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37th Chinese Control Conference (CCC)

作者： Shi, Jingtao Shandong Univ Sch Math Jinan 250100 Shandong Peoples R China

ISBN: (纸本)9789881563958

This paper is concerned with the optimal control problem, where the recursive cost functional is defined as one of the solution to a controlled fully coupled forward-backward stochastic differential equation (FBSDE), and the control domain is convex. Two different approaches-dynamic programming principle (DPP) and maximum principle (MP)-are applied to solve the problem and the relationship between them are studied. Under some differentiable assumptions, relations among the adjoint processes, the value function and the generalized Hamiltonian function are proved, whereas the diffusion term of the forward equation is independent of the state variable z. The general case for the problem is open. A linear example is discussed as the illustration of our main result.

关键词： Stochastic optimal control fully coupled FBSDE dynamic programming principle maximum principle

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Tightness and duality of martingale transport on the Skorokhod space

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STOCHASTIC PROCESSES AND THEIR APPLICATIONS 2017年第3期127卷 927-956页

作者： Guo, Gaoyue Tan, Xiaolu Touzi, Nizar Ecole Polytech CMAP Palaiseau France PSL Res Univ Univ Paris Dauphine CEREMADE Paris France

The martingale optimal transport aims to optimally transfer a probability measure to another along the class of martingales. This problem is mainly motivated by the robust superhedging of exotic derivatives in financial mathematics, which turns out to be the corresponding Kantorovich dual. In this paper we consider the continuous-time martingale transport on the Skorokhod space of cadlag paths. Similar to the classical setting of optimal transport, we introduce different dual problems and establish the corresponding dualities by a crucial use of the S-topology and the dynamic programming principle. (C) 2016 Elsevier B.V. All rights reserved.

关键词： S-topology dynamic programming principle Robust superhedging

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Optimal stopping with random maturity under nonlinear expectations

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STOCHASTIC PROCESSES AND THEIR APPLICATIONS 2017年第8期127卷 2586-2629页

作者： Bayraktar, Erhan Yao, Song Univ Michigan Dept Math Ann Arbor MI 48109 USA Univ Pittsburgh Dept Math Pittsburgh PA 15260 USA

We analyze an optimal stopping problem sup(gamma is an element of T) (xi) over bar 0[y(gamma Lambda tau 0)] with random maturity to under a nonlinear expectation (xi) over bar0[.] := sup(P is an element of P) Eg[.], where P is a weakly compact set of mutually singular probabilities. The maturity tau(0) is specified as the hitting time to level 0 of some continuous index process X at which the payoff process g is even allowed to have a positive jump. When P collects a variety of semimartingale measures, the optimal stopping problem can be viewed as a discretionary stopping problem for a player who can influence both drift and volatility of the dynamic of underlying stochastic flow. We utilize a martingale approach to construct an optimal pair (P-*, y(*)) for sup((P, gamma)is an element of P X T) Ep[y(gamma Lambda tau 0)], in which y(*) is the first time y meets the limit. L of its approximating (xi) over bar -Snell envelopes. To overcome the technical subtleties caused by the mutual singularity of probabilities in P and the discontinuity of the payoff process y, we approximate tau(0) by an increasing sequence of Lipschitz continuous stopping times and approximate y by a sequence of uniformly continuous processes. (C) 2016 Elsevier B.V. All rights reserved.

关键词： Discretionary stopping Random maturity Controls in weak formulation Optimal stopping Nonlinear expectation Weak stability under pasting Lipschitz continuous stopping time dynamic programming principle Martingale approach

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BSDES IN GAMES,COUPLED WITH THE VALUE *** NONLOCAL BELLMAN-ISAACS EQUATIONS

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Acta Mathematica Scientia 2017年第5期37卷 1497-1518页

作者：郝涛李娟 School of Statistics Shandong University of Finance and Economics School of Mathematics and Statistics Shandong University

We establish a new type of backward stochastic differential equations(BSDEs)connected with stochastic differential games(SDGs), namely, BSDEs strongly coupled with the lower and the upper value functions of SDGs, where the lower and the upper value functions are defined through this BSDE. The existence and the uniqueness theorem and comparison theorem are proved for such equations with the help of an iteration method. We also show that the lower and the upper value functions satisfy the dynamic programming principle. Moreover, we study the associated Hamilton-Jacobi-Bellman-Isaacs(HJB-Isaacs)equations, which are nonlocal, and strongly coupled with the lower and the upper value functions. Using a new method, we characterize the pair(W, U) consisting of the lower and the upper value functions as the unique viscosity solution of our nonlocal HJB-Isaacs equation. Furthermore, the game has a value under the Isaacs’ condition.

关键词： Mc Kean-Vlasov SDE BSDE coupled with the lower and the upper value functions dynamic programming principle mean-field BSDE viscosity solution coupled nonlocal HJB-Isaacs equation Isaacs' condition

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A uniform Tauberian theorem in dynamic games

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SBORNIK MATHEMATICS 2018年第1期209卷 122-144页

作者： Khlopin, Dmitrii V. Russian Acad Sci Krasovskii Inst Math & Mech Ural Branch Ekaterinburg Russia

Antagonistic dynamic games including games represented in normal form are considered. The asymptotic behaviour of value in these games is investigated as the game horizon tends to infinity (Cesaro mean) and as the discounting parameter tends to zero (Abel mean). The corresponding Abelian-Tauberian theorem is established: it is demonstrated that in both families the game value uniformly converges to the same limit, provided that at least one of the limits exists. Analogues of one-sided Tauberian theorems are obtained. An example shows that the requirements are essential even for control problems.

关键词： dynamic programming principle games with a saddle point Tauberian theorem

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MULTIVALUED STOCHASTIC DELAY DIFFERENTIAL EQUATIONS AND RELATED STOCHASTIC CONTROL PROBLEMS

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QUAESTIONES MATHEMATICAE 2017年第6期40卷 769-802页

作者： Diomande, Bakarime Maticiuc, Lucian Alexandru Ioan Cuza Univ Fac Math Carol 1 Blvd 11 Iasi 700506 Romania Gheorghe Asachi Tech Univ Dept Math Carol 1 Blvd 11 Iasi 700506 Romania

We study the existence and uniqueness of a solution for the multivalued stochastic differential equation with delay (the multivalued term is of subdifferential type): {dX(t) + partial derivative phi(X(t)) dt is an element of b(t, X(t), Y(t), Z(t)) dt + sigma(t, X(t), Y(t), Z(t)) dW(t), t is an element of(s, T), X(t) = xi(t - s), t is an element of[s - delta, s]. Specify that in this case the coecients at time t depends also on previous values of X(t) through Y (t) and Z(t). Also X is constrained with the help of a bounded variation feedback law K to stay in the convex set Dom(phi). After wards we consider optimal problems where the state X is a soluction of a controlled delay stochastic system as above. We establish the dynamic programming principle for the value function and finally we prove that the value function is a viscosity solution for a suitable Hamilton-Jacobi-Bellman type equation.

关键词： Multivalued SDE with delay dynamic programming principle Hamilton-Jacobi-Bellman equation viscosity solutions

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ON THE ROBUST DYNKIN GAME

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ANNALS OF APPLIED PROBABILITY 2017年第3期27卷 1702-1755页

作者： Bayraktar, Erhan Ya, Song Univ Michigan Dept Math Ann Arbor MI 48109 USA Univ Pittsburgh Dept Math Pittsburgh PA 15260 USA

We analyze a robust version of the Dynkin game over a set P of mutually singular probabilities. We first prove that conservative player's lower and upper value coincide (let us denote the value by V). Such a result connects' the robust Dynkin game with second-order doubly reflected backward stochastic differential equations. Also, we show that the value process V is a submartingale under an appropriately defined nonlinear expectation E up to the first time tau* when V meets the lower payoff process L. If the probability set P is weakly compact, one can even find an optimal triplet (P*, tau*, gamma*,) for the value V-0. The mutual singularity of probabilities in P causes major technical difficulties. To deal with them, we use some new methods including two approximations with respect to the set of stopping times.

关键词： Robust Dynkin game nonlinear expectation dynamic programming principle controls in weak formulation weak stability under pasting martingale approach path dependent stochastic differential equations with controls optimal triplet optimal stopping with random maturity

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Towards Parallelizable Sampling–based Nonlinear Model Predictive Control

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IFAC-PapersOnLine 2017年第1期50卷 13176-13181页

作者： Bobiti, R.V. Lazar, M. Department of Electrical Engineering Eindhoven University of Technology Netherlands

This paper proposes a new sampling–based nonlinear model predictive control (MPC) algorithm, with a bound on complexity quadratic in the prediction horizon N and linear in the number of samples. The idea of the proposed algorithm is to use the sequence of predicted inputs from the previous time step as a warm start, and to iteratively update this sequence by changing its elements one by one, starting from the last predicted input and ending with the first predicted input. This strategy, which resembles the dynamic programming principle, allows for parallelization up to a certain level and yields a suboptimal nonlinear MPC algorithm with guaranteed recursive feasibility, stability and improved cost function at every iteration, which is suitable for real–time implementation. The complexity of the algorithm per each time step in the prediction horizon depends only on the horizon, the number of samples and parallel threads, and it is independent of the measured system state. Comparisons with the fmincon nonlinear optimization solver on a benchmark example indicates that, as the simulation time progresses, the proposed algorithm converges rapidly to the “optimal” solution, even when using a small number of samples. © 2017

关键词： dynamic programming Computational complexity Constrained optimization Cost functions Iterative methods Model predictive control Nonlinear programming Nonlinear systems Predictive control systems Constrained controls dynamic programming principle Embedded model predictive control Nonlinear model predictive control Nonlinear optimization solver Nonlinear predictive control Number of samples Prediction horizon

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Linear-quadratic optimal sampled-data control problems: Convergence result and Riccati theory

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AUTOMATICA 2017年 79卷 273-281页

作者： Bourdin, Loic Trelat, Emmanuel Univ Limoges Inst Rech XLIM Pole Math CNRS UMR 7252 Limoges France UPMC Univ Paris 06 Sorbonne Univ CNRS UMR 7598 Lab Jacques Louis Lions F-75005 Paris France

We consider linear-quadratic optimal sampled-data control problems, where the state evolves continuously in time according to a linear control system and the control is sampled, i.e., is piecewise constant over a subdivision of the time interval, and the cost is quadratic. As a first result, we prove that, as the sampling periods tend to zero, the optimal sampled-data controls converge pointwise to the optimal permanent control. Then, we extend the classical Riccati theory to the sampled-data control framework, by developing two different approaches: the first one uses a recently established version of the Pontryagin maximum principle for optimal sampled-data control problems, and the second one uses an adequate version of the dynamic programming principle. In turn, we obtain a closed-loop expression for optimal sampled-data controls of linear-quadratic problems. (C) 2017 Elsevier Ltd. All rights reserved.

关键词： Optimal control Linear-quadratic problems Sampled-data control Digital control Convergence Pontryagin maximum principle Riccati theory dynamic programming principle

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Optimal consumption-investment strategy under the Vasicek model: HARA utility and Legendre transform

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INSURANCE MATHEMATICS & ECONOMICS 2017年 72卷 215-227页

作者： Chang, Hao Chang, Kai Tianjin Polytech Univ Sch Sci Tianjin 300387 Peoples R China Tianjin Univ Coll Management & Econ Tianjin 300072 Peoples R China Zhejiang Univ Finance & Econ Sch Finance Hangzhou 310018 Zhejiang Peoples R China

This paper studies the optimal consumption-investment strategy with multiple risky assets and stochastic interest rates, in which interest rate is supposed to be driven by the Vasicek model. The objective of the individuals is to seek an optimal consumption-investment strategy to maximize the expected discount utility of intermediate consumption and terminal wealth in the finite horizon. In the utility theory, Hyperbolic Absolute Risk Aversion (HARA) utility consists of CRRA utility, CARA utility and Logarithmic utility as special cases. In addition, HARA utility is seldom studied in continuous-time portfolio selection theory due to its sophisticated expression. In this paper, we choose HARA utility as the risky preference of the individuals. Due to the complexity of the structure of the solution to the original Hamilton-Jacobi-Bellman (HJB) equation, we use Legendre transform to change the original non-linear HJB equation into its linear dual one, whose solution is easy to conjecture in the case of HARA utility. By calculations and deductions, we obtain the closed-form solution to the optimal consumption-investment strategy in a complete market. Moreover, some special cases are also discussed in detail. Finally, a numerical example is given to illustrate our results. (C) 2016 Elsevier B.V. All rights reserved.

关键词： Consumption-investment problem The Vasicek model HARA utility dynamic programming principle Legendre transform Closed-form solution

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