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Stochastic Calculus: An Introduction with Applications

What are the applications of stochastic calculus?
In finance, the stochastic calculus is applied to pricing options by no arbitrage.
In biology, it is applied to populations' models, and in engineering it is applied to filter signal from noise.
Not everything is proved, but enough proofs are given to make it a mathematically rigorous exposition.
What is the introduction of stochastic calculus?
Stochastic calculus is the area of mathematics that deals with processes containing a stochastic component and thus allows the modeling of random systems.
Many stochastic processes are based on functions which are continuous, but nowhere differentiable.
How hard is stochastic calculus?
Stochastic calculus is genuinely hard from a mathematical perspective, but it's routinely applied in finance by people with no serious understanding of the subject.
Two ways to look at it: PURE: If you look at stochastic calculus from a pure math perspective, then yes, it is quite difficult.
Stochastic calculus is not something that you would encounter in an elementary calculus sequence.
It is typically a graduate level course for applied mathematics and statistics majors, and requires a firm grasp of real analysis, differential equations, and probability theory, among other subjects.

What are the applications of stochastic calculus?