arXiv subjects

Michel Mandjes

Publications and source records attributed to Michel Mandjes.

At least 73 records · Page 4Linked to original sources

arXiv2021

A decomposition for Levy processes inspected at Poisson moments

We consider a Lévy process $Y(t)$ that is not permanently observed, but rather inspected at Poisson($ω$) moments only, over an exponentially distributed time $T_β$ with parameter $β$. The focus lies on the analysis of the distribution of the running maximum at such inspection moments up to $T_β$, denoted by $Y_{β,ω}$. Our main result is a decomposition: we derive a remarkable distributional equality that contains $Y_{β,ω}$ as well as the running maximum process $\bar Y(t)$ at the exponentially distributed times $T_β$ and $T_{β+ω}$. Concretely, $\overline{Y}(T_β)$ can be written the sum of the two independent random variables that are distributed as $Y_{β,ω}$ and $\overline{Y}(T_{β+ω})$. The distribution of $Y_{β,ω}$ can be identified more explicitly in the two special cases of a spectrally positive and a spectrally negative Lévy process. As an illustrative example of the potential of our results, we show how to determine the asymptotic behavior of the bankruptcy probability in the Cramér-Lundberg insurance risk model.

math.PR↗

arXiv2021

From reflected Lévy processes to stochastically monotone Markov processes via generalized inverses and supermodularity

It was recently proven that the correlation function of the stationary version of a reflected Lévy process is nonnegative, nonincreasing and convex. In another branch of the literature it was established that the mean value of the reflected process starting from zero is nonnegative, nondecreasing and concave. In the present paper it is shown, by putting them in a common framework, that these results extend to substantially more general settings. Indeed, instead of reflected Lévy processes, we consider a class of more general stochastically monotone Markov processes. In this setup we show monotonicity results associated with a supermodular function of two coordinates of our Markov process, from which the above-mentioned monotonicity and convexity/concavity results directly follow, but now for the class of Markov processes considered rather than just reflected Lévy processes. In addition, various results for the transient case (when the Markov process is not in stationarity) are provided. The conditions imposed are natural, in that they are satisfied by various frequently used Markovian models, as illustrated by a series of examples.

math.PR↗

arXiv2021

Exact and Asymptotic Analysis of General Multivariate Hawkes Processes and Induced Population Processes

This paper considers population processes in which general, not necessarily Markovian, multivariate Hawkes processes dictate the stochastic arrivals. We establish results to determine the corresponding time-dependent joint probability distribution, allowing for general intensity decay functions, general intensity jumps, and general sojourn times. We obtain an exact, full characterization of the time-dependent joint transform of the multivariate population process and its underlying intensity process in terms of a fixed-point representation and corresponding convergence results. We also derive the asymptotic tail behavior of the population process and its underlying intensity process in the setting of heavy-tailed intensity jumps. By exploiting the results we establish, arbitrary joint spatial-temporal moments and other distributional properties can now be readily evaluated using standard transform differentiation and inversion techniques, and we illustrate this in a few examples.

math.PR↗

arXiv2021

Continuous Assortment Optimization with Logit Choice Probabilities under Incomplete Information

We consider assortment optimization over a continuous spectrum of products represented by the unit interval, where the seller's problem consists of determining the optimal subset of products to offer to potential customers. To describe the relation between assortment and customer choice, we propose a probabilistic choice model that forms the continuous counterpart of the widely studied discrete multinomial logit model. We consider the seller's problem under incomplete information, propose a stochastic-approximation type of policy, and show that its regret -- its performance loss compared to the optimal policy -- is only logarithmic in the time horizon. We complement this result by showing a matching lower bound on the regret of any policy, implying that our policy is asymptotically optimal. We then show that adding a capacity constraint significantly changes the structure of the problem: we construct a policy and show that its regret after $T$ time periods is bounded above by a constant times $T^{2/3}$ (up to a logarithmic term); in addition, we show that the regret of any policy is bounded from below by a positive constant times $T^{2/3}$, so that also in the capacitated case we obtain asymptotic optimality. Numerical illustrations show that our policies outperform or are on par with alternatives.

stat.ML↗

arXiv2021

On Capital Allocation for a Risk Measure Derived from Ruin Theory

This paper addresses allocation methodologies for a risk measure inherited from ruin theory. Specifically, we consider a dynamic value-at-risk (VaR) measure defined as the smallest initial capital needed to ensure that the ultimate ruin probability is less than a given threshold. We introduce an intuitively appealing, novel allocation method, with a focus on its application to capital reserves which are determined through the dynamic value-at-risk (VaR) measure. Various desirable properties of the presented approach are derived including a limit result when considering a large time horizon and the comparison with the frequently used gradient allocation method. In passing, we introduce a second allocation method and discuss its relation to the other allocation approaches. A number of examples illustrate the applicability and performance of the allocation approaches.

q-fin.MF↗

arXiv2021

Bounds for expected supremum of fractional Brownian motion with drift

We provide upper and lower bounds for the mean ${\mathscr M}(H)$ of $\sup_{t\geqslant 0} \{B_H(t) - t\}$, with $B_H(\cdot)$ a zero-mean, variance-normalized version of fractional Brownian motion with Hurst parameter $H\in(0,1)$. We find bounds in (semi-)closed-form, distinguishing between $H\in(0,\frac{1}{2}]$ and $H\in[\frac{1}{2},1)$, where in the former regime a numerical procedure is presented that drastically reduces the upper bound. For $H\in(0,\frac{1}{2}]$, the ratio between the upper and lower bound is bounded, whereas for $H\in[\frac{1}{2},1)$ the derived upper and lower bound have a strongly similar shape. We also derive a new upper bound for the mean of $\sup_{t\in[0,1]} B_H(t)$, $H\in(0,\tfrac{1}{2}]$, which is tight around $H=\tfrac{1}{2}$.

math.PR↗

arXiv2021

A diffusion-based analysis of a multi-class road traffic network

This paper studies a stochastic model that describes the evolution of vehicle densities in a road network. It is consistent with the class of (deterministic) kinematic wave models, which describe traffic flows on the basis of conservation laws that incorporate the macroscopic fundamental diagram (a functional relationship between vehicle density and flow). Our setup is capable of handling multiple types of vehicle densities, with general macroscopic fundamental diagrams, on a network with arbitrary topology. Interpreting our system as a spatial population process, we derive, under a natural scaling, fluid and diffusion limits. More specifically, the vehicle density process can be approximated with a suitable Gaussian process, which yield accurate normal approximations to the joint (in the spatial and temporal sense) vehicle density process. The corresponding means and variances can be computed efficiently. Along the same lines, we develop an approximation to the vehicles' travel-time distribution between any given origin and destination pair. Finally, we present a series of numerical experiments that demonstrate the accuracy of the approximations and illustrate the usefulness of the results.

math.PR↗

arXiv2020

A transient Cramér-Lundberg model with applications to credit risk

This paper considers a variant of the classical Cramér-Lundberg model that is particularly appropriate in the credit context, with the distinguishing feature that it corresponds to a finite number of obligors. The focus is on computing the ruin probability, i.e., the probability that the initial reserve, increased by the interest received from the obligors and decreased by the losses due to defaults, drops below zero. Besides an exact analysis (in terms of transforms) of this ruin probability, also an asymptotic analysis is performed, including an efficient importance-sampling based simulation approach. The base model is extended in multiple dimensions: (i) we consider a model in which there may, in addition, be losses that do not correspond to defaults, (ii) then we analyze a model in which the individual obligors are coupled through a regime-switching mechanism, (iii) then we extend the model such that between the losses the reserve process behaves as a Brownian motion rather than a deterministic drift, and (iv) we finally consider a set-up with multiple groups of statistically identical obligors.

math.PR↗

arXiv2020

Hypothesis testing for a Lévy-driven storage system by Poisson sampling

This paper focuses on hypothesis testing for the input of a Lévy-driven storage system by sampling of the storage level. As the likelihood is not explicit we propose two tests that rely on transformation of the data. The first approach uses i.i.d. `quasi-busy-periods' between observations of zero workload. The distribution of the duration of quasi-busy-periods is determined. The second method is a conditional likelihood ratio test based on the Bernoulli events of observing a zero or positive workload, conditional on the previous workload. Performance analysis is presented for both tests along with speed-of-convergence results, that are of independent interest.

math.PR↗

arXiv2020

A sample-path large deviation principle for dynamic Erdős-Rényi random graphs

We consider a dynamic Erdős-Rényi random graph (ERRG) on $n$ vertices in which each edge switches on at rate $λ$ and switches off at rate $μ$, independently of other edges. The focus is on the analysis of the evolution of the associated empirical graphon in the limit as $n\to\infty$. Our main result is a large deviation principle (LDP) for the sample path of the empirical graphon observed until a fixed time horizon. The rate is $\binom{n}{2}$, the rate function is a specific action integral on the space of graphon trajectories. We apply the LDP to identify (i) the most likely path that starting from a constant graphon creates a graphon with an atypically large density of $d$-regular subgraphs, and (ii) the mostly likely path between two given graphons. It turns out that bifurcations may occur in the solutions of associated variational problems.

math.PR↗

arXiv2020

Large deviations for acyclic networks of queues with correlated Gaussian inputs

We consider an acyclic network of single-server queues with heterogeneous processing rates. It is assumed that each queue is fed by the superposition of a large number of i.i.d. Gaussian processes with stationary increments and positive drifts, which can be correlated across different queues. The flow of work departing from each server is split deterministically and routed to its neighbors according to a fixed routing matrix, with a fraction of it leaving the network altogether. We study the exponential decay rate of the probability that the steady-state queue length at any given node in the network is above any fixed threshold, also referred to as the "overflow probability". In particular, we first leverage Schilder's sample-path large deviations theorem to obtain a general lower bound for the limit of this exponential decay rate, as the number of Gaussian processes goes to infinity. Then, we show that this lower bound is tight under additional technical conditions. Finally, we show that if the input processes to the different queues are non-negatively correlated, non short-range dependent fractional Brownian motions, and if the processing rates are large enough, then the asymptotic exponential decay rates of the queues coincide with the ones of isolated queues with appropriate Gaussian inputs.

math.PR↗

arXiv2020

A Gaussian segment-based traffic flow model for the design and control of transport networks

In the setting of a recently developed cellular stochastic traffic flow model, it has shown that the joint per-cell vehicle densities, as a function of time, can be accurately approximated by a Gaussian process, which has the attractive feature that its means and (spatial and temporal) covariances can be efficiently evaluated. The present article demonstrates the rich potential of this methodology in the context of road traffic control and transportation network design. To solidly provide empirical backing for the use of a multivariate Gaussian approximation, we rely on a detailed historical dataset that contains traffic flow data. Then, in the remainder of the paper, we provide a sequence of design and control related example questions that can be analyzed using the Gaussian methodology. These cover the following topics: (i) evaluation of stationary performance measures, (ii) route selection, (iii) control of traffic flows, and (iv) performance of traffic networks with arbitrary topology. In discussing the setup, results, and applications of these examples, we stress the appropriateness of our {stochastic} traffic model over a deterministic counterpart.

math.OC↗

arXiv2020

Performance of the smallest-variance-first rule in appointment sequencing

A classical problem in appointment scheduling, with applications in health care, concerns the determination of the patients' arrival times that minimize a cost function that is a weighted sum of mean waiting times and mean idle times. One aspect of this problem is the sequencing problem, which focuses on ordering the patients. We assess the performance of the smallest-variance-first (SVF) rule, which sequences patients in order of increasing variance of their service durations. While it was known that SVF is not always optimal, it has been widely observed that it performs well in practice and simulation. We provide a theoretical justification for this observation by proving, in various settings, quantitative worst-case bounds on the ratio between the cost incurred by the SVF rule and the minimum attainable cost. We also show that, in great generality, SVF is asymptotically optimal, i.e., the ratio approaches 1 as the number of patients grows large. While evaluating policies by considering an approximation ratio is a standard approach in many algorithmic settings, our results appear to be the first of this type in the appointment scheduling literature.

math.PR↗

arXiv2020

Single-server queues under overdispersion in the heavy-traffic regime

This paper addresses the analysis of the queue-length process of single-server queues under overdispersion, i.e., queues fed by an arrival process for which the variance of the number of arrivals in a given time window exceeds the corresponding mean. Several variants are considered, using concepts as mixing and Markov modulation, resulting in different models with either endogenously triggered or exogenously triggered random environments. Only in special cases explicit expressions can be obtained, e.g. when the random arrival and/or service rate can attain just finitely many values. While for more general model variants exact analysis is challenging, one ${\it can}$ derive limit theorems in the heavy-traffic regime. In some of our derivations we rely on evaluating the relevant Laplace transform in the heavy-traffic scaling using Taylor expansions, whereas other results are obtained by applying the continuous mapping theorem.

math.PR↗

arXiv2020

A Multiplicative Version of the Lindley Recursion

This paper presents an analysis of the stochastic recursion $W_{i+1} = [V_iW_i+Y_i]^+$ that can be interpreted as an autoregressive process of order 1, reflected at 0. We start our exposition by a discussion of the model's stability condition. Writing $Y_i=B_i-A_i$, for independent sequences of non-negative i.i.d.\ random variables $\{A_i\}_{i\in N_0}$ and $\{B_i\}_{i\in N_0}$, and assuming $\{V_i\}_{i\in N_0}$ is an i.i.d. sequence as well (independent of $\{A_i\}_{i\in N_0}$ and $\{B_i\}_{i\in N_0}$), we then consider three special cases: (i) $V_i$ attains negative values only and $B_i$ has a rational LST, (ii) $V_i$ equals a positive value $a$ with certain probability $p\in (0,1)$ and is negative otherwise, and both $A_i$ and $B_i$ have a rational LST, (iii) $V_i$ is uniformly distributed on $[0,1]$, and $A_i$ is exponentially distributed. In all three cases we derive transient and stationary results, where the transient results are in terms of the transform at a geometrically distributed epoch.

math.PR↗

arXiv2020

A roundabout model with on-ramp queues: exact results and scaling approximations

This paper introduces a general model of a single-lane roundabout, represented as a circular lattice that consists of $L$ cells, with Markovian traffic dynamics. Vehicles enter the roundabout via on-ramp queues that have stochastic arrival processes, remain on the roundabout a random number of cells, and depart via off-ramps. Importantly, the model does not oversimplify the dynamics of traffic on roundabouts, while various performance-related quantities (such as delay and queue length) allow an analytical characterization. In particular, we present an explicit expression for the marginal stationary distribution of each cell on the lattice. Moreover, we derive results that give insight on the dependencies between parts of the roundabout, and on the queue distribution. Finally, we find scaling limits that allow, for every partition of the roundabout in segments, to approximate 1) the joint distribution of the occupation of these segments by a multivariate Gaussian distribution; and 2) the joint distribution of their total queue lengths by a collection of independent Poisson random variables. To verify the scaling limit statements, we develop a novel way to empirically assess convergence in distribution of random variables.

nlin.CG↗

arXiv2019

Queues on a dynamically evolving graph

This paper considers a population process on a dynamically evolving graph, which can be alternatively interpreted as a queueing network. The queues are of infinite-server type, entailing that at each node all customers present are served in parallel. The links that connect the queues have the special feature that they are unreliable, in the sense that their status alternates between 'up' and 'down'. If a link between two nodes is down, with a fixed probability each of the clients attempting to use that link is lost; otherwise the client remains at the origin node and reattempts using the link (and jumps to the destination node when it finds the link restored). For these networks we present the following results: (a) a system of coupled partial differential equations that describes the joint probability generating function corresponding to the queues' time-dependent behavior (and a system of ordinary differential equations for its stationary counterpart), (b) an algorithm to evaluate the (time-dependent and stationary) moments, and procedures to compute user-perceived performance measures which facilitate the quantification of the impact of the links' outages, (c) a diffusion limit for the joint queue length process.We include explicit results for a series relevant special cases, such as tandem networks and symmetric fully connected networks.

math.PR↗

arXiv2019

Exact asymptotics in an infinite-server system with overdispersed input

This short communication considers an infinite-server system with overdispersed input. The objective is to identify the exact tail asymptotics of the number of customers present at a given point in time under a specific scaling of the model (which involves both the arrival rate and time). The proofs rely on a change-of-measure approach. The results obtained are illustrated by a series of examples.

math.PR↗