PrefaceDynamic stochastic optimization is the study of dynamical systems subject to random perturbations, and which can be controlled in order to optimize some performance criterion. It arises in decision-making problems under uncertainty, and finds numerous and various applications in economics, management and finance.Historically handled with Bellman's and Pontryagin's optimality principles, the research on control theory has considerably developed over recent years, inspired in particular by problems emerging from mathematical finance. The dynamic programming principle (DPP) to a stochastic control problem for Markov processes in continuous-time leads to a nonlinear partial differential equation (PDE), called the Hamilton-JacobiBellman (HJB) equation, satisfied by the value function. The global approach for studying stochastic control problems by the Bellman DPP has a suitable framework in viscosity solutions, which have become popular in mathematical finance: this allows us to go beyond the classical verification approach by relaxing the lack of regularity of the value function, and by dealing with degenerate singular controls problems arising typically in finance. The stochastic maximum principle found a modern presentation with the concept of backward stochastic differential equations (BSDEs), which led to a very active research area with interesting applications in stochastic analysis, PDE theory and mathematical finance. On the other hand, and motivated by portfolio optimization problems in finance, another approach, called the convex duality martingale method, developed and generated an important literature. It relies on recent results in stochastic analysis and on classical methods in convex analysis and optimization. There exist several monographs dealing with either the dynamic programming approach for stochastic con-. They mainly focus on the theoretical aspects, and are technically of advanced level, and usually difficult to read for a nonexpert in the topic. Moreover, although there are many papers about utility maximization by duality methods, this approach is rarely addressed in graduate and research books, with the exception of the forthcoming one [FBS09].The purpose of this book is to fill in this gap, and to provide a systematic treatment of the different aspects in the resolution of stochastic optimization problems in continuous time with a view towards financial applications. We included recent developments and original results on this field, which appear in monograph form for the first time. We paid VIII Preface attention to the presentation of an accessible version of the theory for those who are not necessarily experts on stochastic control. Although the results are stated in a rather general framework, useful for the various applications, with complete and detailed proofs, we have outlined the intuition behind some advanced mathematical concepts. We also take care to illustrate each of the resolution methodologies using several examples in finance. This monograph is directed towards gra...

We consider the mean-variance hedging problem when the risky assets price process is a continuous semimartingale. The usual approach deals with self-financed portfolios with respect to the primitive assets family. By adding a numéraire as an asset to trade in, we show how self-financed portfolios may be expressed with respect to this extended assets family, without changing the set of attainable contingent claims.We introduce the hedging numéraire and relate it to the variance-optimal martingale measure. Using this numéraire both as a deflator and to extend the primitive assets family, we are able to transform the original mean-variance hedging problem into an equivalent and simpler one; this transformed quadratic optimization problem is solved by the Galtchouk-Kunita-Watanabe projection theorem under a martingale measure for the hedging numéraire extended assets family. This gives in turn an explicit description of the optimal hedging strategy for the original mean-variance hedging problem.We thank the co-editor Freddy Delbaen, an associate editor, and both referees for their careful reading of the manuscript and for many valuable comments that led to an improvement of an earlier version of this paper. Helpful comments from Martin Schweizer are gratefully acknowledged. We also thank Peter Bossaerts for his remarks.

We study the optimal control of general stochastic McKean-Vlasov equation. Such problem is motivated originally from the asymptotic formulation of cooperative equilibrium for a large population of particles (players) in mean-field interaction under common noise. Our first main result is to state a dynamic programming principle for the value function in the Wasserstein space of probability measures, which is proved from a flow property of the conditional law of the controlled state process. Next, by relying on the notion of differentiability with respect to probability measures due to P.L. Lions [35], and Itô's formula along a flow of conditional measures, we derive the dynamic programming Hamilton-Jacobi-Bellman equation, and prove the viscosity property together with a uniqueness result for the value function. Finally, we solve explicitly the linear-quadratic stochastic McKean-Vlasov control problem and give an application to an interbank systemic risk model with common noise.MSC Classification: 93E20, 60H30, 60K35.

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We consider a robust switching control problem. The controller only observes the evolution of the state process, and thus uses feedback (closed-loop) switching strategies, a non standard class of switching controls introduced in this paper. The adverse player (nature) chooses open-loop controls that represent the so-called Knightian uncertainty, i.e., misspecifications of the model. The (half) game switcher versus nature is then formulated as a two-step (robust) optimization problem. We develop the stochastic Perron method in this framework, and prove that it produces a viscosity sub and supersolution to a system of Hamilton-Jacobi-Bellman (HJB) variational inequalities, which envelope the value function. Together with a comparison principle, this characterizes the value function of the game as the unique viscosity solution to the HJB equation, and shows as a byproduct the dynamic programming principle for robust feedback switching control problem.MSC Classification: 60G40, 91A05, 49L20, 49L25.

This paper studies a robust continuous-time Markowitz portfolio selection problem where the model uncertainty affects the covariance matrix of multiple risky assets. This problem is formulated into a min-max mean-variance problem over a set of nondominated probability measures that is solved by a McKean-Vlasov dynamic programming approach, which allows us to characterize the solution in terms of a Bellman-Isaacs equation in the Wasserstein space of probability measures. We provide explicit solutions for the optimal robust portfolio strategies and illustrate our results in the case of uncertain volatilities and ambiguous correlation between two risky assets. We then derive the robust efficient frontier in closed form, and obtain a lower bound for the Sharpe ratio of any robust efficient portfolio strategy. Finally, we compare the performance of Sharpe ratios for a robust investor and for an investor with a misspecified model. K E Y W O R D S ambiguous correlation, continuous-time Markowitz problem, covariance matrix uncertainty, dynamic programming, McKean-Vlasov, Wasserstein space wileyonlinelibrary.com/journal/mafi Mathematical Finance. 2019;29:174-207.