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On the dynamic programming principle for uniformly nondegenerate stochastic differential games in domains and the Isaacs equations

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PROBABILITY THEORY AND RELATED FIELDS 2014年第3-4期158卷 751-783页

作者： Krylov, N. V. Univ Minnesota Minneapolis MN 55455 USA

We prove the dynamic programming principle for uniformly nondegenerate stochastic differential games in the framework of time-homogeneous diffusion processes considered up to the first exit time from a domain. In contrast with previous results established for constant stopping times we allow arbitrary stopping times and randomized ones as well. There is no assumption about solvability of the the Isaacs equation in any sense (classical or viscosity). The zeroth-order "coefficient" and the "free" term are only assumed to be measurable in the space variable. We also prove that value functions are uniquely determined by the functions defining the corresponding Isaacs equations and thus stochastic games with the same Isaacs equation have the same value functions.

关键词： dynamic programming principle Stochastic games Isaacs equations

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INFINITE TIME HORIZON STOCHASTIC RECURSIVE CONTROL PROBLEMS WITH JUMPS: dynamic programming AND STOCHASTIC VERIFICATION THEOREMS

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2025年第2期63卷 796-821页

作者： Luo, Sheng Li, Xun Wei, Qingmeng Hong Kong Polytech Univ Dept Appl Math Kowloon Hong Kong Peoples R China Northeast Normal Univ Sch Math & Stat Changchun 130024 Peoples R China

This paper is devoted to studying an infinite time horizon stochastic recursive control problem with jumps, where an infinite time horizon stochastic differential equation and backward stochastic differential equation with jumps describe the state process and the cost functional, respectively. By establishing the dynamic programming principle, we shed light on the value function of the control problem with an integral-partial differential equation of HJB type in the sense of viscosity solutions. On the other hand, stochastic verification theorems are also studied to provide sufficient conditions to verify the optimality of the given admissible controls. Such a study is carried out within the framework of classical solutions as well as in that of viscosity solutions. Our work emphasizes important differences from the approach for finite time horizon problems. In particular, we have to work in an L-p-setting for p > 4 in order to study the verification theorem in viscosity sense.

关键词： infinite horizon FBSDEs with jumps dynamic programming principle viscosity solution HJB equation integral-differential operators stochastic verification theorem

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dynamic programming approach to reflected backward stochastic differential equations

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ELECTRONIC JOURNAL OF PROBABILITY 2023年第none期28卷 1-20页

作者： Hun, O. Kim, Mun-Chol Kim, Kon-Gun Kim II Sung Univ Fac Math Pyongyang North Korea

By introducing a new type of minimality condition, this paper gives a novel approach to the reflected backward stochastic differential equations (RBSDEs) with cadlag obstacles. Our first step is to prove the dynamic programming principles for nonlinear optimal stopping problems with g-expectations. We then use the nonlinear DoobMeyer decomposition theorem for g-supermartingales to get the existence of the solution. With a new type of minimality condition, we prove a representation formula of solutions to RBSDEs, in an efficient way. Finally, we derive some a priori estimates and stability results.

关键词： reflected backward stochastic differential equation nonlinear optimal stopping dynamic programming principle g-supermartingale decomposition non-Skorohod-type minimality condition second order reflected backward stochastic differential equation

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A dynamic programming APPROACH TO DISTRIBUTION-CONSTRAINED OPTIMAL STOPPING

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ANNALS OF APPLIED PROBABILITY 2022年第3期32卷 1902-1928页

作者： Kallblad, Sigrid Royal Inst Technol Dept Math Stockholm Sweden

We consider an optimal stopping problem where a constraint is placed on the distribution of the stopping time. Reformulating the problem in terms of so-called measure-valued martingales enables us to transform the distributional constraint into an initial condition and view the problem as a stochastic control problem;we establish the corresponding dynamic programming principle. The method offers a systematic approach for solving the problem for general constraints and under weak assumptions on the cost function. In addition, we provide certain continuity results for the value of the problem viewed as a function of its distributional constraint.

关键词： Distribution-constrained optimal stopping measure-valued martingales dynamic programming principle optimal transport measurable selection

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Local regularity estimates for general discrete dynamic programming equations

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JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES 2022年 167卷 225-256页

作者： Arroyo, Angel Blanc, Pablo Parviainen, Mikko Univ Complutense Madrid Interdisciplinary Math Inst Dept Appl Math & Math Anal MOMAT Res Grp Madrid 28040 Spain Univ Jyvaskyla Dept Math & Stat POB 35 FI-40014 Jyvaskyla Finland

We obtain an analytic proof for asymptotic Holder estimate and Harnack's inequality for solutions to a discrete dynamic programming equation. The results also generalize to functions satisfying Pucci-type inequalities for discrete extremal operators. Thus the results cover a quite general class of equations.(c) 2022 The Author(s). Published by Elsevier Masson SAS. This is an open access article under the CC BY license (http://***/licenses/by/4.0/).

关键词： ABP-estimate dynamic programming principle Harnack?s inequality Local H?lder estimate Tug-of-war with noise Elliptic non-divergence form partial differential equation with bounded and measurable coefficients

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Research on investment incorporating both environmental performance and long (short) term financial performance of firms

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INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE 2024年第2期55卷 273-299页

作者： Li, Liang Fei, Chen Fei, Weiyin Anhui Polytech Univ Sch Math Phys & Finance Wuhu Peoples R China Univ Shanghai Sci & Technol Business Sch Shanghai Peoples R China Univ Shanghai Sci & Technol Business Sch Shanghai 200093 Peoples R China

Firstly, with the rapid popularisation of the ESG concept and the deepening of the conception of green and low-carbon development, the performance of a firm in protecting and improving the ecological environment and the level of green technology investment has gradually become the reference criteria for judging whether the firm has social value and sustainability in development. Secondly, to pursue short-term survival and long-term development, the firm will also vigorously strive for short-term profit creation and long-term capital accumulation. Thirdly, due to the existence of information asymmetry, the principal-agent parties in a firm may generate agency conflicts in the pursuit of maximising their own interests, which may lead to losses or even bankruptcy. Therefore, making rational short-term, long-term, and green technology investment decisions that consider both characteristics of firm and agency conflicts is essential for the survival and sustainable development of the firm. From the perspective of contract theory, this paper considers the design of an optimal dynamic financial contract that considers the environment's improvement and the achievement of the firm's long- and short-term financial performance while satisfying incentive compatibility. On this basis, we explore how the optimal investment strategies vary with firm characteristics. The results of the study suggest that the decisions of green technology investment, short-term investment, and long-term capital investment should be appropriately adjusted according to the level of financial slack and the effect of different market shocks to facilitate the achievement of the dual objectives of long- and short-term profitability and environmental improvement.

关键词： Green technology investment (GTI) environmental performance (EP) sustainable development dynamic financial contract dynamic programming principle

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Stochastic control/stopping problem with expectation constraints

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STOCHASTIC PROCESSES AND THEIR APPLICATIONS 2024年 176卷

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

We study a stochastic control/stopping problem with a series of inequality-type and equalitytype expectation constraints in a general non-Markovian framework. We demonstrate that the stochastic control/stopping problem with expectation constraints (CSEC) is independent of a specific probability setting and is equivalent to the constrained stochastic control/stopping problem in weak formulation (an optimization over joint laws of Brownian motion, state dynamics, diffusion controls and stopping rules on an enlarged canonical space). Using a martingale-problem formulation of controlled SDEs in spirit of Stroock and Varadhan (2006), we characterize the probability classes in weak formulation by countably many actions of canonical processes, and thus obtain the upper semi-analyticity of the CSEC value function. Then we employ a measurable selection argument to establish a dynamic programming principle (DPP) in weak formulation for the CSEC value function, in which the conditional expected costs act as additional states for constraint levels at the intermediate horizon. This article extends (El Karoui and Tan, 2013) to the expectation-constraint case. We extend our previous work (Bayraktar and Yao, 2024) to the more complicated setting where the diffusion is controlled. Compared to that paper the topological properties of diffusion-control spaces and the corresponding measurability are more technically involved which complicate the arguments especially for the measurable selection for the super-solution side of DPP in the weak formulation.

关键词： Martingale-problem formulation Enlarged canonical space Polish space of diffusion controls Polish space of stopping times dynamic programming principle Regular conditional probability distribution Measurable selection Stochastic control/stopping problem with expectation constraints

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Transition density function expansion methods for portfolio optimization

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OPTIMAL CONTROL APPLICATIONS & METHODS 2024年第4期45卷 1345-1374页

作者： Lu, Yuxuan Zhou, Qing Wu, Weixing Xiao, Weilin Beijing Univ Posts & Telecommun Sch Sci Minist Educ Key Lab Math & Informat Networks Beijing 100876 Peoples R China Capital Univ Econ & Business Sch Finance Beijing Peoples R China Zhejiang Univ Sch Management Hangzhou Peoples R China

In this study, we introduce transition density function expansion methods inspired from Yang et al. (J Econom. 2019;209(2):256-288.) to stochastic control issues related to utility maximization, without imposing limitations on the variety of asset price models and utility functions. Utilizing Bellman's dynamic programming principle, we initially recast the conditional expectation via the transition density function pertinent to the diffusion process. Subsequently, we employ the Ito-Taylor expansion and Delta expansion techniques to the transition density function associated with the multivariate diffusion process, facilitated by a quasi-Lamperti transformation, aiming to derive explicit recursive expressions for expansion coefficient functions. Our main contributions are that we articulate detailed algorithms, stemming from the backward recursive formulations of the value function and optimal strategies, achieved through discretization methodologies with rigorous proof of expansion convergence in portfolio optimization. Both theoretical and practical demonstrations are presented to validate the convergence of these approximate techniques in addressing stochastic control challenges. To underscore the efficiency and precision of our proposed methods, we apply them to portfolio selection problems within several benchmark models, and highlight the reduced complexity in comparison to the current methodologies. We introduce transition density function expansion methods to stochastic control issues related to utility maximization, without imposing limitations on the variety of asset price models and utility functions. Our main contributions are that we articulate detailed algorithms, stemming from the backward recursive formulations of the value function and optimal strategies with proof of expansion convergence in portfolio optimization. We apply them to portfolio selection problems within several benchmark models, highlighting the reduced complexity in comparison to the

关键词： dynamic programming principle portfolio optimization quasi-Lamperti transformation transition density expansion

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Optimal Claim-Dependent Proportional Reinsurance Under a Self-Exciting Claim Model

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2024年第3期201卷 1229-1255页

作者： Wu, Fan Shen, Yang Zhang, Xin Ding, Kai Southeast Univ Sch Math Nanjing 211189 Jiangsu Peoples R China Univ New South Wales Sch Risk & Actuarial Studies Sydney NSW 2052 Australia

This paper investigates an optimal reinsurance problem for an insurance company with self-exciting claims, where the insurer's historical claims affect the claim intensity itself. We focus on a claim-dependent proportional reinsurance contact, where the term "claim-dependent" signifies that the insurer's risk retention ratio is allowed to depend on claim size. The insurer aims to maximize the expected utility of terminal wealth. By utilizing the dynamic programming principle and verification theorem, we obtain the optimal reinsurance strategy and corresponding value function in closed-form from the Hamilton-Jacobi-Bellman equation under an exponential utility function. We show that the claim-dependent proportional reinsurance is optimal among all types of reinsurance under the exponential utility maximization criterion. In addition, we present several analytical properties and numerical examples of the derived optimal strategy and provide economic insights through analytical and numerical analyses. In particular, we show the optimal claim-dependent proportional reinsurance can be considered as a continuous approximation of the step-wise risk sharing rule between the insurer and the reinsurer.

关键词： Optimal reinsurance problem Self-exciting claim Claim-dependent proportional reinsurance dynamic programming principle Verification theorem

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A stochastic target problem for branching diffusion processes

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STOCHASTIC PROCESSES AND THEIR APPLICATIONS 2024年 170卷

作者： Kharroubi, Idris Ocello, Antonio Sorbonne Univ LPSM UMR CNRS 8001 Paris France Univ Paris Cite Paris France

We consider an optimal stochastic target problem for branching diffusion processes. This problem consists in finding the minimal condition for which a control allows the underlying branching process to reach a target set at a finite terminal time for each of its branches. This problem is motivated by an example from fintech where we look for the super-replication price of options on blockchain-based cryptocurrencies. We first state a dynamic programming principle for the value function of the stochastic target problem. Next, we show that the value function can be simplified into a novel function with the use of a finite-dimensional argument through a concept known as the branching property. Under wide conditions, this last function is shown to be the unique viscosity solution to an HJB variational inequality.

关键词： Stochastic target control Fintech Cryptocurrencies options Branching diffusion process dynamic programming principle Hamilton-Jacobi-Bellman equation Viscosity solution

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