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Regularity properties in a state-constrained expected utility maximization problem

MATHEMATICAL METHODS OF OPERATIONS RESEARCH 2018年第2期88卷 185-240页

作者： Lazgham, Mourad Univ Mannheim Dept Math Mannheim Germany

We consider a stochastic optimal control problem in a market model with temporary and permanent price impact, which is related to an expected utility maximization problem under finite fuel constraint. We establish the initial condition fulfilled by the corresponding value function and show its first regularity property. Moreover, we can prove the existence and uniqueness of an optimal strategy under rather mild model assumptions. This will then allow us to derive further regularity properties of the corresponding value function, in particular its continuity and partial differentiability. As a consequence of the continuity of the value function, we will prove a dynamic programming principle without appealing to the classical measurable selection arguments. This permits us to establish a tight relation between our value function and a nonlinear parabolic degenerated Hamilton-Jacobi-Bellman (HJB) equation with singularity. To conclude, we show a comparison principle, which allows us to characterize our value function as the unique viscosity solution of the HJB equation.

关键词： Expected utility maximization problem Value function Price impact Optimal strategy dynamic programming principle Bellman's principle Hamilton-Jacobi-Bellman equation Viscosity solution Comparison principle

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Discontinuous solutions of the Hamilton-Jacobi equation for exit time problems

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2000年第4期38卷 1067-1085页

作者： Ye, JJ Univ Victoria Dept Math & Stat Victoria BC V8W 3P4 Canada

In general, the value function associated with an exit time problem is a discontinuous function. We prove that the lower (upper) semicontinuous envelope of the value function is a supersolution (subsolution) of the Hamilton Jacobi equation involving the proximal subdifferentials (superdifferentials) with subdifferential-type (superdifferential-type) mixed boundary condition. We also show that if the value function is upper semicontinuous, then it is the maximum subsolution of the Hamilton Jacobi equation involving the proximal superdifferentials with the natural boundary condition, and if the value function is lower semicontinuous, then it is the minimum solution of the Hamilton Jacobi equation involving the proximal subdifferentials with a natural boundary condition. Futhermore, if a compatibility condition is satis ed, then the value function is the unique lower semicontinuous solution of the Hamilton Jacobi equation with a natural boundary condition and a subdifferential type boundary condition. Some conditions ensuring lower semicontinuity of the value functions are also given.

关键词： Hamilton-Jacobi equation dynamic programming principle exit time problems proximal subdifferentials

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A partial history of the early development of continuous-time nonlinear stochastic systems theory

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AUTOMATICA 2014年第2期50卷 303-334页

作者： Kushner, Harold J. Brown Univ Dept Appl Math Providence RI 02912 USA

This article is a survey of the early development of selected areas in nonlinear continuous-time stochastic control. Key developments in optimal control and the dynamic programming principle, existence of optimal controls under complete and partial observations, nonlinear filtering, stochastic stability, the stochastic maximum principle and ergodic control are discussed. Issues concerning wide bandwidth noise for stability, modeling, filtering and ergodic control are dealt with. The focus is on the earlier work, but many important topics are omitted for lack of space. (C) 2013 Elsevier Ltd. All rights reserved.

关键词： Stochastic optimal control Viscosity solutions dynamic programming principle Nisio semigroup Existence of optimal controls Existence under partial observations Compactness methods Wide bandwidth noise Approximations to systems Nonlinear filtering Stochastic stability Stochastic maximum principle Ergodic control Stochastic stability Filtering and control with wide bandwidth noise

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A new existence result for quadratic BSDEs with jumps with application to the utility maximization problem

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STOCHASTIC PROCESSES AND THEIR APPLICATIONS 2010年第10期120卷 1966-1995页

作者： Morlais, Marie-Amelie Univ Maine Lab Manceau Math F-72085 Le Mans France

In this study, we consider the exponential utility maximization problem in the context of a jump-diffusion model. To solve this problem, we rely on the dynamic programming principle to express the value process of this problem in terms of the solution of a quadratic BSDE with jumps. Since the quadratic BSDE1 under study is driven by both a Wiener process and a Poisson random measure having a Levy measure with infinite mass, our main task is therefore to establish a new existence result for the specific BSDE introduced. (C) 2010 Elsevier B.V. All rights reserved.

关键词： Backward stochastic differential equations Levy measure Utility maximization dynamic programming principle

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Randomized filtering and Bellman equation in Wasserstein space for partial observation control problem

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STOCHASTIC PROCESSES AND THEIR APPLICATIONS 2019年第2期129卷 674-711页

作者： Bandini, Elena Cosso, Andrea Fuhrman, Marco Huyen Pham Univ Milano Bicocca Dipartimento Matemat Applicaz Via Roberto Cozzi 55 I-20125 Milan Italy Univ Bologna Dipartimento Matemat Piazza Porta S Donato 5 I-40126 Bologna Italy Univ Milan Dipartimento Matemat Via Saldini 50 I-20133 Milan Italy Univ Paris Diderot CNRS UMR 8001 Lab Probabilites Stat & Modelisat Paris France ENSAE CREST Palaiseau France

We study a stochastic optimal control problem for a partially observed diffusion. By using the control randomization method in Bandini et al. (2018), we prove a corresponding randomized dynamic programming principle (DPP) for the value function, which is obtained from a flow property of an associated filter process. This DPP is the key step towards our main result: a characterization of the value function of the partial observation control problem as the unique viscosity solution to the corresponding dynamic programming Hamilton-Jacobi-Bellman (HJB) equation. The latter is formulated as a new, fully non linear partial differential equation on the Wasserstein space of probability measures. An important feature of our approach is that it does not require any non-degeneracy condition on the diffusion coefficient, and no condition is imposed to guarantee existence of a density for the filter process solution to the controlled Zakai equation. Finally, we give an explicit solution to our HJB equation in the case of a partially observed non Gaussian linear-quadratic model. (C) 2018 Elsevier B.V. All rights reserved.

关键词： Partial observation control problem Randomization of controls dynamic programming principle Bellman equation Wasserstein space Viscosity solutions

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Stochastic recursive optimal control problem of reflected stochastic differential systems

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INTERNATIONAL JOURNAL OF CONTROL 2020年第9期93卷 2187-2198页

作者： Feng, Xinwei Chinese Univ Hong Kong Dept Stat Shatin Hong Kong Peoples R China

In this paper, we study one kind of stochastic recursive optimal control problem in which the control system is stochastic differential equations reflected in a domain and the cost functional is defined by generalised backward stochastic differential equations with reflection. We establish the dynamic programming principle for the value function and show that it is a viscosity solution of the associated obstacle problem for the corresponding Hamilton-Jacobi-Bellman equation with a nonlinear Neumann boundary condition.

关键词： Stochastic recursive optimal control generalised reflected backward stochastic differential equations dynamic programming principle viscosity solution nonlinear Neumann boundary condition

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dynamic Set Values for Nonzero-Sum Games with Multiple Equilibriums

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MATHEMATICS OF OPERATIONS RESEARCH 2022年第1期47卷 616-642页

作者： Feinstein, Zachary Rudloff, Birgit Zhang, Jianfeng Stevens Inst Technol Sch Business Hoboken NJ 07030 USA Vienna Univ Econ & Business Inst Stat & Math A-1020 Vienna Austria Univ Southern Calif Dept Math Los Angeles CA 90089 USA

Nonzero sum games typically have multiple Nash equilibriums (or no equilibrium), and unlike the zero-sum case, they may have different values at different equilibriums. Instead of focusing on the existence of individual equilibriums, we study the set of values over all equilibriums, which we call the set value of the game. The set value is unique by nature and always exists (with possible value 0). Similar to the standard value function in control literature, it enjoys many nice properties, such as regularity, stability, and more importantly, the dynamic programming principle. There are two main features in order to obtain the dynamic programming principle: (i) we must use closed-loop controls (instead of open-loop controls);and (ii) we must allow for path dependent controls, even if the problem is in a state-dependent (Markovian) setting. We shall consider both discrete and continuous time models with finite time horizon. For the latter, we will also provide a duality approach through certain standard PDE (or path-dependent PDE), which is quite efficient for numerically computing the set value of the game.

关键词： nonzero sum game Nash equilibrium set value dynamic programming principle closed-loop controls path dependent PDE

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Regularity for nonlinear stochastic games

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ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE 2018年第6期35卷 1435-1456页

作者： Luiro, Hannes Parviainen, Mikko Univ Jyvaskyla Dept Math & Stat POB 35 FI-40014 Jyvaskyla Finland

We establish regularity for functions satisfying a dynamic programming equation, which may arise for example from stochastic games or discretization schemes. Our results can also be utilized in obtaining regularity and existence results for the corresponding partial differential equations. (C) 2018 Elsevier Masson SAS. All rights reserved.

关键词： dynamic programming principle p-Laplace Tug-of-war Tug-of-war with noise with space dependent probabilities Viscosit solutions

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A d-person differential game with state space constraints

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APPLIED MATHEMATICS AND OPTIMIZATION 2007年第3期56卷 312-342页

作者： Ramasubramanian, S. Indian Stat Inst Stat Math Unit Bangalore 560059 Karnataka India

We consider a network of d companies ( insurance companies, for example) operating under a treaty to diversify risk. Internal and external borrowing are allowed to avert ruin of any member of the network. The amount borrowed to prevent ruin is viewed upon as control. Repayment of these loans entails a control cost in addition to the usual costs. Each company tries to minimize its repayment liability. This leads to a d-person differential game with state space constraints. If the companies are also in possible competition a Nash equilibrium is sought. Otherwise a utopian equilibrium is more appropriate. The corresponding systems of HJB equations and boundary conditions are derived. In the case of Nash equilibrium, the Hamiltonian can be discontinuous;there are d interlinked control problems with state constraints;each value function is a constrained viscosity solution to the appropriate discontinuous HJB equation. Uniqueness does not hold in general in this case. In the case of utopian equilibrium, each value function turns out to be the unique constrained viscosity solution to the appropriate HJB equation. Connection with Skorokhod problem is briefly discussed.

关键词： d-person differential game state space constraints nash equilibrium Utopian equilibrium dynamic programming principle system of HJB equations constrained viscosity solution semicontinuous envelope deterministic Skorokhod problem drift reflection

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Approximation of optimal ergodic dividend strategies using controlled Markov chains

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IET CONTROL THEORY AND APPLICATIONS 2018年第16期12卷 2194-2204页

作者： Jin, Zhuo Yang, Hailiang Yin, George Univ Melbourne Dept Econ Ctr Actuarial Studies Melbourne Vic 3010 Australia Univ Hong Kong Dept Stat & Actuarial Sci Hong Kong Hong Kong Peoples R China Wayne State Univ Dept Math Detroit MI 48202 USA

This study develops a numerical method to find optimal ergodic (long-run average) dividend strategies in a regime-switching model. The surplus process is modelled by a regime-switching process subject to liability constraints. The regime-switching process is modelled by a finite-time continuous-time Markov chain. Using the dynamic programming principle, the optimal long-term average dividend payment is a solution to the coupled system of Hamilton-Jacobi-Bellman equations. Under suitable conditions, the optimal value of the long-term average dividend payment can be determined by using an invariant measure. However, due to the regime switching, getting the invariant measure is very difficult. The objective is to design a numerical algorithm to approximate the optimal ergodic dividend payment strategy. By using the Markov chain approximation techniques, the authors construct a discrete-time controlled Markov chain for the approximation, and prove the convergence of the approximating sequences. A numerical example is presented to demonstrate the applicability of the algorithm.

关键词： continuous time systems approximation theory Markov processes dynamic programming discrete time systems optimal control discrete-time controlled Markov chain optimal ergodic dividend strategies controlled Markov chains numerical method regime-switching model surplus process regime-switching process subject finite-time continuous-time Markov chain dynamic programming principle long-term average dividend payment Hamilton-Jacobi-Bellman equations optimal value invariant measure regime switching optimal ergodic dividend payment strategy Markov chain approximation techniques

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