Search

Our paper proposes an effective way to make decisions based on data for uncertain situations. In simple terms, a data-driven decision is just a choice we make by looking at the available data. We express this choice as the best one according to a model we create from the data. The quality of this decision is judged by how well it performs in situations not seen during training. We also consider how often it disappoints in those situations. The challenge is that we do not know the exact probability of generating the data. An ideal data-driven decision should work well for any possible probability. However, such ideal decisions are usually not possible. Therefore, we look for decisions that work well on unseen data, considering the chances of disappointment. We prove that such effective decisions exist under certain conditions, allowing for practical applications. This approach holds regardless of whether the original problem is simple or complex, and it works even when the data are not uniformly collected. Our study also uncovers how the characteristics of the data-generating process influence the optimal decision-making model. We propose a statistically optimal approach to construct data-driven decisions for stochastic optimization problems. Fundamentally, a data-driven decision is simply a function that maps the available training data to a feasible action. It can always be expressed as the minimizer of a surrogate optimization model constructed from the data. The quality of a data-driven decision is measured by its out-of-sample risk. An additional quality measure is its out-of-sample disappointment, which we define as the probability that the out-of-sample risk exceeds the optimal value of the surrogate optimization model. The crux of data-driven optimization is that the data-generating probability measure is unknown. An ideal data-driven decision should therefore minimize the out-of-sample risk simultaneously with respect to every conceivable probability measure (and thus in particular with respect to the unknown true measure). Unfortunately, such ideal data-driven decisions are generally unavailable. This prompts us to seek data-driven decisions that minimize the in-sample risk subject to an upper bound on the out-of-sample disappointment—again simultaneously with respect to every conceivable probability measure. We prove that such Pareto dominant data-driven decisions exist under conditions that allow for interesting applications: The unknown data-generating probability measure must belong to a parametric ambiguity set, and the corresponding parameters must admit a sufficient statistic that satisfies a large deviation principle. If these conditions hold, we can further prove that the surrogate optimization model generating the optimal data-driven decision must be a distributionally robust optimization problem constructed from the sufficient statistic and the rate function of its large deviation principle. This shows that the optimal method for mapping data to decisions is, in a rigorous statistical sense, to solve a distributionally robust optimization model. Maybe surprisingly, this result holds irrespective of whether the original stochastic optimization problem is convex or not and holds even when the training data are not independent and identically distributed. As a byproduct, our analysis reveals how the structural properties of the data-generating stochastic process impact the shape of the ambiguity set underlying the optimal distributionally robust optimization model. Funding: This research was supported by the Swiss National Science Foundation under the NCCR Automation [Grant Agreement 51NF40_180545]. Supplemental Material: The online appendices are available at https://doi.org/10.1287/opre.2021.0609. [ABSTRACT FROM AUTHOR]

The modeling and identification of time series data with a long memory are important in various fields. The streamflow discharge is one such example that can be reasonably described as an aggregated stochastic process of randomized affine processes where the probability measure, we call it reversion measure, for the randomization is not directly observable. Accurate identification of the reversion measure is critical because of its omnipresence in the aggregated stochastic process. However, the modeling accuracy is commonly limited by the available real‐world data. We resolve this issue by proposing the novel upper and lower bounds of a statistic of interest subject to ambiguity of the reversion measure. Here, we use the Tsallis value‐at‐risk (TsVaR) as a convex risk functional to generalize the widely used entropic value‐at‐risk (EVaR) as a sharp statistical indicator. We demonstrate that the EVaR cannot be used for evaluating key statistics, such as mean and variance, of the streamflow discharge due to the blowup of some exponential integrand. We theoretically show that the TsVaR can avoid this issue because it requires only the existence of some polynomial moment, not exponential moment. As a demonstration, we apply the semi‐implicit gradient descent method to calculate the TsVaR and corresponding Radon–Nikodym derivative for time series data of actual streamflow discharges in mountainous river environments. [ABSTRACT FROM AUTHOR]

A p -adic Brownian motion is a continuous time stochastic process in a p -adic state space that has a Vladimirov operator as its infinitesimal generator. The current work shows that any such process is the scaling limit of a discrete time random walk on a discrete group. Earlier work required the exponent of the Vladimirov operator to be in (1 , ∞) , and the convergence was the weak convergence of probability measures on the Skorohod space of paths on a compact time interval. The current approach simplifies the earlier approach, allows for any positive exponent, eliminates the restriction to compact time intervals, and establishes some moment estimates for the discrete time processes that are of independent interest. [ABSTRACT FROM AUTHOR]

We present a novel method for analysing socio-spatial segregation in cities by considering constraints imposed by transportation networks. Using a multilayered network approach, we model the interaction probabilities of socio-economic groups with random walks and Lévy flights. This method allows for evaluation of new transport infrastructure's impact on segregation while quantifying each network's contribution to interaction opportunities. The proposed random walk segregation index measures the probability of individuals encountering diverse social groups based on their available means of transit via random walks. The index incorporates temporal constraints in urban mobility with a parameter, α ∈ [ 0 , 1) , of the probability of the random walk continuing at each time step. By applying this to a toy model and conducting a sensitivity analysis, we explore how the index changes dependent on this temporal constraint. When the parameter equals zero, the measure simplifies to an isolation index. When the parameter approaches one it represents the city's overall socio-economic distribution by mirroring the steady-state of the random walk process. Using Cuenca, Ecuador as a case study, we illustrate the method's applicability in transportation planning as a valuable tool for policymakers, addressing spatial distribution of socio-economic groups and the connectivity of existing transport networks, thus promoting equitable interactions throughout the city. [ABSTRACT FROM AUTHOR]

Given an -dimensional stochastic process driven by -Brownian motions and Poisson random measures, we search for a probability measure, with minimal relative entropy to, such that the -expectations of some terminal and running costs are constrained. We prove existence and uniqueness of the optimal probability measure, derive the explicit form of the measure change, and characterize the optimal drift and compensator adjustments under the optimal measure. We provide an analytical solution for Value-at-Risk (quantile) constraints, discuss how to perturb a Brownian motion to have arbitrary variance, and show that pinned measures arise as a limiting case of optimal measures. The results are illustrated in a risk management setting—including an algorithm to simulate under the optimal measure—and explore an example where an agent seeks to answer the question what dynamics are induced by a perturbation of the Value-at-Risk and the average time spent below a barrier on the reference process? [ABSTRACT FROM AUTHOR]

Inspired by issues of formal kinetics in materials science, we consider a class of processes with density with respect to an inhomogeneous finite Poisson point process, which may be regarded as a generalization of the classical Strauss hard-core process. We prove expressions for the intensity measure and the void probabilities, together with upper and lower bounds. A discussion on some special cases of interest, links with literature and a comparison between Matérn I and Strauss hard-core process are also provided. We apply such a special class of point processes in modeling patterns of non-overlapping grains and in the study of the mean volume density of particular birth-and-growth processes. [ABSTRACT FROM AUTHOR]

• The epistemic uncertainty in the process of space object tracking is properly modeled by OPMs. • A refined PAR+ algorithm is developed to handle different levels of prior knowledge in IOD. • A rigorous parameter estimation method of Gaussian possibility functions is proposed to initiate the proposed possibilistic filter. • Improved accuracy of orbital state estimation can be achieved by the fusion of TLEs with real radar measurements. Bayesian filtering is a popular class of estimation algorithms for addressing the space object tracking problem. Bayesian filters assume a random physical system with known statistics of various uncertainty sources. The major challenge is that the exact knowledge of some random process may not be available for analysis, preventing us from performing a probabilistic characterization of the epistemic uncertainty components. In this paper, we explore the use of the Outer Probability Measures (OPMs) to achieve a faithful uncertainty representation derived from all available yet imperfect information in the process of space object tracking. Leveraging the concepts of OPMs, a refined Possibilistic Admissible Region approach is proposed, in which the initial orbital state is modeled using a novel parameter estimation method. The OPM filter is employed to integrate different types of data sources in the presence of assumed ignorance. The efficacy of the developed method is validated by several space object tracking scenarios using real radar measurements and two-line elements data. [ABSTRACT FROM AUTHOR]

We study a class of kinetic-type differential equations ∂ ϕ t / ∂ t + ϕ t = Q ̂ ϕ t , where Q ̂ is an inhomogeneous smoothing transform and, for every t ≥ 0 , ϕ t is the Fourier–Stieltjes transform of a probability measure. We show that under mild assumptions on Q ̂ the above differential equation possesses a unique solution and represent this solution as the characteristic function of a certain stochastic process associated with the continuous time branching random walk pertaining to Q ̂. Establishing limit theorems for this process allows us to describe asymptotic properties of the solution, as t → ∞. [ABSTRACT FROM AUTHOR]

We establish a one-to-one correspondence between (i) exchangeable sequences of random variables whose finite-dimensional distributions are minimum (or maximum) infinitely divisible and (ii) non-negative, non-decreasing, infinitely divisible stochastic processes. The exponent measure of an exchangeable minimum infinitely divisible sequence is shown to be the sum of a very simple "drift measure" and a mixture of product probability measures, which uniquely corresponds to the Lévy measure of a non-negative and non-decreasing infinitely divisible process. The latter is shown to be supported on non-negative and non-decreasing functions. In probabilistic terms, the aforementioned infinitely divisible process is equal to the conditional cumulative hazard process associated with the exchangeable sequence of random variables with minimum (or maximum) infinitely divisible marginals. Our results provide an analytic umbrella which embeds the de Finetti subfamilies of many interesting classes of multivariate distributions, such as exogenous shock models, exponential and geometric laws with lack-of-memory property, min-stable multivariate exponential and extreme-value distributions, as well as reciprocal Archimedean copulas with completely monotone generator and Archimedean copulas with log-completely monotone generator. [ABSTRACT FROM AUTHOR]

The dynamic Schrödinger bridge problem seeks a stochastic process that defines a transport between two target probability measures, while optimally satisfying the criteria of being closest, in terms of Kullback-Leibler divergence, to a reference process. We propose a novel sampling-based iterative algorithm, the iterated diffusion bridge mixture (IDBM) procedure, aimed at solving the dynamic Schrödinger bridge problem. The IDBM procedure exhibits the attractive property of realizing a valid transport between the target probability measures at each iteration. We perform an initial theoretical investigation of the IDBM procedure, establishing its convergence properties. The theoretical findings are complemented by numerical experiments illustrating the competitive performance of the IDBM procedure. Recent advancements in generative modeling employ the time-reversal of a diffusion process to define a generative process that approximately transports a simple distribution to the data distribution. As an alternative, we propose utilizing the first iteration of the IDBM procedure as an approximation-free method for realizing this transport. This approach offers greater flexibility in selecting the generative process dynamics and exhibits accelerated training and superior sample quality over larger discretization intervals. In terms of implementation, the necessary modifications are minimally intrusive, being limited to the training loss definition. [ABSTRACT FROM AUTHOR]

Optimization problems with continuous data appear in, e.g., robust machine learning, functional data analysis, and variational inference. Here, the target function is given as an integral over a family of (continuously) indexed target functions--integrated with respect to a probability measure. Such problems can often be solved by stochastic optimization methods: performing optimization steps with respect to the indexed target function with randomly switched indices. In this work, we study a continuous-time variant of the stochas tic gradient descent algorithm for optimization problems with continuous data. This so-called stochastic gradient process consists in a gradient flow minimizing an indexed target function that is coupled with a continuous-time index process determining the index. Index processes are, e.g., reflected diffusions, pure jump processes, or other Lévy processes on compact spaces. Thus, we study multiple sampling patterns for the continuous data space and allow for data simulated or streamed at runtime of the algorithm. We analyze the approximation properties of the stochastic gradient process and study its longtime behavior and ergodicity under constant and decreasing learning rates. We end with illustrating the applicability of the stochastic gradient process in a polynomial regression problem with noisy functional data, as well as in a physics-informed neural network. [ABSTRACT FROM AUTHOR]

We prove the existence of statistical arbitrage opportunities for jump-diffusion models of stock prices when the jump-size distribution is assumed to have finite moments. We show that to obtain statistical arbitrage, the risky asset holding must go to zero in time. Existence of statistical arbitrage is demonstrated via 'buy-and-hold until barrier' and 'short until barrier' strategies with both single and double barrier. In order to exploit statistical arbitrage opportunities, the investor needs to have a good approximation of the physical probability measure and the drift of the stochastic process for a given asset. [ABSTRACT FROM AUTHOR]

Stochastic point process models of event timing are common in many areas, including finance, insurance, and reliability. Monte Carlo simulation is often used to perform computations for these models. The standard sampling algorithm, which is based on a time-change argument, is widely applicable but generates biased simulation estimators. This article develops and analyzes a change of probability measure that can reduce or even eliminate the bias without restricting the scope of the algorithm. A result of independent interest offers new conditions that guarantee the existence of a broad class of point process martingales inducing changes of measure. Numerical results illustrate our approach. [ABSTRACT FROM AUTHOR]

We obtain a large deviations principle for terminating multidimensional compound renewal processes. We also obtain the asymptotics of large deviations for the case where a Gibbs change of the original probability measure takes place. The random processes mentioned in the paper are widely used in polymer pinning models. [ABSTRACT FROM AUTHOR]

Main goal of this paper is to formulate possibly simple and easy to verify criteria on existence of the unique attracting probability measure for stochastic process induced by generalized iterated function systems with probabilities (GIFSPs). To do this, we study the long-time behavior of trajectories of Markov-type operators acting on product of spaces of Borel measures on arbitrary Polish space. Precisely, we get the desired geometric rate of convergence of sequences of measures under the action of such operator to the unique distribution in the Hutchinson–Wasserstein distance. We apply the obtained results to study limiting behavior of random trajectories of GIFSPs as well as stochastic difference equations with multiple delays. [ABSTRACT FROM AUTHOR]

In this paper, we consider the quantitative stability of a class of risk-averse multistage stochastic programs, whose objective functions are defined by multi-period ppth order lower partial moments (LPM) with given targets, and their distributionally robust counterparts. We first derive the upper bounds of feasible solutions as preliminaries. Then, by employing calm modifications, the quantitative stability results are obtained under a special measurable perturbation of stochastic process, which extend the present results under risk-neutral cases to risk-averse ones. Moreover, we recast the risk-averse model by probability measures of stochastic process, and obtain new quantitative stability estimations on the basis of proper probability metrics under the general perturbation of stochastic process. Finally, motivated by the availability of only partial information about probability measures, we further consider the distributionally robust counterpart of our recasting model, and establish the discrepancy of optimal values with respect to the perturbation of ambiguity sets. [ABSTRACT FROM AUTHOR]

Define (X n) on Z / q Z by X n + 1 = 2 X n + b n , where the steps b n are chosen independently at random from - 1 , 0 , + 1 . The mixing time of this random walk is known to be at most 1.02 log 2 q for almost all odd q (Chung, Diaconis, Graham in Ann Probab 15(3):1148–1165, 1987), and at least 1.004 log 2 q (Hildebrand in Proc Am Math Soc 137(4):1479–1487, 2009). We identify a constant c = 1.01136 ⋯ such that the mixing time is (c + o (1)) log 2 q for almost all odd q. In general, the mixing time of the Markov chain X n + 1 = a X n + b n modulo q, where a is a fixed positive integer and the steps b n are i.i.d. with some given distribution in Z , is related to the entropy of a corresponding self-similar Cantor-like measure (such as a Bernoulli convolution). We estimate the mixing time up to a 1 + o (1) factor whenever the entropy exceeds (log a) / 2 . [ABSTRACT FROM AUTHOR]

We study the problem of maximizing terminal utility for an agent facing model uncertainty, in a frictionless discrete‐time market with one safe asset and finitely many risky assets. We show that an optimal investment strategy exists if the utility function, defined either on the positive real line or on the whole real line, is bounded from above. We further find that the boundedness assumption can be dropped, provided that we impose suitable integrability conditions, related to some strengthened form of no‐arbitrage. These results are obtained in an alternative framework for model uncertainty, where all possible dynamics of the stock prices are represented by a collection of stochastic processes on the same filtered probability space, rather than by a family of probability measures. [ABSTRACT FROM AUTHOR]

Uncertainty quantification has become an efficient tool for uncertainty-aware prediction, but its power in yield-aware optimization has not been well explored from either theoretical or application perspectives. Yield optimization is a much more challenging task. On the one side, optimizing the generally nonconvex probability measure of performance metrics is difficult. On the other side, evaluating the probability measure in each optimization iteration requires massive simulation data, especially, when the process variations are non-Gaussian correlated. This article proposes a data-efficient framework for the yield-aware optimization of photonic ICs. This framework optimizes the design performance with a yield guarantee, and it consists of two modules: 1) a modeling module that builds stochastic surrogate models for design objectives and chance constraints with a few simulation samples and 2) a novel yield optimization module that handles probabilistic objectives and chance constraints in an efficient deterministic way. This deterministic treatment avoids repeatedly evaluating probability measures at each iteration, thus it only requires a few simulations in the whole optimization flow. We validate the accuracy and efficiency of the whole framework by a synthetic example and two photonic ICs. Our optimization method can achieve more than $30\times $ reduction of simulation cost and better design performance on the test cases compared with a Bayesian yield optimization approach developed recently. [ABSTRACT FROM AUTHOR]

Random ordinary differential equations (RODEs) are ordinary differential equations (ODEs) that contain a stochastic process in their vector field functions. They have been used for many years in a wide range of applications, but have been a shadow existence to stochastic differential equations (SDEs) despite being able to model a wider and often physically more adequate range of disturbances. In this article, we study the safety verification problem over both finite time horizons and the infinite time horizon for RODEs incorporating Wiener processes. Concretely, we investigate the ${p}$ -safety problem, where we identify the set of initial states from which the probability to satisfy safety specifications is at least ${p}$. Based on identifying a set of sample paths whose probability measure is larger than ${p}$ , we propose a method of reducing stochastic reachability to adversary reachability of ODEs for solving the ${p}$ -safety problem over finite time horizons. This method permits an efficient lifting of reach-set computation methods for perturbed ODEs to RODEs. In this method, the ${p}$ -safety problem over finite time horizons is reduced to the problem of inner-approximating robust backward reachable sets for ODEs with time-varying perturbation inputs. We then extend the method to the ${p}$ -safety problem over the infinite time horizon. Finally, we demonstrate our method on several examples. [ABSTRACT FROM AUTHOR]

In the simplest case, we obtain a general solution to a problem of minimizing an integral of a nondecreasing right continuous stochastic process from zero to some nonnegative random variable τ , under the constraints that for some nonnegative random variable T , τ ∈ [ 0 , T ] almost surely and E τ = α (or E τ ≤ α) for some α. The nondecreasing process and T are allowed to be dependent. In fact a more general setup involving σ finite measures, rather than just probability measures is considered and some consequences for families of stochastic processes are given as special cases. Various applications are provided. [ABSTRACT FROM AUTHOR]

Quantization for probability distributions refers broadly to estimating a given probability measure by a discrete probability measure supported by a finite number of points. We consider general geometric approaches to quantization using stationary processes arising in dynamical systems, followed by a discussion of the special cases of stationary processes: random processes and Diophantine processes. We are interested in how close stationary process can be to giving optimal n-means and nth optimal mean distortion errors. We also consider different ways of measuring the degree of approximation by quantization, and their advantages and disadvantages in these different contexts. [ABSTRACT FROM AUTHOR]

Time-dependent reliability analysis (TRA) has drawn much attention due to its ability in measuring the probability that a system or component keeps safe in the full life cycle. Since it is difficult to efficiently obtain accurate results for TRA problems with expensive simulation demand, many surrogate model-based methods have been proposed to handle this challenge. Moreover, when both random and interval uncertainties are included in these TRA problems simultaneously, the analysis process will be more complicated. In this paper, two methods based on projection outline adaptive Kriging (POK) are proposed to handle TRA and TRA with mixed interval uncertainties (iTRA), respectively. Firstly, POK-TRA method is put forward for the TRA problems with different stochastic processes. Different from current TRA methods, POK-TRA regards the time parameter as a special interval variable, which converts TRA problem into a special hybrid reliability analysis (HRA) problem with one interval variable. Based on the concept of projection outline as well as a correlation condition, an efficient sampling strategy is proposed to refine the Kriging model adaptively. Secondly, POK-TRA is extended to the time-dependent reliability analysis including both random and interval variables (POK-iTRA). By inheriting the processing strategy of time parameter and stochastic processes, the iTRA problem is converted into the HRA problem with multiple interval variables. Finally, four cases are used to show the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

We construct suitable families of basic functions and prove theorems of Baxter type for generalized Gaussian random processes with independent values. These theorems are used to divide families of such processes into classes. The singularity of probability measures corresponding to representatives of different classes is proved. [ABSTRACT FROM AUTHOR]

The set L of infinite-dimensional, symmetric stable tail dependence functions associated with exchangeable max-stable sequences of random variables with unit Fréchet margins is shown to be a simplex. Except for a single element, the extremal boundary of L is in one-to-one correspondence with the set F 1 of distribution functions of non-negative random variables with unit mean. Consequently, each ℓ ∈ 𝔏 is uniquely represented by a pair (b, µ) of a constant b and a probability measure µ on F 1 . A canonical stochastic construction for arbitrary exchangeable max-stable sequences and a stochastic representation for the Pickands dependence measure of finite-dimensional margins of l are immediate corollaries. As by-products, a canonical analytical description and an associated canonical Le Page series representation for non-decreasing stochastic processes that are strongly infinitely divisible with respect to time are obtained. [ABSTRACT FROM AUTHOR]

We study modification properties of stochastic processes under different probability measures in an initially enlarged filtration setup. For this purpose, we consider several pure-jump Lévy processes under two equivalent probability measures and derive the associated martingale compensators with respect to different enlarged filtrations. As our main result, we prove that the obtained martingale processes under different probability measures in our enlarged filtration approach are indistinguishable. In addition, we provide a condition under which the pure-jump result can be carried over to the Brownian motion case. In this context, we show how indistinguishable Brownian motions under different probability measures can be constructed in an enlarged filtration framework. We finally apply our theoretical results to precipitation derivatives pricing under weather forecasts. [ABSTRACT FROM AUTHOR]

In a general jump-diffusion Radon–Nikodym setup with stochastic Girsanov processes, we derive optimal equivalent probability measures. Optimality is measured in terms of minimum relative entropy and also by more general divergence concepts. We further provide an anticipative sufficient stochastic minimum principle and derive optimal equivalent probability measures under various enlarged filtration approaches. [ABSTRACT FROM AUTHOR]

Let {L(t), t ≥ 0} be a Lévy process with representative random variable L(1) defined by the infinitely divisible logarithmic series distribution. We study here the transition probability and Lévy measure of this process. We also define two subordinated processes. The first one, Y(t), is a Negative-Binomial process X(t) directed by Gamma process. The second process, Z(t), is a Logarithmic Lévy process L(t) directed by Poisson process. For them, we prove that the Bernstein functions of the processes L(t) and Y(t) contain the iterated logarithmic function. In addition, the Lévy measure of the subordinated process Z(t) is a shifted Lévy measure of the Negative-Binomial process X(t).We compare the properties of these processes, knowing that the total masses of corresponding Lévy measures are equal. [ABSTRACT FROM AUTHOR]

We study the transition time distribution for a particle moving between two wells of a multidimensional potential in the low-noise limit of overdamped Langevin dynamics. Possible transition paths are restricted to a thin tube surrounding the most probable trajectory. We demonstrate that finding the transition time distribution reduces to a one-dimensional problem. The resulting transition time distribution has a universal and compact form. We suggest that transition barriers can be estimated from a single-temperature experiment if both the life times and the transition times are measured. [ABSTRACT FROM AUTHOR]

We consider a high-dimensional nonlinear computational model of a dynamical system, parameterized by a vector-valued control parameter, in the presence of uncertainties represented by an uncontrolled parameter modeled by a vector-valued random variable, and possibly with stochastic excitation. The objective is to construct a statistical surrogate model where the input is any deterministic value of the control parameter, and the output is a vector-valued observation of the computational model, which is a random vector whose probability measure is updated using a target dataset. To construct this statistical surrogate model, the stochastic response of the computational model must be built, which is a vector-valued time-discretized stochastic process in high dimension, depending on the control parameter. It is assumed that the computational cost of a single evaluation of the deterministic model is high. For the probabilistic updating, we consider a subset of the components of the observation of the computational model, defined as the "identification observation" of the computational model, for which a small target dataset is available. Therefore, the target dataset is associated with partial observability, corresponding to an incomplete data case. Given a prior probability model of the random control and uncontrolled parameters, a training dataset is constructed, consisting of realizations of the random triplet composed of the stochastic response, the random identification observation, and the random control parameter. Since the computational cost of a single evaluation of the deterministic model is assumed to be large, the training dataset is also of small size. The main challenges in this problem are the high dimensionality, partial observability leading to incomplete data in the target dataset for the identification observation of the computational model (which is not sufficient to identify the computational stochastic responses), and the availability of a small training dataset. To address these challenges, we propose a methodology based on statistical methods for constructing necessary reduced representations, direct probabilistic learning under constraints using probabilistic learning on manifolds (PLoM) constrained by the target dataset, and the use of a weak formulation of the Fourier transform of probability measures. Statistical conditioning is also employed to explore the learned dataset. The constructed predictive statistical surrogate model can be implemented in the context of online computation. We apply this approach to a problem of nonlinear stochastic dynamics in high dimensions within the framework of deformable solids mechanics. • Statistical surrogate model devoted to fast online computation for nonlinear dynamical systems. • Small training dataset coming from an uncertain nonlinear computational model. • Partial observability inducing incomplete data for the target dataset of the identification observations. • Kullback divergence minimum for estimating the updated probability measure on the manifold. • Application to the nonlinear stochastic dynamics of a 3D MEMS device. [ABSTRACT FROM AUTHOR]

Summary: This paper examines a codesign problem in industrial networked control systems (NCS) whereby physical systems are controlled over wireless fading channels. The considered wireless channels are assumed to be stochastically dependent on the physical states of moving machineries in the industrial working space. In this paper, the moving machineries are modeled as Markov decision processes whereas the characteristics of the correlated fading channels are modeled as a binary random process whose probability measure depends on both the physical states of moving machineries and the transmission power of communication channels. Under such a state‐dependent fading channel model, sufficient conditions to ensure the stochastic safety of the NCS are first derived. Using the derived safety conditions, a codesign problem is then formulated as a constrained joint optimization problem that seeks for optimal control and transmission power policies which simultaneously minimize an infinite time cost on both communication resource and control effort. This paper shows that such optimal policies can be obtained in a computationally efficient manner using convex programming methods. Simulation results of an autonomous forklift truck and a networked DC motor system are presented to illustrate the advantage and efficacy of the proposed codesign framework for industrial NCS. [ABSTRACT FROM AUTHOR]

This work considers a financial market stochastic model where the uncertainty is driven by a multidimensional Brownian motion. The market price of the risk process makes the transition between real world probability measure and risk neutral probability measure. Traditionally, the martingale representation formulas under the risk neutral probability measure require the market price of risk process to be bounded. However, in several financial models the boundedness assumption of the market price of risk fails; for example a financial market model with the market price of risk following an Ornstein–Uhlenbeck process. This work extends the Clark–Haussmann representation formula to underlying stochastic processes which fail to satisfy the standard requirements. Our methodology is classical, and it uses a sequence of mollifiers. Our result can be applied to hedging and optimal investment in financial markets with unbounded market price of risk. In particular, the mean variance optimization problem can be addressed within our framework. [ABSTRACT FROM AUTHOR]

In this paper, we characterise compatibility of distributions and probability measures on a measurable space. For a set of indices J , we say that the tuples of probability measures (Q i) i ∈ J and distributions (F i) i ∈ J are compatible if there exists a random variable having distribution F i under Q i for each i ∈ J . We first establish an equivalent condition using conditional expectations for general (possibly uncountable) J . For a finite n , it turns out that compatibility of (Q 1 , ... , Q n) and (F 1 , ... , F n) depends on the heterogeneity among Q 1 , ... , Q n compared with that among F 1 , ... , F n . We show that under an assumption that the measurable space is rich enough, (Q 1 , ... , Q n) and (F 1 , ... , F n) are compatible if and only if (Q 1 , ... , Q n) dominates (F 1 , ... , F n) in a notion of heterogeneity order, defined via the multivariate convex order between the Radon–Nikodým derivatives of (Q 1 , ... , Q n) and (F 1 , ... , F n) with respect to some reference measures. We then proceed to generalise our results to stochastic processes, and conclude the paper with an application to portfolio selection problems under multiple constraints. [ABSTRACT FROM AUTHOR]

Highlights • The concept of inertia in physics is introduced into MSSs reliability analysis. • A critical threshold is set for each deteriorating output performance. • Both constant and random thresholds are considered. • Analytical deductions are presented to compare the availability variations. Abstract For the repairable multi-state component (MSC), we introduce a continuous-time Markov process to analyze the availability of the element in this paper. When output performance of the MSC is lower than demand but higher than zero, the component will not fail to work immediately under some circumstances. Based on the concept of inertia in physics, the inertial effect of deteriorating output performances is considered. The influence of inertial effect makes the component operate longer than ever before, resulting in failure delay. To measure the probability influence of inertial effect on availability quantitatively, a critical threshold is set for each deteriorating output state. Both non-negative constants and random variables of the critical thresholds are modeled respectively. Comparing with the original system described with a Markov process, the new situation with inertial effect is recorded as the new system and described with a generic stochastic process. Availability variations of the original system and the new system show the influence of inertial effect on the reliability. A numerical example of a power supply system, which is a typical multi-state system (MSS), is presented to illustrate the results obtained. Relevant studies may be extended in many fields such as queuing theory and quality management. Graphical abstract Image, graphical abstract [ABSTRACT FROM AUTHOR]

This paper deals with the problem of quantifying the impact of model misspecification when computing general expected values of interest. The methodology that we propose is applicable in great generality; in particular, we provide examples involving path-dependent expectations of stochastic processes. Our approach consists of computing bounds for the expectation of interest regardless of the probability measure used, as long as the measure lies within a prescribed tolerance measured in terms of a flexible class of distances from a suitable baseline model. These distances, based on optimal transportation between probability measures, include Wasserstein's distances as particular cases. The proposed methodology is well suited for risk analysis and distributionally robust optimization, as we demonstrate with applications. We also discuss how to estimate the tolerance region nonparametrically using Skorokhod-type embeddings in some of these applications. [ABSTRACT FROM AUTHOR]

We consider the problem of optimal control of a mean-field stochastic differential equation (SDE) under model uncertainty. The model uncertainty is represented by ambiguity about the law of the state X(t) at time t. For example, it could be the law of X(t) with respect to the given, underlying probability measure. This is the classical case when there is no model uncertainty. But it could also be the law with respect to some other probability measure or, more generally, any random measure on with total mass 1. We represent this model uncertainty control problem as a stochastic differential game of a mean-field related type SDE with two players. The control of one of the players, representing the uncertainty of the law of the state, is a measure-valued stochastic process and the control of the other player is a classical real-valued stochastic process u(t). This optimal control problem with respect to random probability processes in a non-Markovian setting is a new type of stochastic control problems that has not been studied before. By constructing a new Hilbert space of measures, we obtain a sufficient and a necessary maximum principles for Nash equilibria for such games in the general nonzero-sum case, and for saddle points in zero-sum games. As an application we find an explicit solution of the problem of optimal consumption under model uncertainty of a cash flow described by a mean-field related type SDE. [ABSTRACT FROM AUTHOR]

This paper focuses on a nonlinear second-order stochastic evolution equations driven by a fractional Brownian motion (fBm) with Poisson jumps and which is dependent upon a family of probability measures. The global existence of mild solutions is established under various growth conditions, and a related stability result is discussed. An example is presented to illustrate the applicability of the theory. [ABSTRACT FROM AUTHOR]

The n -queens problem is to determine Q (n) , the number of ways to place n mutually non-threatening queens on an n × n board. We show that there exists a constant α = 1.942 ± 3 × 10 − 3 such that Q (n) = ((1 ± o (1)) n e − α) n. The constant α is characterized as the solution to a convex optimization problem in P ([ − 1 / 2 , 1 / 2 ] 2) , the space of Borel probability measures on the square. The chief innovation is the introduction of limit objects for n -queens configurations, which we call queenons. These form a convex set in P ([ − 1 / 2 , 1 / 2 ] 2). We define an entropy function that counts the number of n -queens configurations that approximate a given queenon. The upper bound uses the entropy method of Radhakrishnan and Linial–Luria. For the lower bound we describe a randomized algorithm that constructs a configuration near a prespecified queenon and whose entropy matches that found in the upper bound. The enumeration of n -queens configurations is then obtained by maximizing the (concave) entropy function in the space of queenons. Along the way we prove a large deviations principle for n -queens configurations that can be used to study their typical structure. [ABSTRACT FROM AUTHOR]

• Present belief reliability problems of ecological systems under uncertain random environments where uncertainty is subjective indeterminacy and randomness is objective indeterminacy. • The concept of reliability index is defined as chance measure when there are degradation and shock processes. • The reliability index formulas of ecological systems are developed in specific cases. • Apply reliability assessment methods to Caohai nature reserve. Typical failure modes of degradation and shock processes have been applied which are named as soft failures and hard failures, respectively. This paper discusses the reliability of uncertain random systems where randomness is regarded as objective indeterminacy with enough sample data, and uncertainty is referred to epistemic indeterminacy with insufficient sample data. Considering that the combination of internal and external factors, the degradation process is modelled by an uncertain differential equation, and external shocks are driven by an uncertain random renewal process where state variables are uncertain random variables. Chance measure is applied to define the reliability index which is distinguished from traditional reliability assessment methods that utilize probability measure. Three types of reliability models possessed with independent failures are presented, where the general reliability index formulas are provided. The analysis results of Caohai Nature Reserve show that uncertain random modelling method provides an effective approach to system reliability assessment. [ABSTRACT FROM AUTHOR]

This paper is about a class of distributionally robust model predictive controllers (MPC) for nonlinear stochastic processes, which evaluate risk and control performance measures by propagating ambiguity sets in the space of state probability measures. A framework for formulating such ambiguity tube MPC controllers is presented using methods from the field of optimal transport theory. Moreover, an analysis technique based on supermartingales is proposed, leading to stochastic stability results for a large class of distributionally robust controllers. In this context, we also discuss how to construct terminal cost functions for stochastic and distributionally robust MPC that ensure closed-loop stability and asymptotic convergence to robust invariant sets. The corresponding theoretical developments are illustrated by tutorial-style examples and a numerical case study. [ABSTRACT FROM AUTHOR]

In this paper, we first use PDE techniques and probabilistic methods to identify a kind of quasi-continuous random variables. Then we give a characterization of the G -integrable processes and get a kind of quasi-continuous processes by Krylov’s estimates. This result is useful for the development of G -stochastic analysis theory. Moreover, it also provides a tool for the study of the non-Markovian Itô processes. [ABSTRACT FROM AUTHOR]

Several depths suitable for infinite-dimensional functional data that are available in the literature are of the form of an integral of a finite-dimensional depth function. These functionals are characterized by projecting functions into low-dimensional spaces, taking finite-dimensional depths of the projected quantities, and finally integrating these projected marginal depths over a preset collection of projections. In this paper, a general class of integrated depths for functions is considered. Several depths for functional data proposed in the literature during the last decades are members of this general class. A comprehensive study of its most important theoretical properties, including measurability and consistency, is given. It is shown that many, but not all, properties of the integrated depth are shared with the finite-dimensional depth that constitutes its building block. Some pending measurability issues connected with all integrated depth functionals are resolved, a broad new notion of symmetry for functional data is proposed, and difficulties with respect to consistency results are identified. A general universal consistency result for the sample depth version, and for the generalized median, for integrated depth for functions is derived. [ABSTRACT FROM AUTHOR]