Section: Overall Objectives

Stochastic control and backward stochastic differential equations (BSDEs)

Participants : Vlad Bally, Jean-Philippe Chancelier, Marie-Claire Quenez, Agnès Sulem.

B. Øksendal (Oslo University) and Agnès Sulem have written a book on Stochastic control of Jump diffusions [10] ). The types of control problems covered include classical stochastic control, optimal stopping, impulse control and singular control. Both the dynamic programming method and the maximum principle method are discussed, as well as the relation between them. Corresponding verification theorems involving the Hamilton-Jacobi Bellman equation and/or (quasi-)variational inequalities are formulated. There are also chapters on the viscosity solution formulation and numerical methods. In the second edition (2007), a chapter on optimal control of stochastic partial differential equations driven by Lévy processes and a section on optimal stopping with delayed information have been added. Applications to portfolio optimization problems and insurance problems have been studied.

In the context of risk measures, M.C. Quenez (Prof Paris VII) has shown how some dynamic measures of risk can be induced by Backward Stochastic Differential Equations and A. Sulem and B. Øksendal in [90] have studied risk-indifference pricing in incomplete markets with jumps using stochastic control theory and PDE methods.