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Stochastic Optimal Control Problem with Obstacle Constraints in Sublinear Expectation Framework

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2019年第2期183卷 422-439页

作者： Li, Hanwu Wang, Falei Shandong Univ Jinan Shandong Peoples R China Bielefeld Univ Bielefeld Germany

In this paper, we consider a stochastic optimal control problem, in which the cost function is defined through a reflected backward stochastic differential equation in sublinear expectation framework. Besides, we study the regularity of the value function and establish the dynamic programming principle. Moreover, we prove that the value function is the unique viscosity solution of the related Hamilton-Jacobi-Bellman-Isaac equation.

关键词： Sublinear expectation Reflected backward stochastic differential equations dynamic programming principle

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Markov decision processes under model uncertainty

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MATHEMATICAL FINANCE 2023年第3期33卷 618-665页

作者： Neufeld, Ariel Sester, Julian Sikic, Mario NTU Singapore Div Math Sci Singapore Singapore Natl Univ Singapore Dept Math Singapore Singapore Univ Zurich Dept Banking & Finance Zurich Switzerland NTU Singapore Div Math Sci 21 Nanyang Link Singapore 637371 Singapore

We introduce a general framework for Markov decision problems under model uncertainty in a discrete-time infinite horizon setting. By providing a dynamic programming principle, we obtain a local-to-global paradigm, namely solving a local, that is, a one time-step robust optimization problem leads to an optimizer of the global (i.e., infinite time-steps) robust stochastic optimal control problem, as well as to a corresponding worst-case measure. Moreover, we apply this framework to portfolio optimization involving data of the S&P500$S\&P\nobreakspace 500$. We present two different types of ambiguity sets;one is fully data-driven given by a Wasserstein-ball around the empirical measure, the second one is described by a parametric set of multivariate normal distributions, where the corresponding uncertainty sets of the parameters are estimated from the data. It turns out that in scenarios where the market is volatile or bearish, the optimal portfolio strategies from the corresponding robust optimization problem outperforms the ones without model uncertainty, showcasing the importance of taking model uncertainty into account.

关键词： ambiguity dynamic programming principle Markov decision problem portfolio optimization

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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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OPTIMAL RISK CONTROL UNDER MARKED POINT PROCESSES SHOCKS: A dynamic programming DUALITY APPROACH

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INTERNATIONAL JOURNAL OF THEORETICAL AND APPLIED FINANCE 2013年第7期16卷 1350036-1350036页

作者： Mnif, Mohamed Univ Tunis El Manar ENIT BP 37Tunis Belvedere Tunis 1002 Tunisia

We study the stochastic control problem of maximizing expected utility from terminal wealth under a nonbankruptcy constraint. The problem of the agent is to derive the optimal insurance strategy which reduces his exposure to the risk. This optimization problem is related to a suitable dual stochastic control problem in which the delicate boundary constraints disappear. We characterize the dual value function as the unique viscosity solution of the corresponding Hamilton Jacobi Bellman Variational Inequality (HJBVI in short). We characterize the optimal insurance strategy by the solution of the variational inequality which we solve numerically by using an algorithm based on policy iterations.

关键词： Optimal insurance stochastic control duality optional decomposition dynamic programming principle viscosity solution

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Holder estimate for a tug-of-war game with 1 <p <2 from Krylov-Safonov regularity theory

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REVISTA MATEMATICA IBEROAMERICANA 2024年第3期40卷 1023-1044页

作者： Arroyo, Angel Parviainen, Mikko Univ Alicante Dept Matemat Alicante 03690 Spain Univ Jyvaskyla Dept Math & Stat POB 35 FI-40014 Jyvaskyla Finland

We propose a new version of the tug-of-war game and a corresponding dynamic programming principle related to the p -Laplacian with 1 < p < 2 . For this version, the asymptotic Holder continuity of solutions can be directly derived from recent Krylov-Safonov type regularity results in the singular case. Moreover, existence of a measurable solution can be obtained without using boundary corrections. We also establish a comparison principle.

关键词： ABP-estimate elliptic non-divergence form partial differential equation with bounded and measurable coefficients dynamic programming principle local H & ouml lder estimate p-Laplacian Pucci extremal operator tug-of-war with noise

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dynamic APPROACHES FOR SOME TIME-INCONSISTENT OPTIMIZATION PROBLEMS

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ANNALS OF APPLIED PROBABILITY 2017年第6期27卷 3435-3477页

作者： Karnam, Chandrasekhar Ma, Jin Zhang, Jianfeng Univ Southern Calif Dept Math Los Angeles CA 90089 USA

In this paper, we investigate possible approaches to study general time-inconsistent optimization problems without assuming the existence of optimal strategy. This leads immediately to the need to refine the concept of time consistency as well as any method that is based on Pontryagin's maximum principle. The fundamental obstacle is the dilemma of having to invoke the dynamic programming principle (DPP) in a time-inconsistent setting, which is contradictory in nature. The main contribution of this work is the introduction of the idea of the "dynamic utility" under which the original time-inconsistent problem (under the fixed utility) becomes a time-consistent one. As a benchmark model, we shall consider a stochastic controlled problem with multidimensional backward SDE dynamics, which covers many existing time-inconsistent problems in the literature as special cases;and we argue that the time inconsistency is essentially equivalent to the lack of comparison principle. We shall propose three approaches aiming at reviving the DPP in this setting: the duality approach, the dynamic utility approach and the master equation approach. Unlike the game approach in many existing works in continuous time models, all our approaches produce the same value as the original static problem.

关键词： Time inconsistency dynamic programming principle stochastic maximum principle comparison principle duality dynamic utility master equation path derivative

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Impulse control problem with switching technology

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STOCHASTICS-AN INTERNATIONAL JOURNAL OF PROBABILITY AND STOCHASTIC PROCESSES 2012年第2-3期84卷 437-460页

作者： Amami, Rim Univ Toulouse 3 Inst Math Toulouse F-31062 Toulouse France

We consider an impulse control problem with switching technology in infinite horizon. We suppose that the firm decides at certain time (impulse time) to switch the technology and the firm value (e.g. a recapitalization). We show that the value function for such problems satisfies a dynamic programming principle. Our objective is to look for an optimal strategy which maximizes the value function.

关键词： impulse control problem maximum conditional profit dynamic programming principle

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Stochastic modeling and control of bioreactors

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IFAC-PapersOnLine 2017年第1期50卷 12611-12616页

作者： Fontbona, J. Ramírez C., H. Riquelme, V. Silva, F.J. Universidad de Chile Beauchef 851 Casilla 170-3 Santiago 3 Chile Institut de recherche XLIM-DMI UMR-CNRS 7252 Faculté des sciences et techniques Université de Limoges Limoges87060 France

In this work we propose a stochastic model for a sequencing-batch reactor (SBR) and for a chemostat. Both models are described by systems of Stochastic Differential Equations (SDEs), which are obtained as limits of suitable Markov Processes characterizing the microscopic behavior. We study the existence of solutions of the obtained equations as well as some properties, among which the possible extinction of the biomass is the most remarkable feature. The implications of this behavior are illustrated in the problem consisting in maximizing the probability of reaching a desired depollution level prior to biomass extinction. © 2017

关键词： Stochastic models Batch reactors Chemostats Differential equations dynamic programming Markov processes Pollution control Stochastic control systems Stochastic systems Demographic stochasticity Depollution dynamic programming principle Existence of Solutions Microscopic behavior Sequencing batch reactors Stochastic control Systems of stochastic differential equations

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On the dynamic representation of some time-inconsistent risk measures in a Brownian filtration

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MATHEMATICS AND FINANCIAL ECONOMICS 2020年第3期14卷 433-460页

作者： Backhoff-Veraguas, Julio Tangpi, Ludovic Univ Wien Fak Math Oskar Morgenstern Pl 1 A-1090 Vienna Austria Princeton Univ Dept Operat Res & Financial Engn Sherrerd Hall Princeton NJ 08540 USA

It is well-known from the work of Kupper and Schachermayer that most law-invariant risk measures are not time-consistent, and thus do not admit dynamic representations as backward stochastic differential equations. In this work we show that in a Brownian filtration the "Optimized Certainty Equivalent" risk measures of Ben-Tal and Teboulle can be computed through PDE techniques, i.e. dynamically. This can be seen as a substitute of sorts whenever they lack time consistency, and covers the cases of conditional value-at-risk and monotone mean-variance. Our method consists of focusing on the convex dual representation, which suggests an expression of the risk measure as the value of a stochastic control problem on an extended the state space. With this we can obtain a dynamic programming principle and use stochastic control techniques, along with the theory of viscosity solutions, which we must adapt to cover the present singular situation.

关键词： Time-inconsistency Risk measures Optimized certainty equivalent HJB equation Viscosity solution Unbounded stochastic control dynamic programming principle Singular Hamiltonian

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MAXIMAL OPERATORS FOR THE p-LAPLACIAN FAMILY

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PACIFIC JOURNAL OF MATHEMATICS 2017年第2期287卷 257-295页

作者： Blanc, Pablo Pinasco, Juan P. Rossi, Julio D. Univ Buenos Aires FCEyN Dept Matemat Ciudad UnivPabellon 1 RA-1428 Buenos Aires DF Argentina

We prove existence and uniqueness of viscosity solutions for the problem max {-Delta(p1)u(x), -Delta(p2)u(x)} = f(x) in a bounded smooth domain Omega subset of R-N with u = g on partial derivative Omega. Here -Delta(p)u = (N + p)(-1) vertical bar Du vertical bar(2-p) div (vertical bar Du vertical bar(p-2)Du) is the 1-homogeneous p-Laplacian and we assume that 2 <= p(1), p(2) <= infinity. This equation appears naturally when one considers a tug-of-war game in which one of the players (the one who seeks to maximize the payoff) can choose at every step which are the parameters of the game that regulate the probability of playing a usual tug-of-war game (without noise) or playing at random. Moreover, the operator max {-Delta(p1)u(x), -Delta(p2)u(x)} provides a natural analogue with respect to p-Laplacians to the Pucci maximal operator for uniformly elliptic operators. We provide two different proofs of existence and uniqueness for this problem. The first one is based in pure PDE methods (in the framework of viscosity solutions) while the second one is more connected to probability and uses game theory.

关键词： Dirichlet boundary conditions dynamic programming principle p-Laplacian tug-of-war games

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