The fundamental difference between stochastic calculus and ordinary calculus is that stochastic calculus allows the derivative to have a random component determined by a Brownian motion. Ito's Lemma is a stochastic analogue of the chain rule of ordinary calculus. A fundamental tool of stochastic calculus, known as Ito's Lemma allows us to derive it in an alternative manner. The Binomial Model provides one means of deriving the Black-Scholes equation. This process is represented by a stochastic differential equation, which despite its name is in fact an integral equation. The physical process of Brownian motion (in particular, a geometric Brownian motion) is used as a model of asset prices, via the Weiner Process. The main use of stochastic calculus in finance is through modeling the random motion of an asset price in the Black-Scholes model. In quantitative finance, the theory is known as Ito Calculus. Instead, a theory of integration is required where integral equations do not need the direct definition of derivative terms. This rules out differential equations that require the use of derivative terms, since they are unable to be defined on non-smooth functions.
Many stochastic processes are based on functions which are continuous, but nowhere differentiable. Stochastic calculus is the area of mathematics that deals with processes containing a stochastic component and thus allows the modeling of random systems.
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