Ordinary single and multi-variable calculus provides tools for dealing with deterministic (and differentiable) functions. Stochastic calculus provides tools for dealing with stochastic processes which are functions of both the time t and the ‘state of nature’ ω.
The most important result: Ito’s lemma
- Riemann: integrates w.r.t. the Lebesgue measure
- Rieman-Stieltjes: integrates w.r.t. a function
- Ito: integrates w.r.t. a stochastic process