Optimal stochastic control

N. Touzi

The objective of the course is to introduce the essential tools of optimization and stochastic control of distributions. Theoretical results are accompanied by applications to the problem of portfolio management, optimal dividend distribution, and derivative hedging in the presence of market imperfections.

We begin with a review of convex optimization and deterministic optimal control. We develop the dynamic programming approach and the Pontryagin maximum principle.

We then get into the heart of the matter by introducing standard stochastic control problems which we analyse using the dynamic programming approach and the corresponding dynamic programming equation. The verification argument solves the simplest portfolio management problem. We present a similar treatment of optimal stopping problems and the corresponding obstacle PDEs.

We then present the basic notions of viscosity solutions of degenerate elliptic PDEs of the second order which allow a finer analysis of stochastic control problems.

Finally, we introduce the class of stochastic target problems that use a so-called geometric programming principle. The use of viscosity solutions for these problems is crucial. We present applications to derivative hedging problems under portfolio and gamma constraints, and to the quantile hedging problem.


  • W. Fleming et M. Soner, Controlled Markov Processes and Viscosity Solutions, Second Edition. Springer (2006).

  • N. Touzi, Optimization, Stochastic Control, and Applications to Finance. Polycopié.