Dmitrii S. Silvestrov, Jozef L. Teugels
This article is devoted to the investigation of limit theorems for mixed max-sum processes with renewal type stopping indexes. Limit theorems of weak convergence type are obtained as well as functional limit theorems.
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Suppose that i.i.d. random variables $X_{1}, X_{2}, \ldots$ are chosen uniformly from $[0,1]$, and let $f: [0,1] \rightarrow [0,1]$ be an increasing bijection. Define $μ_{f}$ to be the expected value of $f(X_{i})$ for each $i$. Define the random variable $K_{f}$ be to be minimal so that $\sum_{i = 1}^{K_{f}} f(X_{i}) > t$ and let $N_{f}(t)$ be the expected value of $K_{f}$. We prove that if $c_{f} = \frac{\int_{0}^{1} \int_{f^{-1}(u)}^{1} (f(x)-u) dx du}{μ_{f}}$, then $N_{f}(t) = \frac{t+c_{f}}{μ_{f}}+o(1)$. This generalizes a result of Ćurgus and Jewett (2007) on the case $f(x) = x$.
Maxima over random time intervals for heavy-tailed compound renewal and Lévy processes
本文研究了带负漂移的重尾复合更新过程和Lévy过程在随机时间区间上最大值的次指数尾部渐近性，结果对不依赖未来增量的随机时间一致成立，并与随机游走理论建立联系。
We study subexponential tail asymptotics for the distribution of the maximum $M_t:=\sup_{u\in[0,t]}X_u$ of a process $X_t$ with negative drift for the entire range of $t>0$. We consider compound renewal processes with linear drift and Lévy processes. For both we also formulate and prove the principle of a single big jump for their maxima. The class of compound renewal processes particularly includes Cramér-Lundberg risk process.
On discrete-time arrival processes and related random motions
研究离散时间到达过程中的更新过程，包括瞬态、中间态和常返态，关注有缺陷到达间隔时间的瞬态更新过程，通过蒙特卡罗模拟验证结果并应用于随机游走，研究其空间传播子及渐近性，发现有偏游走的超扩散行为和几何分布停止时间的NESS。
Given a matrix of distribution functions and a quasi-stochastic matrix, i.e. an irreducible nonnegative matrix with maximal eigenvalue one and associated unique positive left and right eigenvectors, the article studies the properties of an associated matrix renewal measure and a related integral equation. Unlike earlier work this is done by a purely probabilistic approach based on a simple harmonic transform. Main results include Markov renewal-type theorems and a Stone-type decomposition under an absolute continuity condition. Three applications are given at the end of the paper.
The article describes the limiting distribution of the extremes of observations that arrive in clusters. We start by studying the tail behaviour of an individual cluster and then we apply the developed theory to determine the limiting distribution of $\max\{X_j: j=0,\ldots, K(t)\}$, where $K(t)$ is the number of i.i.d. observations $(X_j)$ arriving up to the time $t$ according to a general marked renewal cluster process. The results are illustrated in the context of some commonly used Poisson cluster models such as the marked Hawkes process.
Discrete time-multidimensional renewal theory and applications
提出离散时间多维更新框架，通过多指标卷积和形式幂级数代数研究多时间更新链，给出更新方程显式解、系数公式和计算方案，并证明比例增长下的强大数律和中心极限定理，应用于保修评估、交替更新可用性计算等。
We study optimal stopping problems related to the pricing of perpetual American options in an extension of the Black-Merton-Scholes model in which the dividend and volatility rates of the underlying risky asset depend on the running values of its maximum and maximum drawdown. The optimal stopping times of exercise are shown to be the first times at which the price of the underlying asset exits some regions restricted by certain boundaries depending on the running values of the associated maximum and maximum drawdown processes. We obtain closed-form solutions to the equivalent free-boundary pro…
Killed resolvents and measure-valued stopping gains for reflected optimal stopping with max-type rewards
针对非光滑最大值型奖励的反射二维扩散最优停止问题，建立条件测度理论框架，证明停止增益为符号测度，并利用被杀死预解式获得正确势表示。
We extend Goldie's (1991) Implicit Renewal Theorem to enable the analysis of recursions on weighted branching trees. We illustrate the developed method by deriving the power tail asymptotics of the distributions of the solutions R to: R =_D sum_{i=1}^N C_i R_i + Q, R =_D max(max_{i=1}^N C_i R_i, Q), and similar recursions, where (Q, N, C_1,..., C_N) is a nonnegative random vector with N in {0, 1, 2, 3, ..., infinity}, and {R_i}_{i >= 1} are iid copies of R, independent of (Q, N, C_1,..., C_N); =_D denotes the equality in distribution.
This note considers a variation of the full-information secretary problem where the random variables to be observed are independent and identically distributed. Consider $X_1,\dots,X_n$ to be an independent sequence of random variables, let $M_n:=\max\{X_1,\dots,X_n\}$, and the objective is to select the maximum of the sequence. What is the maximum probability of "stopping at the maximum"? That is, what is the stopping time $τ$ adapted to $X_1,...,X_n$ that maximizes $P(X_τ=M_n)$? This problem was examined by Gilbert and Mosteller \cite{GilMost} when in addition the common distribution is cont…
Non-zero-sum optimal stopping game with continuous versus periodic exercise opportunities
该研究提出一种非对称行权机会的非零和最优停时博弈，在带正跳的Lévy过程下证明并构造了以尺度函数表示的纳什均衡，还通过数值示例分析策略行为与行权机会价值。
In the literature on optimal stopping, the problem of maximizing the expected discounted reward over all stopping times has been explicitly solved for some special reward functions (including $(x^+)^ν$, $(e^x-K)^+$, $(K-e^{-x})^+$, $x\in\mathbb{R}$, $ν\in(0,\infty)$ and $K>0$) under general random walks in discrete time and Lévy processes in continuous time (subject to mild integrability conditions). All of such reward functions are continuous, increasing and logconcave while the corresponding optimal stopping times are of threshold type (i.e. the solutions are one-sided). In this paper, we sh…
Distribution-constrained maximum stopping of maximum type
本文研究分布约束下布朗运动最大值的最优停时问题，通过变分不等式刻画最优边界，并利用时间反转表示和粘性解方法证明首达时间的最优性。
In this paper, we develop new optional stopping theorems for scenarios where the stopping rules are defined by bounded continuity regions. Moreover, we establish a wide variety of inequalities on the supremums and infimums of functions of stochastic processes over the whole range of time indexes.
The optimal stopping problem for a Hunt processes on $\R$ is considered via the representation theory of excessive functions. In particular, we focus on infinite horizon (or perpetual) problems with one-sided structure, that is, there exists a point $x^*$ such that the stopping region is of the form $[x^*,+\infty)$. Corresponding results for two-sided problems are also indicated. The main result is a spectral representation of the value function in terms of the Green kernel of the process. Specializing in Lévy processes, we obtain, by applying the Wiener-Hopf factorization, a general represent…
This article treats both discrete time and continuous time stopping problems for general Markov processes on the real line with general linear costs. Using an auxiliary function of maximum representation type, conditions are given to guarantee the optimal stopping time to be of threshold type. The optimal threshold is then characterized as the root of that function. For random walks our results condense in the fact that all combinations of concave increasing pay-off functions and convex cost functions lead to a one-sided solution. For Lévy processes an explicit way to obtain the auxiliary func…
Reflected Optimal Stopping with a Max-Type Payoff: Measure-Valued Stopping Gains and Killed Resolvent Representation
研究二维正态反射扩散的无限期最优停止问题，其收益为\(G(x_1,x_2)=x_1\vee \alpha x_2\)。通过制定反射障碍问题等，证明验证定理，推导停止集结构，明确停止收益对象是测度，证明正确势表示为被杀预解式公式。
Previous authors have considered optimal stopping problems driven by the running maximum of a spectrally negative Lévy process $X$, as well as of a one-dimensional diffusion. Many of the aforementioned results are either implicitly or explicitly dependent on Peskir's maximality principle. In this article, we are interested in understanding how some of the main ideas from these previous works can be brought into the setting of problems driven by the maximum of a class of Markov additive processes (more precisely Markov modulated Lévy processes). Similarly to previous works in the Lévy setting, …
Doob-type optional sampling theorems for demimartingales with applications to associated sequences
本文针对半鞅与半下鞅，在停止规则满足特定条件时建立杜布型可选抽样不等式，推导相关极大值不等式、部分和界及广义沃尔德型不等式，为关联序列研究提供了理论支撑。
Time and Killed Resolvents in Reflected Optimal Stopping with a Max Payoff
研究正象限内带最大收益的反射扩散最优停时问题，证明非光滑收益在扭结集上产生奇异停时增益测度，并给出正确值表示需使用首次进入停时集时杀死的预解式。
We consider the optimal stopping problem consisting in, given a strong Markov process, a reward function and a discount rate, finding the stopping time such that the expected reward at the stopping time is maximum. The approach we follow, has two main components: the Dynkin's characterization of the value function as the smallest excessive function dominating the reward; and the Riesz representation of excessive functions in terms of the Green kernel, the main reference being Salminen 85. In the context of one-dimensional diffusions we give a complete characterization of the solution under som…
We consider a zero-sum continuous time stopping game in which the pay-off is revealed in the maximum of the two stopping times instead of the minimum, which is the case in Dynkin games.
We consider renewal processes where events, which can for instance be the zero crossings of a stochastic process, occur at random epochs of time. The intervals of time between events, $τ_{1},τ_{2},...$, are independent and identically distributed (i.i.d.) random variables with a common density $ρ(τ)$. Fixing the total observation time to $t$ induces a global constraint on the sum of these random intervals, which accordingly become interdependent. Here we focus on the largest interval among such a sequence on the fixed time interval $(0,t)$. Depending on how the last interval is treated, we con…
We study the statistics of infima, stopping times and passage probabilities of entropy production in nonequilibrium steady states, and show that they are universal. We consider two examples of stopping times: first-passage times of entropy production and waiting times of stochastic processes, which are the times when a system reaches for the first time a given state. Our main results are: (i) the distribution of the global infimum of entropy production is exponential with mean equal to minus Boltzmann's constant; (ii) we find the exact expressions for the passage probabilities of entropy produ…
A Dynkin game is a zero-sum, stochastic stopping game between two players where either player can stop the game at any time for an observable payoff. Typically the payoff process of the max-player is assumed to be smaller than the payoff process of the min-player, while the payoff process for simultaneous stopping is in between the two. In this paper, we study general Dynkin games whose payoff processes are in arbitrary positions. In both discrete and continuous time settings, we provide necessary and sufficient conditions for the existence of pure strategy Nash equilibria and epsilon-optimal …
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