Continuous-time Stochastic Control and Optimization with Financial Applications

Abstract

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...

Mean-variance hedging with oil futures

“…The same proof as for Proposition 6.6.5 in [15] shows that . Moreover the optimal solution to (3.3) is now given by the optimal control to B(m * ), which by the above observation is π t (λ m * ).…”

“…We claim that E F u, T j e Yu F t = F t, T j )e mj(u−t)+n1(u−t)Yt+n2(u−t)Xt , t ≤ u, (A. 15) for suitable deterministic functions m j (τ ), n 1 (τ ), n 2 (τ ) with m j (0) = n 2 (0) = 0 and n 1 (0) = 1. Indeed, applying Itô's formula to M jb) Plugging M j t (u) = F t, T j q j (t, u) into (A.17) and then using (A.16), we obtain dM j t (u) = q j (t, u) dF t, T j + F t, T j q j (t, u) Applying Itô's formula here and using that V t (λ) and M j t (u) areP -martingales, it follows that dV t (λ) = q j (t, u)du dF t, T j + C 0 t dŴ 0 t + C 1 t dŴ 1 t + dL t with aP -local martingale L t orthogonal to F (t).…”

Mean-Field Games and Ambiguity Aversion

“…The conditions and proof are provided in Pham (2009). The main difference between the solutions provided by the two approaches is that the HJB equation gives us a forwardbackwards partial differential equation, while the Pontryagin maximum principle gives us a forward-backwards stochastic differential equation.…”

Optimal consumption problem in the Vasicek model

“…Its proof is based on the Hamilton-Jacobi-Bellman verification theorem (see [8,Theorem 3.1]) and is rather standard, so we give only a sketch of it. More information on stochastic control theory can be found, for example, in [3] and [10].…”