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We consider a class of time-inhomogeneous optimal stopping problems and we provide sufficient conditions on the data of the problem that guarantee monotonicity of the optimal stopping boundary. In our setting, time-inhomogeneity stems not only from the reward function but, in particular, from the time dependence of the drift coefficient of the one-dimensional stochastic differential equation (SDE) which drives the stopping problem. In order to obtain our results, we mostly employ probabilistic arguments: we use a comparison principle between solutions of the SDE computed at different starting times, and martingale methods of optimal stopping theory. We also show a variant of the main theorem, which weakens one of the assumptions and additionally relies on the renowned connection between optimal stopping and free-boundary problems. [ABSTRACT FROM AUTHOR]

The approach of Li and Zhou (2014) is adopted to find the Laplace transform of occupation time over interval (0, a) and joint occupation times over semi-infinite intervals (−∞, a) and (b, ∞) for a time-homogeneous diffusion process up to an independent exponential time eq for 0 < a < b. The results are expressed in terms of solutions to the differential equations associated with the diffusion generator. Applying these results, we obtain explicit expressions on the Laplace transform of occupation time and joint occupation time for Brownian motion with drift. [ABSTRACT FROM AUTHOR]

In this article, we propose a computer-oriented method of construction of optimal group sequential hypothesis tests with variable group sizes. In particular, for independent and identically distributed observations, we obtain the form of optimal group sequential tests which turn to be a particular case of sequentially planned probability ratio tests (SPPRTs, see Schmitz 1993). Formulas are given for computing the numerical characteristics of general SPPRTs, like error probabilities, average sampling cost, etc. A numerical method of designing the optimal tests and evaluation of the performance characteristics is proposed, and computer algorithms of its implementation are developed. For a particular case of sampling from a Bernoulli population, the proposed method is implemented in R programming language, the code is available in a public GitHub repository. The proposed method is compared numerically with other known sampling plans. [ABSTRACT FROM AUTHOR]

We study a mean-field zero-sum Dynkin game (MF-ZSDG) with time-inconsistent performance functionals adapted to the Brownian filtration. Despite the time-inconsistency of the MF-ZSDG, we show that it admits a value and that the pair of first times the value process hits the upper and lower obstacles, respectively, is a saddle point for the game. We solve the problem by approximating the associated lower and upper value processes with a sequence of value processes of interacting time-consistent zero-sum Dynkin games for which the saddle point of each of the value processes is the pair of first times each of those value processes hits the associated upper and lower obstacles, respectively. Under mild assumptions, we show that this sequence of saddle points converges in probability to the pair of first hitting times of the value process of the upper and lower obstacles, respectively, and that the limit is a saddle point for the time-inconsistent MF-ZSDG. [ABSTRACT FROM AUTHOR]

We consider the propagation of a stochastic SIR-type epidemic in two connected populations: a relatively small local population of interest which is surrounded by a much larger external population. External infectives can temporarily enter the small population and contribute to the spread of the infection inside this population. The rules for entry of infectives into the small population as well as their length of stay are modeled by a general Markov queueing system. Our main objective is to determine the distribution of the total number of infections within both populations. To do this, the approach we propose consists of deriving a family of martingales for the joint epidemic processes and applying classical stopping time or convergence theorems. The study then focuses on several particular cases where the external infection is described by a linear branching process and the entry of external infectives obeys certain specific rules. Some of the results obtained are illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

This article studies deep neural network expression rates for optimal stopping problems of discrete-time Markov processes on high-dimensional state spaces. A general framework is established in which the value function and continuation value of an optimal stopping problem can be approximated with error at most ε by a deep ReLU neural network of size at most κ d q ε − r . The constants κ , q , r ≥ 0 do not depend on the dimension d of the state space or the approximation accuracy ε . This proves that deep neural networks do not suffer from the curse of dimensionality when employed to approximate solutions of optimal stopping problems. The framework covers for example exponential Lévy models, discrete diffusion processes and their running minima and maxima. These results mathematically justify the use of deep neural networks for numerically solving optimal stopping problems and pricing American options in high dimensions. [ABSTRACT FROM AUTHOR]

We study the infinite horizon discrete time N-player nonzero-sum Dynkin game ( N ≥ 2 ) with stopping times as strategies (or pure strategies). We prove existence of an ε -Nash equilibrium point for the game by presenting a constructive algorithm. One of the main features is that the payoffs of the players depend on the set of players that stop at the termination stage which is the minimal stage in which at least one player stops. The existence result is extended to the case of a nonzero-sum game with finite horizon. Finally, the algorithm is illustrated by two explicit examples in the specific case of finite horizon. [ABSTRACT FROM AUTHOR]

In this paper, we study the optimal timing to convert the risk of business for an insurance company in order to improve its solvency. The cash flow of company evolves according to a jump-diffusion process. Business conversion option offers the company an opportunity to transfer the jump risk business out. In exchange for this option, the company needs to pay both fixed and proportional transaction costs. The proportional cost can also be seen as the profit loading of the jump risk business. We formulated this problem as an optimal stopping problem. By solving this stopping problem, we find that the optimal timing of business conversion mainly depends on the profit loading of the jump risk business. A larger profit loading would make the conversion option valueless. The fixed cost, however, only delays the optimal timing of business conversion. In the end, numerical results are provided to illustrate the impacts of transaction costs and environmental parameters to the optimal strategies. [ABSTRACT FROM AUTHOR]

This paper is concerned with the valuation of single and double barrier knock-out call options in a Markovian regime switching model with specific rebates. The integral formulas of the rebates are derived via matrix Wiener-Hopf factorizations and Fourier transform techniques, also, the integral representations of the option prices are constructed. Moreover, the first-passage time density functions in two-state regime model are derived. As applications, several numerical algorithms and numerical examples are presented. [ABSTRACT FROM AUTHOR]

The present article is an essay about mathematical intuition and Artificial intelligence (A.I.), followed by a guided excursion to Robbins' Problem of Optimal Stopping. The latter is still open. The first part of the essay (Sect. 1 to Sect. 4), is written in a narrative style with well-known examples and an easily accessible terminology. It is easy to read. Its goal is to motivate mathematicians to think about certain aspects of A.I., and to prepare people in A.I. for the special features of Robbins' problem. The major objective is presented in the second part (Sect. 5 to Sect. 13). It is to guide readers through the probabilistic intuition behind Robbins' problem and to show why A.I., and in particular Deep Learning, may contribute an essential part in its solution. This article is an essay, because it contains no new mathematical results, and no implementation of deep learning either. However it presents a clear mission statement which seems important to us. We substantiate it by proposing two concrete challenges to view and solve the main part of the problem. [ABSTRACT FROM AUTHOR]

The subject of decisions under uncertainty about future events, if lacking sufficient theory or data to make confident probability assessments, poses a challenge for any quantitative analysis. This article suggests one way to first look at this subject. We give elementary examples within a framework for studying decisions under uncertainty where probabilities are only roughly known. The framework, in gambling terms, is that the size of a bet is proportional to the gambler’s perceived advantage based on their perceived probability, and their accuracy in estimating true probabilities is measured by mean-squared-error. Within this framework one can study the cost of estimation errors, and seek to formalize the “obvious” notion that in competitive interactions between agents whose actions depend on their perceived probabilities, those who are more accurate at estimating probabilities will generally be more successful than those who are less accurate. This article is an extended version of the Brouwer Medal talk at the 2021 Nederlands Mathematisch Congres.

We look at a sequence of Bernoulli random variables where the success rates change from θ1 to θ2. We will assume that both the success rates before and after the change are unknown but increasing. We assume that this change does not happen abruptly but gradually over a period of time η where η is known. We calculate the probability that the change has started and completed. We also look at optimal stopping rules assuming that there is a cost for a false alarm and a cost per time unit to stop late. [ABSTRACT FROM AUTHOR]

We investigate an optimal reinsurance problem when the loss process exhibits jump clustering features and the insurance company has restricted information about the loss process. We maximise expected exponential utility of terminal wealth and show that an optimal strategy exists. By exploiting both the Kushner–Stratonovich and Zakai approaches, we provide the equation governing the dynamics of the (infinite-dimensional) filter and characterise the solution of the stochastic optimisation problem in terms of a BSDE, for which we prove existence and uniqueness of a solution. After discussing the optimal strategy for a general reinsurance premium, we provide more explicit results in some relevant cases. [ABSTRACT FROM AUTHOR]

A functional calculus for an order complete vector lattice E was developed by Grobler (Indag Math (NS) 25(2):275–295, 2014) using the Daniell integral. We show that if one represents the universal completion of E as C ∞ (K) , where K is an extremally disconnected compact Hausdorff topological space, then the Daniell functional calculus for continuous functions is exactly the pointwise composition of functions in C ∞ (K) . This representation allows an easy deduction of the various properties of the functional calculus. Afterwards, we study discrete stopping times and stopped processes in C ∞ (K) . We obtain a representation that is analogous to what is expected in probability theory. [ABSTRACT FROM AUTHOR]

In this article we propose a procedure for selecting a random size subset that contains all experimental treatments that are better than the standard. The comparison is made with regard to two binary endpoints. An experimental treatment is considered to be better than the standard if its two endpoints have successful rates higher than standard rates. We assume responses from different treatments are independent, however, responses from each treatment may be associated. We use the odds ratio as the association parameter and describe the bivariate binomial distribution involved in the computation of the probability of a correct selection for our procedure. We derive the design parameters for three different cases: independent endpoints, dependent endpoints with known association and dependent endpoints with unknown association, and provide tables including the minimum sample size needed for the probability requirements to be satisfied. [ABSTRACT FROM AUTHOR]

This paper discusses a generalized SIR epidemic model that incorporates infectives with or without symptoms, allows arbitrary distributions for infectious periods and assumes infection rates depending on the current size of susceptibles. Our interest lies in the joint distribution of the state of the population and the severity of the disease at the end of infection. The approach is based on a formulation of the epidemic as a so-called collective model. First, a set of martingales is constructed which provides the distribution of this final epidemic outcome. Then, its corresponding distributional transform is expressed in terms of a family of pseudo-polynomials of Abel-Gontcharoff type. Some of the results obtained are illustrated numerically. [ABSTRACT FROM AUTHOR]

We study the first exit time of a fractional Brownian motion with a drift from a parabolic domain. Actually, we explore three different regimes. In the first regime, the role of drift is negligible. In the second regime, the role of drift is dominating. The behavior of exit probability is the same as that of the crossing probability of a certain moving non-random boundary. In particular, the most interesting, intermediate regime, where all factors come into play, has been solved in this paper. Finally, numerical simulations are conducted, providing an approximate range for the asymptotic estimates to illustrate the practical implications and potential applications of our main results. [ABSTRACT FROM AUTHOR]

In this paper, the asmptotics is considered for the distribution tail of a randomly stopped sum ${S_{\nu }}={X_{1}}+\cdots +{X_{\nu }}$ of independent identically distributed consistently varying random variables with zero mean, where ν is a counting random variable independent of $\{{X_{1}},{X_{2}},\dots \}$. The conditions are provided for the relation $\mathbb{P}({S_{\nu }}\gt x)\sim \mathbb{E}\nu \hspace{0.1667em}\mathbb{P}({X_{1}}\gt x)$ to hold, as $x\to \infty $, involving the finiteness of $\mathbb{E}|{X_{1}}|$. The result improves that of Olvera-Cravioto [14], where the finiteness of a moment $\mathbb{E}|{X_{1}}{|^{r}}$ for some $r\gt 1$ was assumed.

In this paper we consider the following optimal stopping problem V A ω (s) = sup τ ∈ T E s [ e - ∫ 0 τ ω (S w) d w g (S τ) ] , where the process S t is a jump-diffusion process, T is a family of stopping times while g and ω are fixed payoff function and discount function, respectively. In a financial market context, if g (s) = (K - s) + or g (s) = (s - K) + and E is the expectation taken with respect to a martingale measure, V A ω (s) describes the price of a perpetual American option with a discount rate depending on the value of the asset process S t . If ω is a constant, the above problem produces the standard case of pricing perpetual American options. In the first part of this paper we find sufficient conditions for the convexity of the value function V A ω (s) . This allows us to determine the stopping region as a certain interval and hence we are able to identify the form of V A ω (s) . We also present a put-call symmetry for American options with asset-dependent discounting. In the case when S t is a spectrally negative geometric Lévy process we give exact expressions using the so-called omega scale functions introduced in [30]. We show that the analysed value function satisfies the HJB equation and we give sufficient conditions for the smooth fit property as well. Finally, we present a few examples for which we obtain the analytical form of the value function V A ω (s) . [ABSTRACT FROM AUTHOR]

In this paper we provide conditions under which the hitting-time problem for Brownian motion is equivalent to solving a heat equation with moving boundary and distributional initial conditions. Motivated by the hitting-time problem, we study the heat equation with absorbing moving boundaries. Using Fourier analysis we develop a procedure to solve this problem for a family of curves that includes the square root, quadratic, and cubic boundaries. As an application of our results, and using Sturm-Liouvile theory, we compute the density of the hitting time of a Brownian motion to a family of quadratic boundaries. [ABSTRACT FROM AUTHOR]

Multiple types of microorganisms impact people with immunological illnesses simultaneously. In the presence of antibiotic resistance and random fluctuations such as rapid changes in the environment, the synchronization growth density varies stochastically from one species to another. Therefore, in this work, we study the multivariate probability distribution of the density of all species at the same time. Using the principle of independence of growth density for each species, we provide the statistical properties of this distribution. Furthermore, we estimate the parameters of the intrinsic rates of antibiotic action and bacterial cell growth for each variety, along with the strength of the intra-competitive pressures influencing this distribution. [ABSTRACT FROM AUTHOR]

Insurance loss distributions are characterized by a high frequency of small claim amounts and a lower, but not insignificant, occurrence of large claim amounts. Composite models, which link two probability distributions, one for the 'body' and the other for the 'tail' of the loss distribution, have emerged in the actuarial literature to take this specificity into account. The parameters of these models summarize the distribution of the losses. One of them corresponds to the breaking point between small and large claim amounts. The composite models are usually fitted using maximum likelihood estimation. A Bayesian approach is considered in this work. Sequential Monte Carlo samplers are used to sample from the posterior distribution and compute the posterior model evidence to both fit and compare the competing models. The method is validated via a simulation study and illustrated on an insurance loss dataset. [ABSTRACT FROM AUTHOR]

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 extend results available in the traditional literature on singular control (Haussmann and Suo in SIAM J Control Optim 33(3):916–936, 1995; SIAM J Control Optim 33(3):937–959, 1995) by relaxing some key assumptions and including the discretionary stopping feature. We also connect with more general versions of the DPP (e.g., Bouchard and Touzi in SIAM J Control Optim 49(3):948–962, 2011) by showing in detail how our class of problems meets the abstract requirements therein. [ABSTRACT FROM AUTHOR]

In this paper, we establish small time large deviation principles for the quasilinear parabolic stochastic partial differential equations with multiplicative noise, which are neither monotone nor locally monotone. [ABSTRACT FROM AUTHOR]

We consider the problem of bounded risk point estimation for the linear combination of the form θ = e α + d β , where α and β are the location and scale parameters of a exponential distribution and e ≥ 0 and d > 0 are constant. We aim to estimate θ under the modified squared error loss function using the constraint that the risk per unit cost is bounded above with fixed preassigned, ω. The two-stage sequential sampling is proposed for estimating θ. The performances of the proposed methodologies are investigated with the help of simulations. Finally, using an actual dataset, the procedure is clearly illustrated. [ABSTRACT FROM AUTHOR]

In this article, we deal with the problem of sequential testing of multiple hypotheses. The main goal is minimizing the expected sample size (ESS) under restrictions on the error probabilities. We take, as a criterion of minimization, a weighted sum of the ESSs evaluated at some points of interest in the parameter space, aiming at its minimization under restrictions on the error probabilities. We use a variant of the method of Lagrange multipliers based on the minimization of an auxiliary objective function (called Lagrangian) combining the objective function with the restrictions, taken with some constants called multipliers. Subsequently, the multipliers are used to make the solution comply with the restrictions. We develop a computer-oriented method of minimization of the Lagrangian function that provides, depending on the specific choice of the parameter points, optimal tests in different concrete settings, as in Bayesian, Kiefer-Weiss, and other settings. To exemplify the proposed methods for the particular case of sampling from a Bernoulli population, we develop a set of computer algorithms for designing sequential tests that minimize the Lagrangian function and for the numerical evaluation of test characteristics like the error probabilities and the ESS. We implement the algorithms in the R programming language. The program code is available in a public GitHub repository. For the Bernoulli model, we made a series of computer evaluations related to the optimality of sequential multi-hypothesis tests, in a particular case of three hypotheses. A numerical comparison with the matrix sequential probability ratio test is carried out. A method of solution of the multi-hypothesis Kiefer-Weiss problem is proposed and is applied for a particular case of three hypotheses in the Bernoulli model. [ABSTRACT FROM AUTHOR]

The problem of the sequential detection of a change in the drift of a one-dimensional Brownian motion is considered under the assumptions that the detection must eventually occur within a finite horizon and the detection delay is exponentially penalized. Our results extend those obtained by Beibel for the infinite horizon sequential detection with exponential penalty (Ann Stat 28:1696–1701, 2000) and by Gapeev and Peskir for the finite horizon sequential detection with linear penalty (Stoch Process Appl 116:1770–1791, 2006). [ABSTRACT FROM AUTHOR]

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 surely optimal Doob-martingale, that is, the martingale part of the Snell envelope, is in a certain sense robust under a particular random perturbation. This new insight leads to a novel randomized dual martingale minimization algorithm that doesn't require nested simulation. As a main feature, in a possibly large family of optimal martingales the algorithm may efficiently select a martingale that is as close as possible to the Doob martingale. As a result, one obtains the dual upper bound for the optimal stopping problem with low variance. [ABSTRACT FROM AUTHOR]

Partial hedging of American options is an interesting minimax problem and in this paper we establish its dual problem that concerns only maximization. The case of a continuous price process is considered under a general incomplete market. Our construction of a duality requires a careful preparation in order to define the dual domain with a compactness property. A key step is an extension of linear functionals preserving norm and positivity. [ABSTRACT FROM AUTHOR]

The objective of this paper is to study a class of zero-sum optimal stopping games of diffusions under a so-called Poisson constraint: the players are allowed to stop only at the arrival times of their respective Poissonian signal processes. These processes can have different intensities, which makes the game setting asymmetric. We give a weak and easily verifiable set of sufficient condition under which we derive a semi-explicit solution to the game in terms of the minimal r-excessive functions of the diffusion. We also study limiting properties of the solutions with respect to the signal intensities and illustrate our main findings with explicit examples. [ABSTRACT FROM AUTHOR]

In this paper, we established a small time large deviation principles for scalar stochastic conservation laws driven by multiplicative noise. The doubling variables method plays a key role. [ABSTRACT FROM AUTHOR]

In this article, we deal with problems of testing hypotheses in the framework of sequential statistical analysis. The main concern is the optimal design and performance evaluation of sampling plans in Kiefer-Weiss problems. The main goal of the Kiefer-Weiss problem is designing hypothesis tests that minimize the maximum average sample number, over all parameter values, as opposed to both the sequential probability tests (SPRTs) minimizing the average sample number only at two hypothesis points and the classical fixed-sample-size test. For observations that follow a distribution from an exponential family of the continuous type, we provide algorithms for optimal design in the modified Kiefer-Weiss problem and obtain formulas for evaluating the performance of sequential tests by calculating the operating characteristic function, the average sample number, and some related characteristics. These formulas cover, as a particular case, the SPRTs and their truncated versions, as well as optimal finite-horizon sequential tests. In the setting of the original Kiefer-Weiss problem we apply the method of our recent work (Sequential Analysis 2022, 41(2), 198–219) for numerical construction of the optimal tests. For the particular case of sampling from a normal distribution with a known variance, we make numerical comparisons of the Kiefer-Weiss solution with the SPRT and the fixed-sample-size test provided that the three tests have the same levels of the error probabilities. All of the algorithms are implemented in the form of computer code written in the R programming language and are available at the GitHub public repository (). Guidelines on the adaptation of the program code to other exponential family distributions are provided. [ABSTRACT FROM AUTHOR]

This article addresses the sequential detection of transient changes by using the finite moving average (FMA) test. It is assumed that a change occurs at an unknown (but nonrandom) change point and the duration of postchange period is finite and known. We relax the assumption that the profile of a transient change is chosen so that the log-likelihood ratios of the observations are associated random variables (r.v.s) in the prechange mode. Hence, the profile of the transient change is arbitrary and it is not necessarily of constant sign for a distribution with monotone likelihood ratio. A new upper bound for the worst-case probability of false alarm is proposed. It is shown that the optimization of the window-limited cumulative sum (CUSUM) test again leads to the FMA test. Three particular transient changes are considered: in the Gaussian mean, in the Gaussian variance, and in the parameter of exponential distribution. In the first case, a comparison between the bounds for the FMA test operating characteristics and the exact operating characteristics calculated by numerical integration is used to estimate the sharpness of the bounds. In the second and third cases, special attention is paid to the calculation of the FMA distribution in the case of arbitrary profile. The method of convolution is used to solve the problem. [ABSTRACT FROM AUTHOR]

In this paper we study a discrete-time switching control problem when the dynamic of the system evolves as a random walk. The payoff function is a discounted-type total cost whose discount factor may depend on the state and/or the action variables. This flexibility includes cases when the controller can stop the dynamics of the system whether is optimal to do it. To provide optimality results, we use the well-known dynamic programming method that leads to the study of certain functional equations. Under suitable conditions, we prove the existence of solutions to these equations, and we also show that the optimal value of the control problem becomes a minimal solution of these equations. Finally, we illustrate our results through an example about the control of epidemic processes. [ABSTRACT FROM AUTHOR]

In this paper, we study the stochastic zero-sum differential game in finite horizon in a general case. We first prove that the BSDE associated with a specific generator (the Hamiltonian function for the game) has a unique solution. Then we characterize the value function as that solution to prove the existence of a saddle point for the game. Finally, in the Markovian framework, we show that the value function is the unique viscosity solution for the related partial differential equation. [ABSTRACT FROM AUTHOR]