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IMA New Directions Short Course
New Mathematical Models in Economics and Finance
June 7-18, 2010

Rene CarmonaPrinceton University
Ivar EkelandUniversity of British Columbia

From June 7-18, 2010 the IMA will host an intensive short course designed to efficiently provide researchers in the mathematical sciences and related disciplines the basic knowledge prerequisite to undertake research in mathematical finance and economics. The course will be taught by Rene Carmona, Department of Operations Research & Financial Engineering, Princeton University, Nizar Touzi, Ecole Polytechnique, Paris, and Guillaume Carlier, CEREMADE, University Paris Dauphine. The primary audience for the course is mathematics faculty. Some background in probability and stochastic processes are expected. Participants will receive full travel and lodging support during the workshop.

The course will consist of lectures on mathematical models for the energy and emission markets, stochastic differential equations, and optimal transport. There will be two lectures each day, with additional planned activities daily including guest lecturers. The tentative syllabus for the course is as follows:

Lecturer: Rene Carmona
Topic: Mathematical Models for Climate Change and the Energy and Emission Markets

  1. Energy and emissions markets, and the existing cap-and-trade schemes
  2. Discrete time competitive equilibrium models for cap-and-trade schemes and the carbon tax
  3. Mathematical models for allocation mechanisms and cost distribution
  4. Discrete time competitive equilibrium models for cap-and-trade schemes and the Clean Development Mechanism
  5. Stochastic optimization and first continuous time models of cap-and-trade schemes
  6. Binary martingales and option pricing:
    • 1) Reduced form models;
      2) Perturbation methods
  7. Singular BSDEs appearing in cap-and-trade models
  8. Game theory, Nash equilibrium, and electricity prices with strategic market players
  9. Stochastic games: Pontryagin maximum principle and the Isaacs conditions
  10. Examples of linear-quadratic stochastic games in environmental finance


Lecturer: Nizar Touzi
Topic: Backward Stochastic Differential Equations and Fully Nonlinear Partial Differential Equations

  1. Stochastic calculus and martingale representation theorems
  2. Backward Stochastic Differential Equations (BSDEs): existence theory and connection with Partial Differential Equations (PDEs)
  3. Optimal Stochastic Control: dynamic programming principle and BSDE formulation
  4. Numerical methods for BSDEs and applications
  5. Fully nonlinear PDEs.


Lecturer: Guillaume Carlier
Topic: Optimal Transport

  1. Monge-Kantorovich optimal transport problem
  2. Strictly convex transportation costs
  3. The case cost=distance
  4. Economic applications of optimal transport
  5. Congested transport