Stochastic calculus is a branch of mathematics that extends classical calculus to handle random processes. Unlike ordinary functions, these processes have continuous but non-differentiable paths, like Brownian motion shown here. The foundation includes probability theory, stochastic processes, Brownian motion, stochastic integration, stochastic differential equations, and Itô's lemma.
Brownian motion has several key properties that make it fundamental to stochastic calculus. It starts at zero, has continuous but non-differentiable paths, and has independent Gaussian increments. The variance of increments equals the time difference, and its quadratic variation grows linearly with time. These properties distinguish it from ordinary functions and require new mathematical tools.
The Itô integral extends integration to stochastic processes like Brownian motion. Unlike ordinary integrals, we must use left-endpoint evaluation in Riemann sums because future values are unknown. The integral is constructed by partitioning time, forming Riemann sums with left endpoints, and taking limits. This creates a well-defined integral for non-anticipating functions that don't peek into the future.
Itô's lemma is the fundamental chain rule for stochastic calculus. When differentiating a function of a stochastic process, we get an extra second-order term due to the quadratic variation of Brownian motion. This is because d W squared equals d t, not zero as in ordinary calculus. For example, when computing d of X squared, we get the usual chain rule term plus an additional sigma squared d t term.
To summarize, stochastic calculus provides the mathematical framework for analyzing random processes. Brownian motion serves as the cornerstone with its continuous but non-differentiable paths. The Itô integral extends integration to stochastic processes, while Itô's lemma modifies the chain rule to account for quadratic variation. These fundamental tools enable us to model complex systems in finance, physics, and engineering where randomness plays a crucial role.