BOOK REVIEW

Convex Stochastic Optimization: Dynamic Programming and Duality
in Discrete Time

by Teemu Pennanen and Ari-Pekka Perkkiö

reviewed by Miklós Rásonyi (Rényi Institute)

Convex stochastic optimization is a classic research area, there are several monographs available. What can this new advanced textbook then give to us, people committed to financial mathematics?

The maximization of concave utility functions has been object of intensive research in past decades. In particular, a sophisticated framework has been developed for semimartingale models: besides the primal problem of finding the best portfolio, also the dual problem of minimizing a convex functional over the set of martingale densities has been thoroughly studied. In subsequent work, markets with imperfections (transaction costs, price impact, etc.) could also be covered adopting similar techniques.

Most of this progress has been achieved with little reliance on the huge literature of convex optimization. As prominent experts of the latter, the authors of the present book followed another path: they recast financial problems using the machinery of normal integrands and general convex duality theory. In a series of papers they have demonstrated that optimal investment problems can be successfully integrated in a more general, non-finance-specific framework.

In this book, Teemu and Ari-Pekka lead us into the realm of convex stochastic optimization with an eye constantly fixed on financial applications. The sections provide a clean presentation of all major topics in a logical succession. The material shows a deceiving simplicity.

Behind a facade of seemingly innocent, unassuming results, a powerful theory is slowly developed in its sharpest form. I mention just three instances: Theorem 2.118 proves perhaps the most general existence theorem available for optimal investments in a frictionless discrete-time market, even including portfolio constraints, deduced from very general considerations; the corresponding dual problem is studied in Subsections 3.4.5 and 4.3.5; in Theorem 4.29, the Dalang-Morton-Willinger theorem is proved all of a sudden, again in a comprehensive duality framework.

There are also novelties in the general theory. Subsection 4.2.2 develops duality for not necessarily bounded from below convex objectives: at this level of generality such results were missing from available literature.

The Appendix is a compendium of all the results you may need in such a setting from convex analysis and probability. I also mention Chapter 5, where a presentation of Orlicz spaces is given: in this domain one can hardly find suitable reference works.

Recent efforts to theoretize robust finance resulted in a spectacular amount of papers with discrete-time models about arbitrage and optimal investment under model uncertainty. I think researchers working in that area will find useful the viewpoint taken in this new monograph.

To sum up, Teemu and Ari-Pekka provide us not only with an excellent manual but, actually, a rich source of inspiration to apply the deep theory developed here to further fields of portfolio choice problems in financial mathematics.