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OPTIMAL CONTROL OF BRANCHING DIFFUSION PROCESSES: A FINITE HORIZON PROBLEM

ANNALS OF APPLIED PROBABILITY 2018年第1期28卷 1-34页

作者： Claisse, Julien Ecole Polytech Ctr Math Appl F-91128 Palaiseau France

In this paper, we aim to develop the stochastic control theory of branching diffusion processes where both the movement and the reproduction of the particles depend on the control. More precisely, we study the problem of minimizing the expected value of the product of individual costs penalizing the final position of each particle. In this setting, we show that the value function is the unique viscosity solution of a nonlinear parabolic PDE, that is, the Hamilton-Jacobi-Bellman equation corresponding to the problem. To this end, we extend the dynamic programming approach initiated by Nisio [J. Math. Kyoto Univ. 25 (1985) 549-575] to deal with the lack of independence between the particles as well as between the reproduction and the movement of each particle. In particular, we exploit the particular form of the optimization criterion to derive a weak form of the branching property. In addition, we provide a precise formulation and a detailed justification of the adequate dynamic programming principle.

关键词： Stochastic control branching diffusion process dynamic programming principle Hamilton-Jacobi-Bellman equation viscosity solution

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Near-Optimal Control of Stochastic Recursive Systems Via Viscosity Solution

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2018年第2期178卷 363-382页

作者： Zhang, Liangquan Zhou, Qing Beijing Univ Posts & Telecommun Sch Sci Beijing Peoples R China

In this paper, we study the near-optimal control for systems governed by forward-backward stochastic differential equations via dynamic programming principle. Since the nonsmoothness is inherent in this field, the viscosity solution approach is employed to investigate the relationships among the value function, the adjoint equations along near-optimal trajectories. Unlike the classical case, the definition of viscosity solution contains a perturbation factor, through which the illusory differentiability conditions on the value function are dispensed properly. Moreover, we establish new relationships between variational equations and adjoint equations. As an application, a kind of stochastic recursive near-optimal control problem is given to illustrate our theoretical results.

关键词： dynamic programming principle Forward-backward stochastic differential equations Near-optimal control Super-/subdifferentials

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Optimal Reinsurance-Investment Strategy Under Risks of Interest Rate, Exchange Rate and Inflation

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METHODOLOGY AND COMPUTING IN APPLIED PROBABILITY 2018年第4期20卷 1477-1502页

作者： Guo, Chang Zhuo, Xiaoyang Constantinescu, Corina Pamen, Olivier Menoukeu Nankai Univ Sch Finance Tianjin 300350 Peoples R China Nankai Univ Sch Business Tianjin 300071 Peoples R China Univ Liverpool Inst Financial & Actuarial Math Dept Math Sci Liverpool L69 7ZL Merseyside England African Inst Math Sci Accra Ghana Univ Ghana Accra Ghana

In this paper, we pursue the optimal reinsurance-investment strategy of an insurer who can invest in both domestic and foreign markets. We assume that both the domestic and the foreign nominal interest rates are described by extended Cox-Ingersoll-Ross (CIR) models. In order to hedge the risk associated to investments, rolling bonds, treasury inflation protected securities and futures are purchased by the insurer. We use the dynamic programming principles to explicitly derive both the value function and the optimal reinsurance-investment strategy. As a conclusion, we analyze the impact of the model parameters on both the optimal strategy and the optimal utility.

关键词： Optimal reinsurance-investment strategy Foreign exchange market Extended CIR Stochastic inflation dynamic programming principle 49L20 91G80

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Data-driven surrogates of value functions and applications to feedback control for dynamical systems

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IFAC-PapersOnLine 2018年第2期51卷 307-312页

作者： Schmidt, A. Haasdonk, B. University of Stuttgart Institute for Applied Analysis and Numerical Simulation Pfaffenwaldring 57 Stuttgart70569 Germany

Dealing with high-dimensional feedback control problems is a difficult task when the classical dynamic programming principle is applied. Existing techniques restrict the application to relatively low dimensions since the discretizations typically suffer from the curse of dimensionality. In this paper we introduce a novel approximation technique for the value function of an infinite horizon optimal control. The method is based on solving optimal open loop control problems on a finite horizon with a sampling of the global value function along the generated trajectories. For the interpolation we choose a kernel orthogonal greedy strategy, because these methods are able to produce extreme sparse surrogates and enable rapid evaluations in high dimensions. Two numerical examples prove the performance of the approach and show that the method is able to deal with high-dimensional feedback control problems, where the dimensionality prevents the approximation by most existing methods. © 2018

关键词： dynamic programming dynamical systems Feedback control Numerical methods Approximation techniques Curse of dimensionality dynamic programming principle Feedback control problem Generated trajectories greedy techniques Kernel approximation Optimal feedback control

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Tauberian Theorem for Value Functions

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dynamic GAMES AND APPLICATIONS 2018年第2期8卷 401-422页

作者： Khlopin, Dmitry Russian Acad Sci Krasovskii Inst Math & Mech Ural Branch 16 S Kovalevskaja St Ekaterinburg 620990 Russia Ural Fed Univ Inst Math & Comp Sci Chair Appl Math 4 Turgeneva St Ekaterinburg 620083 Russia

For two-person dynamic zero-sum games (both discrete and continuous settings), we investigate the limit of value functions of finite horizon games with long-run average cost as the time horizon tends to infinity and the limit of value functions of -discounted games as the discount tends to zero. We prove that the dynamic programming principle for value functions directly leads to the Tauberian theorem-that the existence of a uniform limit of the value functions for one of the families implies that the other one also uniformly converges to the same limit. No assumptions on strategies are necessary. To this end, we consider a mapping that takes each payoff to the corresponding value function and preserves the sub- and superoptimality principles (the dynamic programming principle). With their aid, we obtain certain inequalities on asymptotics of sub- and supersolutions, which lead to the Tauberian theorem. In particular, we consider the case of differential games without relying on the existence of the saddle point;a very simple stochastic game model is also considered.

关键词： dynamic programming principle Zero-sum games Abel mean Cesaro mean Differential games

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dynamic programming FOR OPTIMAL CONTROL OF STOCHASTIC MCKEAN-VLASOV dynamicS

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2017年第2期55卷 1069-1101页

作者： Pham, Huyen Wei, Xiaoli CNRS UMR 7599 Lab Probabilites & Modele Aleatoires Paris France Univ Paris Diderot Paris France CREST ENSAE Paris France Univ Paris Diderot CNRS Lab Probabilites & Modeles Aleatoires UMR 7599 Paris France

We study optimal control of the general stochastic McKean-Vlasov equation. Such a 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, Cours au College de France : Theorie des jeux a champ moyens, (2012), pp. 2006-2012] and Ito'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.

关键词： stochastic McKean-Vlasov SDEs dynamic programming principle Bellman equation Wasserstein space viscosity solutions

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An exactly solvable multiple stochastic optimal stopping problem

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ADVANCES IN DIFFERENCE EQUATIONS 2018年第1期2018卷 1-9页

作者： Yoshioka, Hidekazu Faculty of Life and Environmental Science Shimane University Matsue Japan

A new kind of multiple stochastic optimal stopping problem is formulated and its associated recursive variational inequalities are derived. We show that these variational inequalities can be solved exactly in a cascading manner. The relevance of the present problem in analyzing animal migration, which is an ecologically important problem, is also briefly discussed.

关键词： Multiple optimal stopping Geometric Brownian motion dynamic programming principle Variational inequality

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Data-driven surrogates of value functions and applications to feedback control for dynamical systems

Data-driven surrogates of value functions and applications t...

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9th Vienna International Conference on Mathematical Modelling (MATHMOD)

作者： Schmidt, A. Haasdonk, B. Univ Stuttgart Inst Appl Anal & Numer Simulat Pfaffenwaldring 57 D-70569 Stuttgart Germany

关键词： Kernel approximation optimal feedback control dynamic programming principle greedy techniques

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Stochastic control over networks

Stochastic control over networks

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作者： Isaac Wahbi Universite de recherche Paris Sciences et Lettres

学位级别：博士

This thesis consists of three parts which deal with quasi linear parabolic PDE on a junction, stochastic diffusion on a junction and stochastic control on a junction with control at the junction point. We begin in the first Chapter by introducing and studying a new class of non degenerate quasi linear parabolic PDE on a junction, satisfying a Neumann (or Kirchoff) non linear and non dynamical condition at the junction point. We prove the existence and the uniqueness of a classical solution. The main motivation of studying this new mathematical object is the analysis of stochastic control problems with control at the junction point, and the characterization of the value function of the problem in terms of Hamilton Jacobi Bellman equations. For this end, in the second Chapter we give a proof of the existence of a diffusion on a junction. The process is characterized by its local time at the junction point, whose quadratic approximation is centrally related to the ellipticty assumption of the second order terms around the junction point. We then provide an It 's formula for this process. Thanks to the previous results, in the last Chapter we study a problem of stochastic control on a junction, with control at the junction point. The set of controls is the set of the probability measures (admissible rules) satisfying a martingale problem. We prove the compactness of the admissible rules and the dynamic programming principle.

关键词： Non linear parabolic partial differential equations Neumann boundary condition Junction Hamilton Jacobi Bellman equations stochastic diffusion local time stochastic control martingale problem dynamic programming principle

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The Corporate Optimal Portfolio and Consumption Choice Problem in Trade Project with Borrowing Rate Higher than Deposit Rate

The Corporate Optimal Portfolio and Consumption Choice Probl...

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第三十八届中国控制会议

作者： Panpan Zhang School of Control Science and Engineering Shandong University

This paper is concerned with a kind of optimal portfolio and consumption choice problem, where an investor can invest his wealth in a trade project and foreign exchange deposit. The trade project earns profit by buying the merchandise and selling it with a higher price. The bank pays at an interest rate for any deposit, and vice takes at a large rate for any loan. The optimal strategy is obtained by Hamilton-Jacobi-Bellman(HJB) equation, which is derived from dynamic programming principle. For the specific Hyperbolic Absolute Risk Aversion(HARA) case, we get the explicit form of optimal portfolio and consumption solution, and we give some simulation results.

关键词： Consumption/Investment Optimization dynamic programming principle HJB Equation HARA

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