Ordinary calculus, as in the first calculus people learn, is deterministic. There are functions in this calculus, whether trig functions like cosine or polynomial functions like f(x) = x3 + 2x2 , and these have definite answers/outputs. In other words, their paths are known with certainty.
Stochastic calculus instead deals with processes whose paths are inherently uncertain. Imagine you’re at a campfire. A log is burning. Which direction will the smoke from those flames take? Nobody could precisely determine that. There are way too many variables with uncertain values from wind speed to wind direction to the temperature of adjacent logs and so on.
Stochastic calculus is for scenarios like these, in which no matter how elegant or sophisticated your model of a process is, it is simply impossible to predict the exact and precise path something like smoke or a stock price will take. You can come up with reasonable bounds for the path, but you cannot derive an exact path.
The most common way to describe these bounds for a random path is what’s commonly known as Brownian motion. This is sometimes also called a Wiener process. Within this realm of uncertainty, we can define boundaries for a path that are likely to be accurate some percentage of the time.
So, stochastic calculus involves processes that describe random motion, which is most commonly referred to as a random walk. Using these processes is more complicated than using deterministic functions, but conceptually the same calculation tools you learned in ordinary calculus still apply in stochastic calculus. There are integrals. There is a chain rule. There are Taylor expansions and differential equations.
How is this used? Stochastic calculus is used within finance to describe the movement/path of stock prices. Stock prices follow a path that is well described as geometric Brownian motion.
What does that path look like? What does the path of smoke from a burning log look like? Each smoke stream tends to bend and wiggle upwards at some angle from the burning log. Mostly they go upwards but not straight as an arrow. You’re also not likely to see smoke bend downwards nor are you likely to see it shoot out horizontally. There are probable boundaries to their directional paths, but pinpoint accurate predictions of their paths are beyond us. Stochastic calculus helps us to map those boundaries.