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This class covers analytical and computational PDE-based methods for models arising in Financial Mathematics, and Economics. Topics include: linear PDEs and pricing and hedging of financial derivatives; optimal stopping and variational inequalities motivated by American options; Hamilton-Jacobi-Bellman equations coming from stochastic control problems such as portfolio optimization, optimal execution and transaction costs; and stochastic differential game systems of PDEs related to exhaustible resources and oligopoly models, as well as related mean field games. Emphasis is on exact, approximate and numerical (finite difference) solutions.