Branches of maths a quant actually uses

What kinds of mathematics appear in quantitative finance? Cover both the ways problems are modelled and the tools used to solve the resulting models.

Modelling uses probability and stochastic calculus, PDEs, statistics and occasionally discrete or game-theoretic models; solving uses closed forms, numerical PDE methods, simulation and optimisation. Prices are framed either as expectations under a risk-neutral measure (stochastic calculus, Itô, martingales) or as solutions of PDEs; statistics and time-series methods fit parameters. Solutions come from explicit formulas where possible, otherwise finite differences, trees, Monte Carlo, numerical integration and linear algebra.