W sub t, where W sub t represents Brownian motion, shown here as an irregular, continuous path.
Why do we need Itô integrals? Standard Riemann or Lebesgue integrals cannot handle Brownian motion because it has three problematic properties. First, Brownian motion is nowhere differentiable, meaning it has no smooth tangent lines anywhere. Second, it has infinite total variation, as shown by these red line segments that would sum to infinity. Third, despite infinite total variation, it has finite quadratic variation. These unique properties require a special integration theory, which is exactly what Itô integrals provide.
The construction of Itô integrals follows a three-step process. First, we partition the time interval from zero to T into smaller subintervals, shown here as red dashed lines. Second, we evaluate the integrand at the left endpoints of each subinterval, marked by green dots. This creates a Riemann-like sum where each term is X at t sub i times the Brownian increment. Finally, we take the limit as the partition becomes infinitely fine. The key difference from standard integrals is using left endpoints, which ensures the integral is well-defined for stochastic processes.
Itô's Lemma is the stochastic calculus version of the chain rule. For a function f of time and a stochastic process X sub t, the differential includes three terms: the partial derivative with respect to time, the first-order term with the stochastic differential, and crucially, a second-order term involving the quadratic variation. This extra term, one-half times the second partial derivative times d X sub t squared, distinguishes stochastic calculus from ordinary calculus. Here we see Brownian motion in blue and its square in red, where Itô's Lemma would apply to compute the differential of the squared process.
Itô integrals have revolutionized mathematics and science. They form the foundation of stochastic differential equations, enable the famous Black-Scholes model in finance, help model random systems in physics, advance signal processing theory, and serve as essential tools in modern probability. This powerful mathematical framework allows us to rigorously handle integration with respect to random processes like Brownian motion.