SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1615-1642, June 2024. Abstract. We introduce the notion of mean viability for controlled stochastic differential equations and establish counterparts of Nagumo’s classical viability theorems (necessary and sufficient conditions for mean viability). As an application, we provide a purely probabilistic proof of a comparison principle and of existence for contingent and viscosity solutions of second-order fully nonlinear path-dependent Hamilton–Jacobi–Bellman equations. We do not use compactness and optimal stopping arguments, which are usually employed in the literature on viscosity solutions for second-order path-dependent PDEs.
{"title":"Mean Viability Theorems and Second-Order Hamilton–Jacobi Equations","authors":"Christian Keller","doi":"10.1137/23m1550438","DOIUrl":"https://doi.org/10.1137/23m1550438","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1615-1642, June 2024. <br/> Abstract. We introduce the notion of mean viability for controlled stochastic differential equations and establish counterparts of Nagumo’s classical viability theorems (necessary and sufficient conditions for mean viability). As an application, we provide a purely probabilistic proof of a comparison principle and of existence for contingent and viscosity solutions of second-order fully nonlinear path-dependent Hamilton–Jacobi–Bellman equations. We do not use compactness and optimal stopping arguments, which are usually employed in the literature on viscosity solutions for second-order path-dependent PDEs.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141548536","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1590-1614, June 2024. Abstract. In this paper, we study the optimal stopping problem in the so-called exploratory framework, in which the agent takes actions randomly conditioning on the current state and a regularization term is added to the reward functional. Such a transformation reduces the optimal stopping problem to a standard optimal control problem. For the American put option model, we derive the related HJB equation and prove its solvability. Furthermore, we give a convergence rate of policy iteration and compare our solution to the classical American put option problem. Our results indicate a trade-off between the convergence rate and bias in the choice of the temperature constant. Based on the theoretical analysis, a reinforcement learning algorithm is designed and numerical results are demonstrated for several models.
{"title":"Randomized Optimal Stopping Problem in Continuous Time and Reinforcement Learning Algorithm","authors":"Yuchao Dong","doi":"10.1137/22m1516725","DOIUrl":"https://doi.org/10.1137/22m1516725","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1590-1614, June 2024. <br/> Abstract. In this paper, we study the optimal stopping problem in the so-called exploratory framework, in which the agent takes actions randomly conditioning on the current state and a regularization term is added to the reward functional. Such a transformation reduces the optimal stopping problem to a standard optimal control problem. For the American put option model, we derive the related HJB equation and prove its solvability. Furthermore, we give a convergence rate of policy iteration and compare our solution to the classical American put option problem. Our results indicate a trade-off between the convergence rate and bias in the choice of the temperature constant. Based on the theoretical analysis, a reinforcement learning algorithm is designed and numerical results are demonstrated for several models.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141523846","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1569-1589, June 2024. Abstract. The G-Brownian-motion-driven stochastic differential equations (G-SDEs) as well as the G-expectation, which were seminally proposed by Peng and his colleagues, have been extensively applied to describing a particular kind of uncertainty arising in real-world systems modeling. Mathematically depicting long-time and limit behaviors of the solution produced by G-SDEs is beneficial to understanding the mechanisms of system’s evolution. Here, we develop a new G-semimartingale convergence theorem and further establish a new invariance principle for investigating the long-time behaviors emergent in G-SDEs. We also validate the uniqueness and the global existence of the solution of G-SDEs whose vector fields are only locally Lipschitzian with a linear upper bound. To demonstrate the broad applicability of our analytically established results, we investigate its application to achieving G-stochastic control in a few representative dynamical systems.
{"title":"Invariance Principles for [math]-Brownian-Motion-Driven Stochastic Differential Equations and Their Applications to [math]-Stochastic Control","authors":"Xiaoxiao Peng, Shijie Zhou, Wei Lin, Xuerong Mao","doi":"10.1137/23m1564936","DOIUrl":"https://doi.org/10.1137/23m1564936","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1569-1589, June 2024. <br/> Abstract. The G-Brownian-motion-driven stochastic differential equations (G-SDEs) as well as the G-expectation, which were seminally proposed by Peng and his colleagues, have been extensively applied to describing a particular kind of uncertainty arising in real-world systems modeling. Mathematically depicting long-time and limit behaviors of the solution produced by G-SDEs is beneficial to understanding the mechanisms of system’s evolution. Here, we develop a new G-semimartingale convergence theorem and further establish a new invariance principle for investigating the long-time behaviors emergent in G-SDEs. We also validate the uniqueness and the global existence of the solution of G-SDEs whose vector fields are only locally Lipschitzian with a linear upper bound. To demonstrate the broad applicability of our analytically established results, we investigate its application to achieving G-stochastic control in a few representative dynamical systems.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141191228","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Concentration in Gossip Opinion Dynamics over Random Graphs","authors":"Yu Xing, K. H. Johansson","doi":"10.1137/23m1545823","DOIUrl":"https://doi.org/10.1137/23m1545823","url":null,"abstract":"","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141112754","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We consider the 3D problem of hemolysis minimization in blood flows, namely the minimization of red blood cells damage, through the shape optimization of moving domains. Such a geometry is adopted to take into account the modeling of rotating systems and blood pumps. The blood flow is described by generalized Navier-Stokes equations, in the particular case of shear thinning flows. The velocity and stress fields are then used as data for a transport equation governing the hemolysis index, aimed to measure the red blood cells damage rate. For a sequence of converging moving domains, we show that a sequence of associated solutions to blood equations converges to a solution of the problem written on the limit moving domain. Thus, we extended the result given in (Sokołowski, Stebel, 2014, in Evol. Eq. Control Theory ) for q ≥ 11 / 5, to the range 6 / 5 < q < 11 / 5, where q is the exponent of the rheological law. We then show that the sequence of hemolysis index solutions also converges to the limit solution. This shape continuity properties allows us to show the existence of minimal shapes for a class of functionals depending on the hemolysis index.
.我们考虑的是血液流动中溶血最小化的三维问题，即通过移动域的形状优化使红细胞损伤最小化。采用这种几何形状是为了考虑到旋转系统和血泵的建模。在剪切稀化流的特殊情况下，血液流动由广义纳维-斯托克斯方程描述。然后，速度和应力场被用作控制溶血指数的传输方程的数据，旨在测量红细胞的损伤率。对于一连串收敛的移动域，我们证明了血液方程的一连串相关解收敛于写在极限移动域上的问题解。因此，我们将（Sokołowski, Stebel, 2014, in Evol. Eq. Control Theory ）中给出的 q ≥ 11 / 5 的结果扩展到 6 / 5 < q < 11 / 5 的范围，其中 q 是流变规律的指数。然后，我们证明溶血指数解序列也收敛于极限解。这种形状连续性使我们能够证明一类取决于溶血指数的函数存在最小形状。
{"title":"Shape Optimization of Hemolysis for Shear Thinning Flows in Moving Domains","authors":"V. Calisti, Š. Nečasová","doi":"10.1137/23m1595485","DOIUrl":"https://doi.org/10.1137/23m1595485","url":null,"abstract":". We consider the 3D problem of hemolysis minimization in blood flows, namely the minimization of red blood cells damage, through the shape optimization of moving domains. Such a geometry is adopted to take into account the modeling of rotating systems and blood pumps. The blood flow is described by generalized Navier-Stokes equations, in the particular case of shear thinning flows. The velocity and stress fields are then used as data for a transport equation governing the hemolysis index, aimed to measure the red blood cells damage rate. For a sequence of converging moving domains, we show that a sequence of associated solutions to blood equations converges to a solution of the problem written on the limit moving domain. Thus, we extended the result given in (Sokołowski, Stebel, 2014, in Evol. Eq. Control Theory ) for q ≥ 11 / 5, to the range 6 / 5 < q < 11 / 5, where q is the exponent of the rheological law. We then show that the sequence of hemolysis index solutions also converges to the limit solution. This shape continuity properties allows us to show the existence of minimal shapes for a class of functionals depending on the hemolysis index.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141108680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Filippo De Feo, Salvatore Federico, Andrzej Święch
SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1490-1520, June 2024. Abstract. In this manuscript we consider a class of optimal control problems of stochastic differential delay equations. First, we rewrite the problem in a suitable infinite-dimensional Hilbert space. Then, using the dynamic programming approach, we characterize the value function of the problem as the unique viscosity solution of the associated infinite-dimensional Hamilton-Jacobi-Bellman equation. Finally, we prove a [math]-partial regularity of the value function. We apply these results to path dependent financial and economic problems (Merton-like portfolio problem and optimal advertising).
{"title":"Optimal Control of Stochastic Delay Differential Equations and Applications to Path-Dependent Financial and Economic Models","authors":"Filippo De Feo, Salvatore Federico, Andrzej Święch","doi":"10.1137/23m1553960","DOIUrl":"https://doi.org/10.1137/23m1553960","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1490-1520, June 2024. <br/> Abstract. In this manuscript we consider a class of optimal control problems of stochastic differential delay equations. First, we rewrite the problem in a suitable infinite-dimensional Hilbert space. Then, using the dynamic programming approach, we characterize the value function of the problem as the unique viscosity solution of the associated infinite-dimensional Hamilton-Jacobi-Bellman equation. Finally, we prove a [math]-partial regularity of the value function. We apply these results to path dependent financial and economic problems (Merton-like portfolio problem and optimal advertising).","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141148201","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Role of Correlation in Diffusion Control Ranking Games","authors":"S. Ankirchner, N. Kazi-Tani, J. Wendt","doi":"10.1137/23m1574336","DOIUrl":"https://doi.org/10.1137/23m1574336","url":null,"abstract":"","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140964810","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Nash Equilibria for Exchangeable Team-Against-Team Games, Their Mean-Field Limit, and the Role of Common Randomness","authors":"Sina Sanjari, Naci Saldi, Serdar Yüksel","doi":"10.1137/22m1534055","DOIUrl":"https://doi.org/10.1137/22m1534055","url":null,"abstract":"","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140968245","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Andrea Monti, Benita Nortmann, Thulasi Mylvaganam, Mario Sassano
SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1417-1436, June 2024. Abstract. We consider dynamic games defined over an infinite horizon, characterized by linear, discrete-time dynamics and quadratic cost functionals. Considering such linear-quadratic dynamic games, we focus on their solutions in terms of Nash equilibrium strategies. Both feedback (F-NE) and open-loop (OL-NE) Nash equilibrium solutions are considered. The contributions of the paper are threefold. First, our detailed study reveals some interesting structural insights in relation to F-NE solutions. Second, as a stepping stone toward our consideration of OL-NE strategies, we consider a specific infinite-horizon discrete-time (single-player) optimal control problem, wherein the dynamics are influenced by a known exogenous input and draw connections between its solution obtained via dynamic programming and Pontryagin’s minimum principle. Finally, we exploit the latter result to provide a characterization of OL-NE strategies of the class of infinite-horizon dynamic games. The results and key observations made throughout the paper are illustrated via a numerical example.
{"title":"Feedback and Open-Loop Nash Equilibria for LQ Infinite-Horizon Discrete-Time Dynamic Games","authors":"Andrea Monti, Benita Nortmann, Thulasi Mylvaganam, Mario Sassano","doi":"10.1137/23m1579960","DOIUrl":"https://doi.org/10.1137/23m1579960","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1417-1436, June 2024. <br/> Abstract. We consider dynamic games defined over an infinite horizon, characterized by linear, discrete-time dynamics and quadratic cost functionals. Considering such linear-quadratic dynamic games, we focus on their solutions in terms of Nash equilibrium strategies. Both feedback (F-NE) and open-loop (OL-NE) Nash equilibrium solutions are considered. The contributions of the paper are threefold. First, our detailed study reveals some interesting structural insights in relation to F-NE solutions. Second, as a stepping stone toward our consideration of OL-NE strategies, we consider a specific infinite-horizon discrete-time (single-player) optimal control problem, wherein the dynamics are influenced by a known exogenous input and draw connections between its solution obtained via dynamic programming and Pontryagin’s minimum principle. Finally, we exploit the latter result to provide a characterization of OL-NE strategies of the class of infinite-horizon dynamic games. The results and key observations made throughout the paper are illustrated via a numerical example.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140936172","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1390-1416, June 2024. Abstract. We study an optimal reinsurance problem under a diffusion risk model for an insurer who aims to minimize the probability of lifetime ruin. To rule out moral hazard issues, we only consider moral-hazard-free reinsurance contracts by imposing the incentive compatibility constraint on indemnity functions. The reinsurance premium is calculated under an extended distortion premium principle, in which the distortion function is not necessarily concave or continuous. We first show that an optimal reinsurance contract always exists and then derive two sufficient and necessary conditions to characterize it. Due to the presence of the incentive compatibility constraint and the nonconcavity of the distortion, the optimal contract is obtained as a solution to a double obstacle problem. At last, we apply the general result to study four examples and obtain the optimal contract in (semi-)closed form.
{"title":"Optimal Moral-Hazard-Free Reinsurance Under Extended Distortion Premium Principles","authors":"Zhuo Jin, Zuo Quan Xu, Bin Zou","doi":"10.1137/23m1556046","DOIUrl":"https://doi.org/10.1137/23m1556046","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1390-1416, June 2024. <br/> Abstract. We study an optimal reinsurance problem under a diffusion risk model for an insurer who aims to minimize the probability of lifetime ruin. To rule out moral hazard issues, we only consider moral-hazard-free reinsurance contracts by imposing the incentive compatibility constraint on indemnity functions. The reinsurance premium is calculated under an extended distortion premium principle, in which the distortion function is not necessarily concave or continuous. We first show that an optimal reinsurance contract always exists and then derive two sufficient and necessary conditions to characterize it. Due to the presence of the incentive compatibility constraint and the nonconcavity of the distortion, the optimal contract is obtained as a solution to a double obstacle problem. At last, we apply the general result to study four examples and obtain the optimal contract in (semi-)closed form.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140936173","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号