The purpose of this paper is two-fold:
We extend the well-known relation between optimal stopping and randomized stopping of a given stochastic process to a situation where the available information flow is a filtration with no a priori assumed relation to the filtration of the process. We call these problems optimal stopping and randomized stopping with general information.
Following an idea of Krylov [K] we introduce a special singular stochastic control problem with general information and show that this is also equivalent to the partial information optimal stopping and randomized stopping problems. Then we show that the solution of this singular control problem can be expressed in terms of partial information variational inequalities.
In this paper we study randomized optimal stopping problems and consider corresponding forward and backward Monte Carlo based optimisation algorithms. In particular we prove the convergence of the proposed algorithms and derive the corresponding convergence rates.
UCLA Anderson School of Management, University of California, Los Angeles（加州大学洛杉矶分校安德森管理学院）
Optimal stopping is the problem of determining when to stop a stochastic system in order to maximize reward, which is of practical importance in domains such as finance, operations management and healthcare. Existing methods for high-dimensional optimal stopping that are popular in practice produce deterministic linear policies -- policies that deterministically stop based on the sign of a weighted sum of basis functions -- but are not guaranteed to find the optimal policy within this policy class given a fixed basis function architecture. In this paper, we propose a new methodology for optima…
In this paper we establish a new connection between a class of 2-player nonzero-sum games of optimal stopping and certain $2$-player nonzero-sum games of singular control. We show that whenever a Nash equilibrium in the game of stopping is attained by hitting times at two separate boundaries, then such boundaries also trigger a Nash equilibrium in the game of singular control. Moreover a differential link between the players' value functions holds across the two games.
In this article we study and classify optimal martingales in the dual formulation of optimal stopping problems. In this respect we distinguish between weakly optimal and surely optimal martingales. It is shown that the family of weakly optimal and surely optimal martingales may be quite large. On the other hand it is shown that the Doob-martingale, that is, the martingale part of the Snell envelope, is in a certain sense the most robust surely optimal martingale under random perturbations. This new insight leads to a novel randomized dual martingale minimization algorithm that doesn't require …
The purpose of this paper is to study the following topics and the relation between them: (i) Optimal singular control of mean-field stochastic differential equations with memory, (ii) reflected advanced mean-field backward stochastic differential equations, and (iii) optimal stopping of mean-field stochastic differential equations. More specifically, we do the following: - We prove the existence and uniqueness of the solutions of some reflected advanced memory backward stochastic differential equations (AMBSDEs), - we give sufficient and necessary conditions for an optimal singular control of…
In this paper we study the optimization problem of an economic agent who chooses a job and the time of retirement as well as consumption and portfolio of assets. The agent is constrained in the ability to borrow against future income. We transform the problem into a dual two-person zero-sum game, which involves a controller, who is a minimizer and chooses a non-increasing process, and a stopper, who is a maximizer and chooses a stopping time. We derive the Hamilton-Jacobi- Bellman quasi-variational inequality(HJBQV) of a max-min type arising from the game. We provide a solution to the HJBQV an…
We consider the game-theoretic approach to time-inconsistent stopping of a one-dimensional diffusion where the time-inconsistency is due to the presence of a non-exponential (weighted) discount function. In particular, we study (weak) equilibria for this problem in a novel class of mixed (i.e., randomized) stopping times based on a local time construction of the stopping intensity. For a general formulation of the problem we provide a verification theorem giving sufficient conditions for mixed (and pure) equilibria in terms of a set of variational inequalities, including a smooth fit condition…
A Non Convex Singular Stochastic Control Problem and its Related Optimal Stopping Boundaries
已知具有凸性能准则的奇异随机控制（SSC）的问题与最优停止的相关问题之间的等价性，参见例如Karatzas和Shreve [SIAM J. Control Optim. 22（1984）]。本文的目的是调查多远这种类型的连接推广到一个非凸的问题，购买电力。在经典的等价打破，我们提供了替代连接到最优停止问题。我们考虑了一个非凸的无限时域SSC问题，其状态由一个代表实值商品价格的不受控扩散和一个代表库存的受控递增有界过程组成。我们分析的几何形状的动作和不动作区域的特征（最佳）的边界。与凸SSC问题的情况下，我们发现，最佳的边界可能是反射和排斥，这是很自然的解释问题与任意停止SSC之一。
We prove the dynamic programming principle (DPP) in a class of problems where an agent controls a $d$-dimensional diffusive dynamics via both classical and singular controls and, moreover, is able to terminate the optimisation at a time of her choosing, prior to a given maturity. The time-horizon of the problem is random and it is the smallest between a fixed terminal time and the first exit time of the state dynamics from a Borel set. We consider both the cases in which the total available fuel for the singular control is either bounded or unbounded. We build upon existing proofs of DPP and e…
Given an initial (resp., terminal) probability measure $μ$ (resp., $ν$) on $\mathbb{R}^d$, we characterize those optimal stopping times $τ$ that maximize or minimize the functional $\mathbb{E} |B_0 - B_τ|^α$, $α> 0$, where $(B_t)_t$ is Brownian motion with initial law $B_0\sim μ$ and with final distribution --once stopped at $τ$-- equal to $B_τ\sim ν$. The existence of such stopping times is guaranteed by Skorohod-type embeddings of probability measures in "subharmoic order" into Brownian motion. This problem is equivalent to an optimal mass transport problem with certain constraints, namely t…
Singular control with state-dependent costs for Lévy processes
Centro de Matemática, Facultad de Ciencias, Universidad de la República（乌拉圭共和国大学理学院数学中心）；Departamento de Matemática y Estadística, Universidad Torcuato di Tella（托尔夸托·迪泰拉大学数学与统计系）；Ingeniería Forestal, Centro Universitario de Tacuarembó, Universidad de la República（乌拉圭共和国大学塔夸伦博大学中心森林工程系）
We study the optimal dividend problem for a firm's manager who has partial information on the profitability of the firm. The problem is formulated as one of singular stochastic control with partial information on the drift of the underlying process and with absorption. In the Markovian formulation, we have a 2-dimensional degenerate diffusion, whose first component is singularly controlled and it is absorbed as it hits zero. The free boundary problem (FBP) associated to the value function of the control problem is challenging from the analytical point of view due to the interplay of degeneracy…
Randomized Kaczmarz is a simple iterative method for finding solutions of linear systems $Ax = b$. We point out that the arising sequence $(x_k)_{k=1}^{\infty}$ tends to converge to the solution $x$ in an interesting way: generically, as $k \rightarrow \infty$, $x_k - x$ tends to the singular vector of $A$ corresponding to the smallest singular value. This has interesting consequences: in particular, the error analysis of Strohmer \& Vershynin is optimal. It also quantifies the `pre-convergence' phenomenon where the method initially seems to converge faster. This fact also allows for a fast co…
We study a class of optimal stopping games (Dynkin games) of preemption type, with uncertainty about the existence of competitors. The set-up is well-suited to model, for example, real options in the context of investors who do not want to publicly reveal their interest in a certain business opportunity. We show that there exists a Nash equilibrium in randomized stopping times which is described explicitly in terms of the corresponding one-player game.
We study the problem of stopping a Brownian motion at a given distribution $ν$ while optimizing a reward function that depends on the (possibly randomized) stopping time and the Brownian motion. Our first result establishes that the set $\mathcal{T}(ν)$ of stopping times embedding $ν$ is weakly dense in the set $\mathcal{R}(ν)$ of randomized embeddings. In particular, the optimal Skorokhod embedding problem over $\mathcal{T}(ν)$ has the same value as the relaxed one over $\mathcal{R}(ν)$ when the reward function is semicontinuous, which parallels a fundamental result about Monge maps and Kanto…
We analyze an optimal stopping problem with random maturity under a nonlinear expectation with respect to a weakly compact set of mutually singular probabilities $\mathcal{P}$. The maturity is specified as the hitting time to level $0$ of some continuous index process at which the payoff process is even allowed to have a positive jump. When $\mathcal{P}$ is a collection of semimartingale measures, the optimal stopping problem can be viewed as a {\it discretionary} stopping problem for a player who can influence both drift and volatility of the dynamic of underlying stochastic flow.
Consider the problem of a government that wants to reduce the debt-to-GDP (gross domestic product) ratio of a country. The government aims at choosing a debt reduction policy which minimises the total expected cost of having debt, plus the total expected cost of interventions on the debt ratio. We model this problem as a singular stochastic control problem over an infinite time-horizon. In a general not necessarily Markovian framework, we first show by probabilistic arguments that the optimal debt reduction policy can be expressed in terms of the optimal stopping rule of an auxiliary optimal s…
We consider a stochastic game of control and stopping specified in terms of a process $X_t=-θΛ_t+W_t$, representing the holdings of Player 1, where $W$ is a Brownian motion, $θ$ is a Bernoulli random variable indicating whether Player 2 is active or not, and $Λ$ is a non-decreasing process representing the accumulated "theft" or "fraud" performed by Player 2 (if active) against Player 1. Player 1 cannot observe $θ$ or $Λ$ directly, but can merely observe the path of the process $X$ and may choose a stopping rule $τ$ to deactivate Player 2 at a cost $M$. Player 1 thus does not know if she is th…
We combine the fields of heuristic optimization and optimal stopping. We propose a strategy for benchmarking randomized optimization algorithms that minimizes the expected total cost for obtaining a good solution with an optimal number of calls to the solver. To do so, rather than letting the objective function alone define a cost to be minimized, we introduce a further cost-per-call of the algorithm. We show that this problem can be formulated using optimal stopping theory. The expected cost is a flexible figure of merit for benchmarking probabilistic solvers that can be computed when the opt…
Zhejiang University（浙江大学）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
We propose and study kernel conjugate gradient methods (KCGM) with random projections for least-squares regression over a separable Hilbert space. Considering two types of random projections generated by randomized sketches and Nyström subsampling, we prove optimal statistical results with respect to variants of norms for the algorithms under a suitable stopping rule. Particularly, our results show that if the projection dimension is proportional to the effective dimension of the problem, KCGM with randomized sketches can generalize optimally, while achieving a computational advantage. As a co…
