Some mathematical concepts encountered in Finance

It is needless to say that sophisticated mathematical tools are nowadays commonly used in finance, from derivatives pricing and hedging to statistical arbitrage. In this talk I will focus on the classical problem of derivatives pricing. The mathematical treatment of this problem lies at the intersection between several disciplines of mathematics, and can be viewed under many angles. Starting with the Brownian Motion and the stochastic integral as fundamental modelling objects, the problem moves to stochastic calculus and representation of random variables by stochastic integrals, the celebrated Feynman-Kac representation formula, problems of stochastic control and dynamic programming, viscosity solutions of parabolic equations, game theory, and even more recently optimal transportation. I will finally mention a recent research direction, the so-called "linear market impact models" where the price dynamic is also driven by the agents trading activity, and where the pricing equation becomes a fully non-linear parabolic equation, that relates to the well known porous media equations of physics.