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# "Multidimensional Queues, Risk, and Finance"

## Oct 1 - Oct 3

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#### Summary

The aim of the "Multidimensional Queues, Risk, and Finance" workshop is to bring together researchers from several fields within applied probability theory. The overarching theme is multidimensional stochastic analysis, including topics in queueing networks, limit order books and (re)insurance. A wide range of methods and techniques will be presented: exact, approximate, asymptotic and simulation.

#### Organizers

Onno Boxma | TU Eindhoven |

Stella Kapodistria | TU Eindhoven |

David Koops | University of Amsterdam |

#### Speakers

**Keynotes**

Hansjörg Albrecher | Université de Lausanne |

Søren Asmussen | Aarhus University |

Rama Cont | University of Oxford |

Frank Kelly | University of Cambridge |

Costis Maglaras | Columbia University |

Zbigniew Palmowski | Wroclaw University of Science and Technology |

Jamol Pender | Cornell University |

Ioane Muni Toke | Centrale Supélec |

**Invited (confirmed)**

Ivo Adan | TU Eindhoven |

Urtzi Ayesta | Institut de Recherch en Informatique de Toulouse |

Anita Behme | TU Dresden |

Richard Boucherie | University of Twente |

Xinyun Chen | Wuhan University |

Corina Constantinescu | University of Liverpool |

Guusje Delsing | University of Amsterdam; Rabobank |

Esther Frostig | University of Haifa |

Jevgenijs Ivanovs | Aarhus University |

Roger Laeven | University of Amsterdam |

Michel Mandjes | University of Amsterdam |

Joshua Reed | New York University |

Mayank Saxena | TU Eindhoven |

Seva Shneer | Heriot-Watt University |

Erik Tornij | NN Group (Insurance) |

#### Programme

#### Abstracts

**Ivo Adan (TU Eindhoven)**

**Mean waiting times in multi-server queues with skill based service under FCFS-ALIS**

We study a queue of multi-skilled servers, serving multi-type customers under the service policy FCFS-ALIS. For exponential services, we present an exact expression for the Laplace transform of the waiting time for each customer type. This expression has an interesting probabilistic interpretation. For non-exponential services, we propose approximations for the mean waiting time by exploiting this interpretation.

**Hansjörg Albrecher (Université de Lausanne)**

**On multi-dimensional ruin and dividend problems**

We give an overview of some recent developments in insurance risk theory in several dimensions. Among the topics covered are optimal stochastic control problems for the optimal dividend problem in higher dimensions, as well as challenges in the identification of dependence structures in multivariate portfolios.

**Søren Asmussen (Aarhus University)**

**Markov Renewal Structures: Some Applications and Some Theory**

A task moves between the different states in a system inspired from

series/parallel and *k*-out-of-*n* structures in classical reliability theory, and is served at rate *ri* in state *i*. Some states may be failed (*ri* = 0) and from such a state, the task needs to be restarted from scratch the next time it enters a state* j* with *rj* > 0.

The study of distribution of the total task time *X* leads to defective Markov renewal equations, and or the tail of *X* one needs some extensions of the classical theory, namely a perturbation result for light tails and a study of the heavy-tailed case.

S. Asmussen, L. Lipsky & S. Thompson (2015) Markov renewal methods in restart problems in complex systems. In The Fascination of Probability, Statistics, and their Applications. In Honour of Ole E. Barndorff–Nielsen (M. Podolskij, R. Stelzer, S. Thorbjørnsen & A.E.D. Veraart, eds.), pp 501–527. Springer-Verlag.

S. Asmussen & J. Thøgersen (2017) Markov dependence in renewal equations dom sums with heavy tails. Stochastic Models 33, 617–632.

**Urtzi Ayesta (Institut de Recherch en Informatique de Toulouse)**

**On the performance and stability of redundancy models
**Motivated by empirical evidence suggesting that redundancy can help improve the performance in real-world applications, in recent years there has been a big interest in studying redundancy queueing models. While there are several variants of a redundancy-based system, the general notion of redundancy is to create multiple copies of the same job that will be sent to a subset of servers. By allowing for redundant copies, the aim is to minimize the system latency by exploiting the variability in the queue lengths and the capacity of the different servers. We first analyze a parallel queueing model with state dependent capacities and we characterize sufficient conditions for the steady-state to be of product form. We then show that redundancy models of type

cancel-on-start (c.o.s.) and cancel-on-complete (c.o.c.) can be seen as particular instances of the above model. We compare the performance of c.o.s. and c.o.c. and show that, unexpectedly, c.o.s.

is worse in terms of mean number of jobs. Under the assumption that copies are i.i.d., it is known from literature that the stability region does not reduce. We consider the case of identical copies,

and show that the stability region depends on the scheduling discipline implemented in the servers, and that in the particular case of PS and FCFS it significantly reduces.

(joint work with: E. Anton, T. Bodas, M. Jonckheere, I.M. Verloop, J.P. Dorsman)

**Anita Behme (TU Dresden)**

**Hitting probabilities for compound Poisson processes in a bipartite network**

We present hitting probabilies of a constant barrier for single and multiple components of a multivariate compound Poisson process. The components of the process are allowed to be dependent, with the dependency structure of the components induced by a random bipartite network. In analogy with the non-network scenario, a network Pollaczek-Khintchine parameter P is introduced. This random parameter, which depends on the bipartite network, is crucial for the hitting probabilities. Under certain conditions on the network and for light-tailed jump distributions we obtain Lundberg bounds and, for exponential jump distributions, exact results for the hitting probabilities. For large sparse networks, the parameter P is approximated by a function of independent Poisson variables.

As application, risk balancing networks in ruin theory are discussed.

**Richard Boucherie (University of Twente)**

**Networks of finite queues with multiple customer classes and fixed routes
**We consider networks of finite capacity queues in which customers of multiple classes route among the queues following fixed routes. We modify the network using several product-form preserving blocking protocols and use these protocols to obtain upper and lower bounds on various performance measures for the original network.

(joint work with Nico M. van Dijk)

**Xinyun Chen (Wuhan University)**

**Unraveling Limit Order Books Using Just Bid/Ask Prices
**How much of the structure of a Limit Order Book (LOB) can be recovered by only observing the trade and quote (TAQ) sequence? In this paper we study the queueing dynamics in the LOB which, surprisingly, allows us to recover the LOB of stocks having relatively large spread with reasonable empirical accuracy. In particular, we are able to estimate the time-average order depth on almost all price levels of interest in the LOB by observing only the bid/ask price at the times of trades. As applications, we can use the result to estimate the price impact of trades and size of hidden orders. Our approach starts from a Markovian queueing model for the LOB dynamics. We apply a multi-scale analysis on the model to obtain a closed-form expression connecting the trade price change distribution and the LOB structure, which enables the recovery of LOB from the TAQ sequence. Our approach is also applicable to extended models with autocorrelated and state-dependent order flows in the LOB.

**Rama Cont (University of Oxford)**

**Dynamics of limit order books across time scales: queueing models, diffusion limits and stochastic PDEs
**The advent of electronic trading in financial markets has generated a heterogeneous market landscape leading to a heterogeneous environment where market participants interact across a wide range of trading frequencies through a central

*limit order book*. Based on a detailed empirical study of high frequency order flow in equity and futures markets, we propose a stochastic model for dynamics of price and order flow in a limit order market, which captures the coexistence of high frequency and low frequency order flow and examines the consequences of this heterogeneity on price dynamics, volatility and liquidity.

At the highest frequency, the order book may be described as a multi-scale and multi-class queueing system. Over longer time scales, such as minutes, the effective dynamics of the order book is described by a stochastic moving boundary problem, with a boundary between the buy and sell orders which follows diffusive dynamics. Our model provides insights into how the interaction of different types of order flow affects the dynamics of supply, demand and prices in limit order markets.

**Corina Constantinescu (University of Liverpool)**

**Fractional differential equations in risk theory
**In this talk, we will exemplify how fractional differential equations (FDEs) could be use in a risk theory setting. Ruin probabilities will be derived in a few instances, for models with distributions described as solutions of FDEs.

**Guusje Delsing (University of Amsterdam; Rabobank)**

**Capital reserve management for a multi-dimensional risk model
**Firms should keep capital to offer sufficient protection against the risks they are facing. In the insurance context methods have been developed to determine the minimum capital level required, but less so in the context of firms with multiple business lines including allocation. This research focusses on the calculation of finite-time ruin probabilities and capital reserves for a multi-dimensional risk model. The individual reserves of these lines of business are modelled by means of a Cramér–Lundberg model with constant incoming premiums and outgoing claims that arrive according to a Poisson process. To allow for dependence between business lines we introduce a common (latent) environmental factor. This environmental factor impacts the claim inter-occurrence times as well as the claim sizes.

**Considering a fixed environmental process over time, we present a novel Bayesian approach**

to calibrate the latent environmental state distribution based on observations concerning the claim

processes. We then we allow for the distribution of individual claims to change over time by

using a Markov environmental process. For the latter, we present two approximations for the

finite-time multi- variate survival/ruin probabilities: a diffusion approximation and a single-switch approximation.

Finally, we point out how to determine the optimal initial capital of the different business lines under specific constraints on the ruin/survival probability of subsets of business lines.

(joint work with Erik Winands, Michel Mandjes and Peter Spreij)

**Esther Frostig (University of Haifa)**

**The dual risk model with dividends taken at arrival**

The dual risk model is a Lévy process without negative jumps. It describes the surplus of a company with fixed expense rate and occasional gains. It describes also the workload in an M/G/1 queueing system.

We consider the dual risk model where at arrival epochs a certain part of the gain is paid as dividends or taxes. Two models are considered :

1. If an arriving gain finds the surplus above a barrier b or if it would bring the surplus above that level, a certain part of the gain is paid as dividends or taxes.

2. A part of the gain is paid as dividend only when upon the gain arrival the surplus is above b. We obtain expressions for the joint Laplace-Stieltjes transform of the time to ruin and the amount of dividends paid until ruin, and for the expected discounted dividend paid until ruin. We consider the case where the dividend paid from each gain is a general function of the gain. More explicit results are obtained when the dividend is a given percentage of the gain amount.

(joint work with Onno Boxma)

**Jevgenijs Ivanovs (Aarhus University)**

**Probabilistic interpretations in bivariate queueing and risk models with mutual assistance - an open problem
**Consider two classical risk reserve processes with the special feature that each insurance company covers the deficit of the other. It is assumed that the capital transfers between the companies are instantaneous and incur a certain proportional cost, and that ruin occurs when neither company can cover the deficit of the other. The bivariate transform of the corresponding survival probability is identified in [3] using arguments from complex analysis inspired by [2] and [1]. The expression is in terms of the Wiener-Hopf factors of two auxiliary Lévy processes constructed from descending ladder time processes corresponding to the original drivers. This result cries out for a probabilistic interpretation, but none has been found so far.

The probabilistic approach would explain the form of the result and would possibly allow for various extensions. Furthermore, [1] provides a somewhat similar expression for the joint transform of the stationary workload in a coupled processor model and in a two-node fluid network. Importance of this open problem is underlined by the scarcity of tractable bivariate queueing and risk models.

[1] O. Boxma and J. Ivanovs (2013). Two coupled Lévy queues with independent input.

Stochastic Systems 3(2), p. 574–590.

[2] J. Cohen and O. Boxma (1983). Boundary Value Problems in Queueing System Analysis.

North-Holland Mathematics Studies 79.

[3] J. Ivanovs and O. Boxma (2015). A bivariate risk model with mutual deficit

coverage. Insurance: Mathematics and Economics 64, p. 126–134.

**Frank Kelly (University of Cambridge)**

**A Markov model of a limit order book: thresholds, recurrence, and trading strategies
**This talk concerns an analytically tractable model of a limit order book (due originally to Stigler and Luckock) where the dynamics are driven by stochastic fluctuations between supply and demand. The model has a natural interpretation for a highly traded market on short time-scales where there is a separation between the time-scale of trading, represented in the model, and a longer time-scale on which fundamentals change.

We describe our main result for the model, which is the existence of an explicit limiting distribution for the highest bid, and for the lowest ask, where the limiting distributions are confined between two thresholds. Fluid limits play an important role in establishing the recurrence properties of the model.

We use the model to analyze various high-frequency trading strategies (for example market-making, sniping and mixtures of these), and comment on the Nash equilibria that emerge between high-frequency traders when a market in continuous time is replaced by frequent batch auctions.

(joint work with Elena Yudovina)

**Roger Laeven (University of Amsterdam)**

**Excitation
**In this talk, I will discuss a class of semimartingale models designed to capture the dynamics of asset returns and P&L’s, with periods of crises that are characterized by financial contagion. In the models, a jump in one region of the world or financial institution increases the intensity of jumps both in the same region or institution (self-excitation) as well as in other regions or institutions (cross-excitation), generating episodes of highly clustered jumps across world markets or institutions that mimic the observed features of the data. Estimation and testing procedures for these models are briefly discussed. The models are amenable to a wide variety of quantitative risk management applications such as measuring market stress, CDS and derivative pricing, and portfolio choice. A few illustrations of such applications will be provided.

**Costis Maglaras (Columbia University)**

**The role of queueing phenomena in electronic limit order book markets
**Many financial markets are operated as electronic limit order books (LOB). Over short time scales, seconds to minutes, LOBs can be best understood and modeled as queueing systems. I will offer a brief overview of algorithmic trading in a limit order book, and highlight how queueing phenomena play an important role in trade execution, and, as a consequence, in market behavior.

**Michel Mandjes (University of Amsterdam)**

**Infinite server queues with shot noise modulated Poisson arrivals
**In this talk I’ll consider a network of infinite-server queues where the input process is a Cox process of the following form. The arrival rate is a vector valued linear transform of a multivariate generalized (i.e., being driven by a subordinator rather than a compound Poisson process) shot-noise process. I first derive some distributional properties of the multivariate generalized shot-noise process. Then these are exploited to obtain the joint transform of the numbers of customers, at various time epochs, in a single infinite-server queue fed by the above mentioned Cox process. This allows the derivation of transforms pertaining to the joint stationary arrival rate and queue length processes (thus facilitating the analysis of the corresponding departure process), as well as their means and covariance structure. I show how to extend to the setting of a network of infinite-server queues. Finally, I'll look into a number of scaling limits and asymptotic results.

(joint work with Onno Boxma, Offer Kella en Mayank Saxena)

**Zbigniew Palmowski (Wroclaw University of Science and Technology)**

**Two-dimensional ruin probabilities: exact and asymptotic results**

Ruin theory concerns the study of stochastic processes that represent the time evolution of the surplus of a non-life insurance company.

The initial goal of early researchers of the field, Lundberg (1903) and Cramér (1930), was to determine the probability for the surplus to become negative. In those pioneer works, the authors show that the ruin probability decreases exponentially fast to zero with initial reserve tending to infinity when the net profit condition is satisfied and claim sizes are light-tailed. During the lecture we focus on the two-dimensional counterpart of the above mentioned theory. We analyse the ruin probabilities of two insurance companies (or two branches of the same company) that divide between them both claims and premia in some specified proportions. We present exact expressions when the generic claim size has some specific phase-type distribution. We also give the asymptotics of these ruin probabilities when the initial reserves of both companies tend to infinity. We consider two regimes of claim size distributions: light-tailed and heavy-tailed. We demonstrate the main techniques used in the proofs as well: change of measure and the principle of one big jump.

**Jamol Pender (Cornell University)**

**The Queue-Hawkes Process: Finite Self-Excitement
**In this talk, we propose a new a self-exciting model that couples a Hawkes process and a queueing system which we call the Queue-Hawkes process. In this process the intensity responds to the queue: increasing at arrivals, dropping upon departures, and decaying between. Hence the influence of each arrival is ephemeral, as the excitement only lasts for the duration of the entity's time in system. We study this process both individually and by comparison to other processes, primarily the Hawkes process and the Affine Queue-Hawkes process, which we define as the zero decay case of the Queue-Hawkes. We will demonstrate that a batch-scaling of the Affine Queue-Hawkes process will converge to the Hawkes process. Additionally, we provide all moments of the Hawkes process and the Affine Queue-Hawkes process via a novel matrix structure. Finally, we prove fluid and diffusion limits for the Queue-Hawkes process using moment generating function techniques.

**Josh Reed (New York University)**

**A Dirichlet Process Characterization of RBM in a Wedge**

Reflected Brownian motion (RBM) in a wedge is a 2-dimensional stochastic process Z whose state space in is given in polar coordinates by S = {(r, θ) : r ≥ 0, 0 ≤ θ ≤ ξ} for some 0 < ξ < 2π. Let α(θ1 + θ2)/ξ, where π/2 < θ1, θ2 < π/2 are the directions of reflection of Z off each of the two edges of the wedge as measured from the corresponding inward facing normal. We prove that in the

case of 1 < α < 2, RBM in a wedge is a Dirichlet process. Specifically, its unique Doob-Meyer type decomposition is given by Z = X + Y , where X is a two-dimensional Brownian motion and Y is a continuous process of zero energy. Furthermore, we show that for p > α, the strong p-variation of the sample paths of Y is finite on compact intervals, and, for 0 < p α, the strong p-variation of Y is infinite on [0, T ] whenever Z has been started from the origin. We also show that on excursion intervals of Z away from the origin, (Z, Y ) satisfies the standard Skorokhod problem for X. However, on the entire time horizon (Z, Y ) does not satisfy the standard Skorokhod problem for X, but nevertheless we show that it satisfies the extended Skorokhod problem.

This is joint work with Peter Lakner and Bert Zwart.

**Mayank Saxena (TU Eindhoven)**

**Analysis of a random time-limited polling model**

In this talk, we analyze a single server polling model, with a special service discipline. In particular, we assume a variation of the randomly timed gated service discipline: the server only switches to another queue when an exponential timer expires, irrespective of whether it has been serving or the queues have been empty. In addition, we assume that the server serves both queues in preemptive resume manner. There are two advantages of this model. It enables to: i) keep the frequency of switching at a predetermined level (thus controlling the total cost, if there is a switching cost), ii) balance the time that the server spends in each queue (since, contrary to exhaustive or gated service disciplines, this discipline does not depend on the number of customers present in the various queues).

For this polling model, we derive the steady-state marginal workload distribution and prove a decomposition property. Since this polling model violates the branching property, a direct analytic derivation of the steady-state joint workload/queue-length distribution turns out to be very difficult. One can numerically analyze this model using either perturbation analysis or the power series algorithm. However, both approaches fail in the heavy-traffic regime. In this talk, we illustrate how boundary value methods can be used to derive the explicit joint heavy-traffic limiting distribution of the workload.

(joint work with Onno Boxma, Stella Kapodistria, and Sindo Núñez Queija)

**Seva Shneer (Heroit-Watt University)**

**Stability of some decentralised multi-access algorithms in single- and multi-hop scenarios
**In the first part of the talk we will discuss stability properties of a standard decentralised medium access protocol in a simple single-hop network. We will highlight the main difficulties in the analysis. We will then consider a simpler version of the algorithm in single- and multi-hop networks.

(joint work with Sasha Stolyar (UIUC))

**Ioane Muni Toke (Centrale Supélec)**

**Some contributions to the modeling limit order books
**Limit order books are at the center of modern electronic financial exchanges. In a first part of the talk, we recall basic facts on their functioning rules and some empirical results. From a modeling point of view, limit order books are a system of coupled queues with multiple levels of priority. In a second part of the talk, we briefly review some of the financial literature that directly takes advantage of queueing representations to derive some financial results. We take a special look at simulations results and point out limitations and caveats of basic models. In a third part, we propose a closer look at some intensity-based models of limit orders books. Processes with Hawkes-type intensities and Cox-type intensities are examples of point process models that can be used to model complex behaviours of limit order books. We provide some elements of comparison between these models, focusing on their modeling power. We also introduce a ratio model of Cox-type intensities that can be used as an efficient econometric tool for the analysis of high-frequency limit order books data.

**Erik Tornij (Nationale-Nederlanden)**

**Modelling mortality rates for life insurance portfolios
**For life insurance companies and pension funds, it is fundamental to have clear insights in the mortality risk of their insured lives. A bulk of research has shown that the mortality of a given group insured lives can significantly differ from the mortality of the country’s population. In actuarial practice, this difference in mortality, which can be due to factors such as socioeconomic inequality, is known as experience mortality. It is generally accepted that mortality rates are dependent from age, gender and year of observation. Furthermore, one should take into account that someone’s pension or salary has a significant impact on the mortality risk; insureds with higher salaries live longer. During the presentation it will be discussed that this relation can’t be ignored within provision calculations.

All and all, one may conclude that there are at least for 4 dimensions that have to take into account (age, gender, time, insured pension). Cash flow models as used in the actuarial practice for provision calculations are in general not designed for a great variety of mortality assumptions (as input into the model). Given these practical limitations, the central challenge (for the actuary) is to account for the heterogeneity in a practical but sound way.

The presentation will discuss this challenge from a theoretical as well as from a practical perspective.

**Bert Zwart (CWI - TU Eindhoven)**

**Heavy tails: asymptotics, algorithms, applications**

We present new sample-path large deviations principles for random walks with regularly varying jumps and Weibullian jumps.

We apply these results to develop the first universally applicable importance sampling scheme, and to answer a conjecture dating back to Foss (2009) and Whitt (1998) on the number of large jobs needed to create a large queue length in the many-server queue.

Time permitting, we present ongoing work on Markov additive processes, such as ARCH processes.

(joint work with Mihail Bashba, Jose Blanchet, Bohan Chen, and Chang-Han Rhee)

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