# Convexity, optimization and stochastic control

I. Karroubi

The aim of the course is to introduce the essential tools of convex analysis, finite-dimensional optimization and stochastic control of diffusion processes.

The course is divided into two parts. The first deals with convex analysis and finite-dimensional optimization. The second introduces the theory of optimal control of diffusion processes.

In the first part, we recall elementary results on convex sets and functions. We then present topological properties of convex sets and separation theorems used in particular in arbitrage valuation theory. We then turn to the regularity properties of convex functions. When convex functions are differentiable, we study their optimization. Finally, we conclude this first part with results on the duality of convex functions.

In the second part, we begin by recalling some results on diffusion processes that are relevant to our study: the existence and uniqueness of solutions to stochastic differential equations, the Markov property and the regularity of the flow with respect to the initial condition. We then present the optimal diffusion control problem, as well as the central tool of our study: the dynamic programming principle. We also introduce its infinitesimal version: the dynamic programming equation. We then focus on the case of regular solutions to the dynamic programming equation using the verification approach. As applications, we present examples from financial theory. We then study the case of irregular value functions with the introduction of viscosity solution theory. We show viscosity properties for the value function of a control problem, as well as a uniqueness result by comparison. Finally, we end this section by presenting an alternative approach to dynamic programming: the maximum principle. This approach gives a necessary condition for optimality in the general case and a sufficient condition in the convex case.

## References

1. L. D. Berkovitz (2001) Convexity and Optimization in R^n. Wiley-Interscience.

2. J.-B. Hiriart-Urruty, C. Lemaréchal (2001) Fundamentals of Convex Analysis, Grundlehren Text Editions, Springer.

3. B. Oksendal (2003) Stochastic Differential Equations, An Introduction with Applications. Universitext, Springer

4. H. Pham (2010) Continuous-time Stochastic Control and Optimization with Financial Applications, Stochastic Modelling and Applied Probability, Springer.

5. R. T. Rockafellar (1996) Convex Analysis. Princeton University Press.

6. N. Touzi (2016) Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE. Fields Institute Monographs.