Istvan Gyongy, David Siska
A general result on the method of randomized stopping is proved. It is applied to optimal stopping of controlled diffusion processes with unbounded coefficients to reduce it to an optimal control problem without stopping. This is motivated by recent results of Krylov on numerical solutions to the Bellman equation.
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We study two-player zero-sum stopping games in continuous time and infinite horizon. We prove that the value in randomized stopping times exists as soon as the payoff processes are right-continuous. In particular, as opposed to existing literature, we do not assume any conditions on the relations between the payoff processes. We also show that both players have simple epsilon- optimal randomized stopping times; namely, randomized stopping times which are small perturbations of non-randomized stopping times.
On differentiability of reward functionals corresponding to Markovian randomized stopping times
本文研究了马尔可夫随机停时相关回报泛函的可微性与连续性，重点指出可微性是推导回报函数解析表达式常用方法的关键要素。
Existence of Strong Randomized Equilibria in Mean-Field Games of Optimal Stopping with Common Noise
本文借助 Bank-El Karoui 表示问题研究带共同噪声的最优停止平均场博弈，证明强随机均衡存在性，并分析严格停止均衡集合的比较静态性质。
In this work we consider optimal stopping problems with conditional convex risk measures called optimised certainty equivalents. Without assuming any kind of time-consistency for the underlying family of risk measures, we derive a novel representation for the solution of the optimal stopping problem. In particular, we generalise the additive dual representation of Rogers (2002) to the case of optimal stopping under uncertainty. Finally, we develop several Monte Carlo algorithms and illustrate their power for optimal stopping under Average Value at Risk.
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 …
Markovization of Randomized Stopping Times
本文研究随机停时的马尔可夫化，证明任意可容许强度存在保持关键分布且不增熵的马尔可夫代表，从而将变分问题简化为对马尔可夫强度的优化，并给出唯一最小熵停时及布朗停时的实现结果。
In a classical optimal stopping problem the aim is to maximize the expected value of a functional of a diffusion evaluated at a stopping time. This note considers optimal stopping problems beyond this paradigm. We study problems in which the value associated to a stopping rule depends on the law of the stopped process. If this value is quasi-convex on the space of attainable laws then it is a well known result that it is sufficient to restrict attention to the class of threshold strategies. However, if the objective function is not quasi-convex, this may not be the case. We show that, nonethel…
On the existence of Markovian randomized equilibria in Dynkin games of war-of-attrition-type
本文针对消耗战型非零和 Dynkin 博弈，通过构建马尔可夫随机化停止时间的适当拓扑，证明了马尔可夫完美均衡的存在性，填补了随机化情形下马尔可夫停止博弈均衡理论的空白。
General Markovian randomized equilibrium existence and construction in zero-sum Dynkin games for diffusions
本文研究一维扩散下的零和Dynkin停时博弈，提出基于随机化概念的新方法，在无收益序条件下构造出任意ε>0的马尔可夫随机全局ε-Nash均衡，还给出ε=0时纯策略与随机化Nash均衡的存在条件，可显式求解均衡停时与对应值。
On the time consistent solution to optimal stopping problems with expectation constraint
针对带停时期望约束的一维扩散最优停时问题，本文以随机化且纯状态依赖的停时定义弱均衡，给出验证定理、必要条件及结构性质，并用示例说明猜测—验证方法。
We prove that every two-player non-zero-sum Dynkin game in continuous time admits an epsilon-equilibrium in randomized stopping times. We provide a condition that ensures the existence of an epsilon-equilibrium in non-randomized stopping times.
A new probabilistic approach for mean field games of optimal stopping
该研究为最优停止平均场博弈提出新概率公式，用耦合反射正倒向随机微分方程刻画均衡，证明其与均衡的等价性，通过不动点定理建立均衡存在性与唯一性，还给出序理论方法及与分析方法的联系，以及对多人博弈的作用。
Optimal Stopping via Randomized Neural Networks
本文提出用隐藏层随机生成、仅训练最后一层的随机化神经网络来近似最优停止问题的延续价值，在高维美式期权定价和分数布朗运动最优停止任务上以更低计算成本达到与最先进方法相当的结果，并能高效计算Greeks。
In this paper, we will study the behavior of the space of positive harmonic functions associated with the random walk on a discrete group under the change of probability measure by a randomized stopping time. We show that this space remains unchanged after applying a bounded randomized stopping time.
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…
It is proved that the law of a possibly killed Lévy process $X$, seen up to and including (resp. up to strictly before) a stopping time, determines already the law of $X$ (resp. up to a compound Poisson component and killing).
This paper attempts to study the optimal stopping time for semi-Markov processes (SMPs) under the discount optimization criteria with unbounded cost rates. In our work, we introduce an explicit construction of the equivalent semi-Markov decision processes (SMDPs). The equivalence is embodied in the value functions of SMPs and SMDPs, that is, every stopping time of SMPs can induce a policy of SMDPs such that the value functions are equal, and vice versa. The existence of the optimal stopping time of SMPs is proved by this equivalence relation. Next, we give the optimality equation of the value …
We study the optimal stopping problem of McKean-Vlasov diffusions when the criterion is a function of the law of the stopped process. A remarkable new feature in this setting is that the stopping time also impacts the dynamics of the stopped process through the dependence of the coefficients on the law. The mean field stopping problem is introduced in weak formulation in terms of the joint marginal law of the stopped underlying process and the survival process. This specification satisfies a dynamic programming principle. The corresponding dynamic programming equation is an obstacle problem on…
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…
Neural Optimal Stopping Boundary
本文提出基于深度神经网络和经验风险最小化的方法计算最优停止边界，通过参数化边界并引入模糊松弛规则优化，验证了其在高维金融工具中的有效性并证明了边界存在性。
On the applicability of the maximality principle in optimal stopping: an example involving a diffusion and its running maximum
本文构造一个定制的最优停止问题，证明极大性原理在该问题中不适用，从而揭示其适用性的边界条件。
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 propose a new method for solving optimal stopping problems (such as American option pricing in finance) under minimal assumptions on the underlying stochastic process $X$. We consider classic and randomized stopping times represented by linear and non-linear functionals of the rough path signature $\mathbb{X}^{<\infty}$ associated to $X$, and prove that maximizing over these classes of signature stopping times, in fact, solves the original optimal stopping problem. Using the algebraic properties of the signature, we can then recast the problem as a (deterministic) optimization problem depen…
Optimal stopping of Gauss-Markov bridges
本文通过时间-空间变换方法求解高斯-马尔可夫桥的非贴现有限时域最优停止问题，证明最优停止边界的Lipschitz连续性并用积分方程唯一解刻画，同时实现Picard迭代算法进行数值计算。
Stopping Times of Boundaries: Relaxation and Continuity
该研究在离散时间最优停止语境下，证明了边界到停止策略期望值映射的连续性，研究了击中时间及其松弛的联系，并结合相关概念为特定算法提供收敛性分析。
Convergence is proved for solutions of Dirichlet problems in regions with many small excluded sets (holes), as the holes become smaller and more numerous. The problem is formulated in the context of Markov processes associated with general Dirichlet forms, for random and nonrandom excluded sets. Sufficient conditions are given under which the sequence of entrance times or hitting times of the excluded sets converges in the stable topology. Convergence in the stable topology is a strengthened form of convergence in distribution, introduced by Renyi. Stable convergence of the entrance times impl…
Let X_t, 0<=t<=T be a one-dimensional stochastic process with independent and stationary increments. This paper considers the problem of stopping the process X_t "as close as possible" to its eventual supremum M_T:=sup{X_t: 0<=t<=T}, when the reward for stopping with a stopping time tau<=T is a nonincreasing convex function of M_T-X_tau. Under fairly general conditions on the process X_t, it is shown that the optimal stopping time tau is of "bang-bang" form: it is either optimal to stop at time 0 or at time T. For the case of random walk, the rule tau=T is optimal if the steps of the walk stoc…
Convergence of the Backward Deep BSDE Method with Applications to Optimal Stopping Problems
本文针对传统深度BSDE方法无法求解最优停时问题的局限，为后向深度BSDE方法提供严格理论支撑，推导后验误差估计与损失函数上界，并通过数值算例验证了理论一致性。
Optimal Prediction of Resistance and Support Levels under Constant Elasticity of Variance Processes
研究在资产价格\(X\)遵循恒定方差弹性过程时，针对与\(X\)无关的隐藏期望水平\(\ell\)的最优预测问题。采用期望水平假设，得出最优交易边界，证明其存在并推导方程刻画，正、负漂移情况停止集有不同描述。
Loewner Evolution as Itô Diffusion
本文将随机化 Loewner 链转化为 Itô 扩散的已有结果推广至向量情形，并证明该转化方式具有唯一性。
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