On the One-Dimensional Optimal Switching Problem
Erhan Bayraktar, Masahiko Egami
我们直接利用动态规划原理和值函数的过度特征化，显式地解决了一维扩散的最优切换问题。然后利用凹函数的性质证明了值函数的形状和光滑拟合原理。
We explicitly solve the optimal switching problem for one-dimensional diffusions by directly employing the dynamic programming principle and the excessive characterization of the value function. The shape of the value function and the smooth fit principle then can be proved using the properties of concave functions.
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This paper studies a discrete-time optimal switching problem on a finite horizon. The underlying model has a running reward, terminal reward and signed (positive and negative) switching costs. Using the martingale approach to optimal stopping problems, we extend a well known explicit dynamic programming method for computing the value function and the optimal strategy to the case of signed switching costs.
This paper is concerned with optimal switching over multiple modes in continuous time and on a finite horizon. The performance index includes a running reward, terminal reward and switching costs that can belong to a large class of stochastic processes. Particularly, the switching costs are modelled by right-continuous with left-limits processes that are quasi-left-continuous and can take both positive and negative values. We provide sufficient conditions leading to a well known probabilistic representation of the value function for the switching problem in terms of interconnected Snell envelo…
Optimal Switching at Poisson Random Intervention Times
本文引入了一类新的最优切换问题，其中局中人被允许在一个外生Poisson到达时间序列上进行切换，而切换系统由一个无穷时域后向随机微分方程系统控制.值函数和最优切换策略的特征在于底层切换系统的解决方案。在马尔可夫背景下，本文利用比较原理给出了开关区域结构的完整描述。
In this paper we study continuous-time two-player zero-sum optimal switching games on a finite horizon. Using the theory of doubly reflected BSDEs with interconnected barriers, we show that this game has a value and an equilibrium in the players' switching controls.
The Finite Horizon Optimal Multi-Modes Switching Problem: the Viscosity Solution Approach
本文证明了一类具有连通障碍的m个变分偏微分不等式组解的存在唯一性。该系统是马尔可夫最优m状态切换问题的验证定理的确定性版本。切换费用函数是任意的。这个问题与金融市场中公司的估值有关。
A Finite Horizon Optimal Multiple Switching Problem
考虑了有限时间内系统的最优多次切换问题，当系统的状态（包括切换成本）是一般适应随机过程时。该问题被制定为一个扩展的脉冲控制问题，并完全解决使用概率工具，如Snell包络的过程和反射向后随机微分方程。最后，当系统的状态是一个马尔可夫扩散过程，我们证明了最优问题的值函数的向量是一个粘性解的变分不等式系统与互联的障碍。
We address the problem of making a managerial decision when the investment project is subsidized, which results in the resolution of an infinite-horizon optimal stopping problem of a switching diffusion driven by either an homogeneous or an inhomogeneous continuous-time Markov chain. We provide a characterization of the value function (and optimal strategy) of the optimal stopping problem. On the one hand, broadly, we can prove that the value function is the unique viscosity solution to a system of HJB equations. On the other hand, when the Markov chain is homogeneous and the switching diffusi…
This paper investigates the stochastic linear quadratic (LQ, for short) optimal control problem of Markov regime switching system. The representation of the cost functional for the stochastic LQ optimal control problem of Markov regime switching system is derived using the technique of It{ô}'s formula. For the stochastic LQ optimal control problem of Markov regime switching system, we establish the equivalence between the open-loop (closed-loop) solvability and the existence of an adapted solution to the corresponding forward-backward stochastic differential equation with constraint (the exist…
In this paper, we investigate the open-loop and weak closed-loop solvabilities of stochastic linear quadratic (LQ, for short) optimal control problem of Markovian regime switching system. Interestingly, these two solvabilities are equivalent. We first provide an alternative characterization of the open-loop solvability of the LQ problem using the perturbation approach. Then, we study the weak closed-loop solvability of the LQ problem of Markovian regime switching system, and establish the equivalent relationship between open-loop and weak closed-loop solvabilities. Finally, we present an examp…
Uniqueness of Viscosity Solutions for Optimal Multi-Modes Switching Problem with Risk of default
本文研究了具有违约风险的有限时间和无限时间内的最优m状态转换问题。我们允许转换成本泛函和违约成本是多项式增长的和任意的。本文证明了一类具有连通障碍的m个变分偏微分不等式系统解的唯一性。该系统是具有违约风险的马尔可夫最优m状态切换问题的验证定理的确定性版本。这个问题与能源市场中发电厂的估值有关。
We solve two stochastic control problems in which a player tries to minimize or maximize the exit time from an interval of a Brownian particle, by controlling its drift. The player can change from one drift to another but is subject to a switching cost. In each problem, the value function is written as the solution of a free boundary problem involving second order ordinary differential equations, in which the unknown boundaries are found by applying the principle of smooth fit. For both problems, we compute the value function, we exhibit the optimal strategy and we prove its generic uniqueness…
In this work we provide explicit conditions on the existence of optimal feedback controls for stochastic processes with regime-switching. We use the compactification method which needs less regularity conditions on the coefficients of the studied stochastic systems. Two different kinds of controls have been considered: one is the control on the coefficients of the diffusion processes, another is the control on the transition rate matrices of the continuous-time Markov chains. Moreover, the dynamic programming principle is established after showing the continuity of the value function.
Optimal Switching of One-Dimensional Reflected BSDEs, and Associated Multi-Dimensional BSDEs with Oblique Reflection
研究了一维反射倒向随机微分方程（RBSDEs）的最优切换问题，其中生成元、终值和障碍都以正代价进行切换.价值过程的特点是一个系统的多维RBSDES斜反射，其存在性和唯一性绝不是平凡的，因此仔细检查。存在性使用皮卡德迭代和惩罚的方法，但在一些不同的条件下。唯一性是通过表示证明，无论是作为价值过程，我们的最佳切换问题的一维RBSDES，或作为平衡值过程的随机微分对策的切换和停止。最后，切换RBBACK被解释为一个真正的选择。
Optimal Multi-Modes Switching Problem in Infinite Horizon
本文研究了在马尔可夫框架下，任意切换费用函数的无限时域最优m状态切换的确定性验证定理问题.该问题被制定为一个扩展的脉冲控制问题和解决的概率工具，如Snell包络的过程和反射向后随机微分方程。利用黏性解的方法对m个具有连通障碍物的变分不等式系统进行了精细分析。我们证明了最优问题的值函数向量是系统唯一的黏性解。这个问题与金融市场中公司的估值有关。
The paper studies a class of multidimensional optimal stopping problems with infinite horizon for linear switching diffusions. There are two main novelties in the optimal problems considered: the underlying stochastic process has discontinuous paths and the cost function is not necessarily integrable on the entire time horizon, where the latter is often a key assumption in classical optimal stopping theory for diffusions, cf. [22, Corollary 2.9]. Under relatively mild conditions, we show, for the class of multidimensional optimal stopping problems under consideration, that the first entry time…
This paper studies a system of $m$ variational inequalities with interconnected obstacles in infinite horizon associated to optimal multi-modes switching problems. Our main result is the existence and uniqueness of a continuous solution in viscosity sense, for that system. The proof of the main result strongly relies on the connection between the systems of variational inequalities and reflected backward stochastic differential equations (RBSDEs) with oblique reflection, which will be characterized through a Feynman-Kac's formula. The main feature of our system of infinite horizon RBSDEs is th…
Optimal Switching Problems under Partial Information
本文提出并研究了一个部分信息下的最优切换问题。在我们的模型中，代理人/经理/投资者试图通过在不同的状态/投资之间切换来最大化预期回报。然而，他并不完全了解他的环境，并且仅可访问包含关于环境/底层的部分信息的观察过程。所有的决策都必须基于这个观察过程所携带的部分信息。我们提出了一种基于动态规划、回归蒙特卡罗方法和随机滤波理论的概率数值算法来计算值函数。本文在观测过程和基础过程满足线性Kalman-Bucy条件下，得到了值函数的逼近和相应的收敛性结果。最后给出了一个数值例子来说明部分信息的一些特性。
In this paper, we present some properties of time optimal controls for linear ODEs with the ball-type control constraint. More precisely, for an optimal control, we build up an upper bound for the number of its switching points; show that it jumps from one direction to the reverse direction at each switching point; give its dynamic behaviour between two consecutive switching points; and study its switching directions.
This paper uses recent results on continuous-time finite-horizon optimal switching problems with negative switching costs to prove the existence of a saddle point in an optimal stopping (Dynkin) game. Sufficient conditions for the game's value to be continuous with respect to the time horizon are obtained using recent results on norm estimates for doubly reflected backward stochastic differential equations. This theory is then demonstrated numerically for the special cases of cancellable call and put options in a Black-Scholes market.
We consider an optimal switching problem with random lag and possibility of component failure. The random lag is modeled by letting the operation mode follow a regime switching Markov-model with transition intensities that depend on the switching mode. The possibility of failures is modeled by having absorbing components. We show existence of an optimal control for the problem by applying a probabilistic technique based on the concept of Snell envelopes.
In this paper, we study the $m$-states optimal switching problem in finite horizon, when the switching cost functions are arbitrary and can be positive or negative. This has an economic incentive in terms of central evaluation in cases where such organizations or state grants or financial assistance to power plants that promotes green energy in their production activity or what uses less polluting modes in their production. We show existence for optimal strategy via a verification theorem then we show existence and uniqueness of the value processes by using an approximation scheme. In the mark…
A Direct Method for Solving Optimal Switching Problems of One-Dimensional Diffusions
本文提出了一维扩散最优切换问题的一种直接解法。该方法不需要对值函数的形式和切换策略进行说明，也不需要通过拟变分不等式证明最优性。直接法使用一维扩散的最优停止问题的一般理论，并将值函数描述为它们各自变换空间中的最小线性优变量集。
We address a general optimal switching problem over finite horizon for a stochastic system described by a differential equation driven by Brownian motion. The main novelty is the fact that we allow for infinitely many modes (or regimes, i.e. the possible values of the piecewise-constant control process). We allow all the given coefficients in the model to be path-dependent, that is, their value at any time depends on the past trajectory of the controlled system. The main aim is to introduce a suitable (scalar) backward stochastic differential equation (BSDE), with a constraint on the martingal…
We employ the viscosity solution technique to analyze optimal stopping problems with regime switching. Specifically, we obtain the viscosity property of value functions, the uniqueness of viscosity solutions, the regularity of value functions and the form of optimal stopping intervals. Finally, we provide an application of the results.
An optimal boundary control problem for the one-dimensional heat equation is considered. The objective functional includes a standard quadratic terminal observation, a Tikhonov regularization term with regularization parameter $ν$, and the $L^1$-norm of the control that accounts for sparsity. The switching structure of the optimal control is discussed for $ν\ge 0$. Under natural assumptions, it is shown that the set of switching points of the optimal control is countable with the final time as only possible accumulation point. The convergence of switching points is investigated for $ν\searrow …
This paper studies the time optimal control problem for systems of heat equations coupled by a pair of constant matrices. The control constraint is of the ball-type, while the target is the origin of the state space. We obtain an upper bound for the number of switching points of the optimal control over each interval with a fixed length. Also, we prove that at each switching point, the optimal control jump from one direction to the reverse direction.
Optimal switching control design for polynomial systems: an LMI approach
提出了一种新的LMI方法来设计状态约束下多项式动力系统的最优开关序列。我们制定的开关设计问题作为一个最优控制问题，然后放松到一个线性规划（LP）问题的空间占用措施。这个无限维LP可以用凸有限维线性矩阵不等式的层次近似地数值求解。与大多数现有的工作LMI方法相比，我们有一个保证的全局最优性，在这个意义上说，我们获得了一个渐近收敛（即消失保守性）层次的下界上可实现的性能。我们还解释了如何构建一个几乎最优的开关序列。
Optimal control of risk process in a regime-switching environment
本文研究的是一家保险公司的成本优化问题。保险公司的盈余模型由受控状态转换扩散，其中状态转换机制提供了随机环境的波动。我们的目标是找到一个最优控制，使总成本最小化到一个随机的退出时间。得到了一个比Fleming和Soner 2006，第V.2节弱的值函数连续的充分条件.此外，值函数是一个黏性解的Hamilton-Jacobian-Bellman方程。
We solve non-Markovian optimal switching problems in discrete time on an infinite horizon, when the decision maker is risk aware and the filtration is general, and establish existence and uniqueness of solutions for the associated reflected backward stochastic difference equations. An example application to hydropower planning is provided.
Optimal control of McKean-Vlasov systems under partial observation and hidden Markov switching
本文研究了在部分观测下麦金-沃拉斯科夫系统最优控制问题，通过概率变换框架处理分布依赖性，推导出扎卡伊方程和贝尔曼方程，解决均场控制问题。
A System of BSDEs with Singular Terminal Values Arising in Optimal Liquidation with Regime Switching
本文研究带制度切换和暗池的最优清算问题，首次引入带跳和奇异终端值的BSDE系统，证明了存在性与唯一性，解决了该随机控制问题。
We consider a type of optimal switching problems with non-uniform execution delays and ramping. Such problems frequently occur in the operation of economical and engineering systems. We first provide a solution to the problem by applying a probabilistic method. The main contribution is, however, a scheme for approximating the optimal control by limiting the information in the state-feedback. In a numerical example the approximation routine gives a considerable computational performance enhancement, when compared to a conventional algorithm.
This paper studies finite-time optimal consumption-investment problems with power, logarithmic and exponential utilities, in a regime switching market with random coefficients, subject to coupled constraints on the consumption and investment strategies. We provide explicit optimal consumption-investment strategies and optimal values for the problems in terms of the solutions to some diagonally quadratic backward stochastic differential equation (BSDE) systems and linear BSDE systems with unbound coefficients. Some of these BSDEs are new in the literature and solving them is one of the main the…
We study local structure of time-optimal controls and trajectories for a 3-dimensional control-affine system with a 2-dimensional control parameter with values in the disk. In particular, we give sufficient conditions, in terms of Lie bracket relations, for optimal controls to be smooth or to have only isolated jump discontinuities.
Stochastic Optimal Multi-Modes Switching with a Viscosity Solution Approach
针对有限时域内任意切换成本函数的最优多模式切换问题，证明了最优策略的存在性，并通过验证定理给出了最优策略有限的条件；当系统状态为马尔可夫过程时，证明了最优问题的值函数向量是带有相互连接障碍的m个变分偏微分不等式系统的唯一粘性解。
Explicit solution to an optimal two-player switching game in infinite horizon
本文针对无限时域双人切换博弈问题，采用粘性方法刻画切换区域（转化为寻找触发切换的状态阈值）并推导显式解，还提出数值程序计算值函数并通过模拟验证，状态为一维伊藤扩散且切换成本可非正。
We consider the optimization of a dynamical system by switching at discrete time points between abstract evolution equations composed by nonlinearly perturbed strongly continuous semigroups, nonlinear state reset maps at mode transition times and Lagrange-type cost functions including switching costs. In particular, for a fixed sequence of modes, we derive necessary optimality conditions using an adjoint equation based representation for the gradient of the costs with respect to the switching times. For optimization with respect to the mode sequence, we discuss a mode-insertion gradient. The t…
We study a finite-dimensional continuous-time optimal control problem on finite horizon for a controlled diffusion driven by Brownian motion, in the linear-quadratic case. We admit stochastic coefficients, possibly depending on an underlying independent marked point process, so that our model is general enough to include controlled switching systems where the switching mechanism is not required to be Markovian. The problem is solved by means of a Riccati equation, which a backward stochastic differential equation driven by the Bronwian motion and by the random measure associated to the marked …
Optimal control problems involving hybrid binary-continuous control costs are challenging due to their lack of convexity and weak lower semicontinuity. Replacing such costs with their convex relaxation leads to a primal-dual optimality system that allows an explicit pointwise characterization and whose Moreau-Yosida regularization is amenable to a semismooth Newton method in function space. This approach is especially suited for computing switching controls for partial differential equations. In this case, the optimality gap between the original functional and its relaxation can be estimated a…
An Optimal Switching Approach for Bird Migration
本文提出最优切换模型，将鸟类迁徙表示为不同随机微分方程间的切换，并在完全与部分环境信息下结合数值方法、有限差分和随机模拟分析其最优策略。
We consider an optimal switching problem where the terminal reward depends on the entire control trajectory. We show existence of an optimal control by applying a probabilistic technique based on the concept of Snell envelopes. We then apply this result to solve an impulse control problem for stochastic delay differential equations driven by a Brownian motion and an independent compound Poisson process. Furthermore, we show that the studied problem arises naturally when maximizing the revenue from operation of a group of hydro-power plants with hydrological coupling.
Non-homogeneous stochastic linear-quadratic optimal control problems with multi-dimensional state and regime switching
本文研究多维状态与状态切换下的非齐次随机LQ最优控制问题，通过压缩映射方法求解随机Riccati方程和伴随BSDE，从而得到闭环最优控制与最优值。
The benefits of transmission line switching are well-known in terms of reducing operational cost and improving system reliability of power systems. However, finding the optimal power network configuration is a challenging task due to the combinatorial nature of the underlying optimization problem. In this work, we identify a certain "node-based" set that appears as substructure of the optimal transmission switching problem and then conduct a polyhedral study of this set. We construct an extended formulation of the integer hull of this set and present the inequality description of the integer h…
Turnpike Property of Stochastic Linear-Quadratic Optimal Control Problems in Large Horizons with Regime Switching I: Homogeneous Cases
本文研究大时域带机制切换的线性齐次随机LQ最优控制，通过证明微分Riccati方程解收敛到代数Riccati方程，建立最优对的turnpike性质，并改进无切换情形的已有估计。
A neural network approach to high-dimensional optimal switching problems with jumps in energy markets
本文提出一种向后推进的机器学习算法，利用序列神经网络解决能源市场中带随机跳跃的高维最优切换与调度问题，实验证明其准确且随维度呈线性至次线性减速。
Optimal control of uncertain stochastic systems with Markovian switching and its applications to portfolio decisions
本文首先描述了一类具有马尔可夫切换的不确定随机控制系统，并导出了马尔可夫调制过程的Itô-Liu公式。在此基础上，我们刻画了一个满足广义Hamilton-Jacobi-Bellman（HJB）方程的带有马尔可夫切换的最优控制律。然后，利用广义HJB方程，推导出了不确定随机金融市场中马尔可夫转换下的最优消费和投资组合策略。最后，对于常数相对风险厌恶（CRRA）的幸福函数，我们明确地得到了最优消费和投资组合政策。并通过算例进行了经济分析。
In this paper we set up an optimal control framework for a hybrid stochastic system with dual or multiple Markov switching diffusion processes, while Markov chains governing these switching diffusions are not identical as assumed by the existing literature. As an application and illustration of this model, we solve a portfolio choice problem for an investor facing financial and labor markets that are both regime switching. In continuous time context we combine two separate Markov chains into one synthetic Markov chain and derive its corresponding generator matrix, then state the HJB equations …
In this paper we analyse local regularity of time-optimal controls and trajectories for an n-dimensional affine control system with a control parameter, taking values in a k-dimensional closed ball. In the case of k equal to n-1, we give sufficient conditions in terms of Lie bracket relations for all optimal controls to be smooth or to have only isolated jump discontinuities.
Turnpike Property of a Linear-Quadratic Optimal Control Problem in Large Horizons with Regime Switching II: Non-Homogeneous Cases
本文研究大时域非齐次线性随机系统带机制切换的二次最优控制问题，将强收费公路性质从齐次情形推广至非齐次情形，并发现部分结果在无切换时亦为新结论。
Feedback Solution to Optimal Switching Problems with Switching Cost
本文研究了在非线性自主子系统间切换的最优问题，通过引入切换成本项，提出反馈解法以近似最优解，并通过数值例子验证了方法的有效性。
This paper is concerned with the problem of finding the optimal of extraction policies of an oil field in light of various financial and economical restrictions and constraints. Taking into account the fact that the oil price in worldwide commodity markets fluctuates randomly following global and seasonal macro-economic parameters, we model the evolution of the oil price as a mean reverting regime switching jump diffusion process. We formulate this problem as finite-time horizon optimal control problem. We solve the control problem using the method of viscosity solutions. Moreover, we construc…
In this paper, we establish a general stochastic maximum principle for optimal control for systems described by a continuous-time Markov regime-switching stochastic recursive utilities model. The control domain is postulated not to be convex, and the diffusion terms depend on control variables. To this end, we first study a kind of classical forward stochastic optimal control problems. Afterwards, based on previous results, we introduce two groups of new first and second-order adjoint equations. The corresponding variational equations for forward-backward stochastic differential equations are …
A Problem of Finite-Horizon Optimal Switching and Stochastic Control for Utility Maximization
本文研究带强制退休日期和换工作选择权的效用最大化问题，采用对偶鞅方法将其转化为有限时域纯最优切换问题，利用双障碍问题理论分析自由边界，建立对偶定理并给出最优切换策略的财富边界刻画及数值结果。
Robustness of optimal control for controlled regime-switching diffusions with incorrect models
本文研究了受控切换扩散过程的鲁棒性，通过分析价值函数和最优策略的连续性，扩展了鲁棒性框架至更广泛的混合系统。
In this paper, we study a model of network adaptation mechanism to control spreading processes over switching contact networks, called adaptive susceptible-infected-susceptible model. The edges in the network model are randomly removed or added depending on the risk of spread through them. By analyzing the joint evolution of the spreading dynamics "in the network" and the structural dynamics "of the network", we derive conditions on the adaptation law to control the dynamics of the spread in the resulting switching network. In contrast with the results in the literature, we allow the initial t…
In optimal control problems of control-affine systems, whose solutions are bang-bang or singular type, verification of optimality using the Hamilton-Jacobi-Bellman (HJB) equation involves the computation of partial derivatives of switching times and switching states with respect to initial conditions (time and state). In this paper, we establish a formula for the partial derivatives of ordinary differential equations (ODE) flows with respect to initial conditions, which is more suitable for using in HJB equation than such provided by the classical theory of ODE. We apply the obtained results t…
The mathematical theory for optimal switching is by now relatively well developed, but the number of concrete applications of this theoretical framework remains few. In this paper, we bridge parts of this gap by applying optimal switching theory to a set of production planning problems related to hydropower plants. In particular, we study two different cases involving small run-of-river hydropower plants and show how optimal switching can be used to create fully automatic production schemes in these cases, with non-zero cost of switching between different states of production. Along the way of…
Stochastic linear-quadratic optimal control problems with multi-dimensional state and non-Markovian regime switching
本文研究多维状态下非马尔可夫机制切换的随机LQ控制，通过扩展Riccati方程、单调Picard迭代及FBSDE的$L^p$估计证明解的存在性并完成问题求解。
Linear-Quadratic Optimal Control for Mean-Field Stochastic Differential Equations in Infinite-Horizon with Regime Switching
本文研究切换状态下含条件平均场项的无限时域随机线性二次最优控制问题，通过正交分解推导出代数Riccati方程和倒向随机微分方程，并利用其可解性刻画闭环与开环最优策略。
Viscosity solutions for mean field optimal switching with a two-time-scale Markov chain
本文研究带双时间尺度马尔可夫链的平均场最优切换问题，建立值函数与动态规划原理，刻画为Wasserstein空间障碍方程并证明粘性解的相关性质，得到极限系统收敛性，还应用于体制切换市场的股票交易问题。
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