(PDF) ‎Generalized Gini-type Index

Workshop “Stochastic Models And Related Topics” 2016

BOOK OF ABSTRACTS

January 21-22, 2016

Dipartimento di Matematica – Universit`a di Salerno

Via Giovanni Paolo II, n. 132, 84084 Fisciano (SA), Italy

Workshop SMART 2016 Book of Abstracts

FOREWORD

The Workshop “Stochastic Models And Related Topics” 2016 is a program of lectures and

poster session for a group of specialists in various ﬁelds of Probability and Mathematical

Statistics. Main topics of the Workshop include stochastic models in dynamical systems,

reliability theory, biomathematics, mathematical ﬁnance, queueing systems. Some talks

also focus on current problems in applications of mathematics, probability and statistics,

and on related computational aspects.

We aim to have a scientiﬁcally interesting and stimulating opportunity to exchange infor-

mation and to promote fruitful interactions.

Fisciano

January 2016

The Organizing Committee

Antonio Di Crescenzo, Virginia Giorno, Barbara Martinucci, Alessandra Meoli, Amelia

G. Nobile, Serena Spina

Sponsor

Dipartimento di Matematica, Universit`a di Salerno

Workshop SMART 2016 Book of Abstracts

TABLE OF CONTENTS

On the inverse ﬁrst-passage-time problems for one-dimensional diﬀusions . . . . . . . . . . . . . . 1

Mario Abundo

New classes of priors based on stochastic orders and distortion functions . . . . . . . . . . . . . . 2

Jos´e Pablo Arias-Nicol´as, Fabrizio Ruggeri and Alfonso Su´arez-Llorens

Comparing residual lives and inactivity times by transform stochastic orders . . . . . . . . . . 3

Antonio Arriaza, Miguel Angel Sordo and Alfonso Su´arez-Llorens

L´evy processes with Poisson and Gamma times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Luisa Beghin

Approximation of Markov Chains by hybrid switching jump diﬀusion processes . . . . . . . .5

Enrico Bibbona and Roberta Sirovich

Some notes on stochastic diﬀusion equations for neurons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Stefano Bonaccorsi

Mathematical modeling of tumor-driven angiogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7

Vincenzo Capasso

The F¨ollmer-Schweizer decomposition under incomplete information and ﬁnancial

applications...........................................................................8

Claudia Ceci, Katia Colaneri and Alessandra Cretarola

On a fractional growth process with two kinds of jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Antonio Di Crescenzo, Barbara Martinucci and Alessandra Meoli

Multivariate Gini-type index.........................................................10

Antonio Di Crescenzo and Motahareh Parsa

A new model of population growth .................................................. 11

Antonio Di Crescenzo and Serena Spina

Stochastic variability of synaptic responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 12

Vito Di Maio and Silvia Santillo

Sample estimations and numerical evaluations of ﬁring neural activity . . . . . . . . . . . . . . . .13

Giuseppe D’Onofrio, Enrica Pirozzi and Marcelo O. Magnasco

Revisiting some macroscopic limit results for particle systems under the view of

modelling tumor growth.............................................................14

Franco Flandoli

Workshop SMART 2016 Book of Abstracts

Some examples of the interplay between Probability and Number Theory . . . . . . . . . . . . 15

Rita Giuliano

Asymptotic results for runs and empirical cumulative entropies . . . . . . . . . . . . . . . . . . . . . . .16

Rita Giuliano, Claudio Macci and Barbara Pacchiarotti

Size biased couplings and the spectral gap for random regular graphs . . . . . . . . . . . . . . . . 17

Larry Goldstein

Stochastic single vehicle routing problem with ordered customers. . . . . . . . . . . . . . . . . . . . . 18

Epaminondas G. Kyriakidis, Theodosis D. Dimitrakos and

Constantinos C. Karamatsoukis

The impact of degree variability on connectivity properties of large networks . . . . . . . . . 19

Lasse Leskel¨a

Two stochastic dominance criteria based on tail comparisons . . . . . . . . . . . . . . . . . . . . . . . . . 20

Julio Mulero, Miguel Angel Sordo, Marilia C. de Souza and Alfonso Su´arez-Llorens

Comparisons of residual lifetimes of coherent systems under dependence. . . . . . . . . . . . . .21

Jorge Navarro

Gradient ﬂow to the optimal transport plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 22

Giovanni Pistone

An unpredictable talk on prediction, the hot hand in basketball, and some

statistical principles ................................................................. 23

Yosef Rinott

First passage times of two-dimensional correlated diﬀusion processes with

application to neural modelling......................................................24

Laura Sacerdote

Inference in the early phase of an epidemic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Gianpaolo Scalia Tomba and Tom Britton

Some remarks on multivariate conditional hazard rates and dependence modeling . . . . 26

Fabio L. Spizzichino

Workshop SMART 2016 Book of Abstracts

On the inverse ﬁrst-passage-time problems for one-dimensional

diﬀusions

Mario Abundo

Dipartimento di Matematica, Universit`a Tor Vergata, Roma, Italy

We consider a temporally homogeneous, one-dimensional diﬀusion process X(t) driven by

the stochastic diﬀerential equation (SDE):

dX(t) = b(X(t))dt +σ(X(t))dBt, X(0) = η

where Btis a standard Brownian motion and the functions band σare regular enough,

so that a unique solution exists. If S(t) is a continuous function of t≥0,let τS(η) =

inf{t > 0 : X(t) = S(t)|X(0) = η}be the ﬁrst-passage time (FPT) of X(t) through the

boundary S(t),with the condition that X(0) = η, and let f(t|η) denote its density. In the

direct FPT problem, the initial position ηand the barrier Sare assigned, so it consists

in ﬁnding f(t|η); on the contrary, various kinds of inverse FPT (IFPT) problems can be

considered: in the ﬁrst one (IFPT1), one focuses on determining the boundary shape S,

when the FPT density fand the starting point ηare assigned; in the second one (IFPT2)

one assumes that the boundary is known, while the initial position ηis random, thus,

for a given FPT-distribution F, the IFPT2 problem consists in ﬁnding the density of η

s.t. P(τS≤t) = F(t).The IFPT1 problem has important applications e.g. in diﬀusion

models for neural activity ([3], [6]); the IFPT2 problem has interesting applications in

Mathematical Finance, in particular in credit risk modeling, where the FPT represents a

default event of an obligor ([4]) and also in diﬀusion models for neural activity ([5]).

We give an overview on IFPT problems (see [1], [2]); in particular, we present some new

results on the IFPT2 problem for diﬀusions with holding and jumps from a boundary.

References

1. Abundo M. (2006) Limit at zero of the ﬁrst-passage time density and the inverse problem

for one-dimensional diﬀusions. Stochastic Anal. Appl. 24, 1119–1145.

2. Abundo M. (2014) One-dimensional reﬂected diﬀusions with two boundaries and an inverse

ﬁrst-hitting problem. Stochastic Anal. Appl. 32, 975–991.

3. Giorno V., Nobile A.G., Ricciardi L.M. (1987) Single neurons activity: uncertain problems

of modeling and interpretation. Biosystems 40, 65–74.

4. Jackson K., Kreinin A., Zhang W. (2009) Randomization in the ﬁrst hitting problem.

Statist. Probab. Lett. 79, 2422–2428.

5. Lanska V., Smith C.E. (1989) The eﬀect of a random initial value in neural ﬁrst- passage-

time models. Math. Biosci. 93(2), 191–215.

6. Sacerdote L., Zucca C. (2003) Threshold shape corresponding to a Gamma ﬁring distri-

bution in an Ornstein–Uhlenbeck neuronal model. Sci. Math. Japon. 2, 295–305.

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Workshop SMART 2016 Book of Abstracts

New classes of priors based on stochastic orders and distortion

functions

Jos´e Pablo Arias-Nicol´asa, Fabrizio Ruggeriband Alfonso Su´arez-Llorensc

aFacultad de Matem´aticas, Universidad de Extremadura, Spain

bCNR IMATI, Milano, Italy

cDpto. Estad´ıstica e I.O. Universidad de C´adiz, Spain

In the context of robust Bayesian analysis, we introduce a new class of prior distribu-

tions based on stochastic orders and distortion functions. We provide the new deﬁnition,

its interpretation and the main properties and we also study the relationship with other

classical classes of prior beliefs. We also consider Kolmogorov and Kantorovich metrics

to measure the uncertainty induced by such class, as well as its eﬀect on the set of corre-

sponding Bayes actions. Finally, we conclude the paper with some numerical examples.

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Workshop SMART 2016 Book of Abstracts

Comparing residual lives and inactivity times by transform

stochastic orders

Antonio Arriaza, Miguel Angel Sordo and Alfonso Su´arez-Llorens

Dpto. Estad´ıstica e Investigaci´on Operativa, Universidad de C´adiz, Spain

Comparisons of residual lives and inactivity times is an important topic in reliability. In

this framework, our purpose is twofold. First, we interpret a family of stochastic orders,

known in the literature as transform stochastic orderings, in terms of stochastic compar-

isons among residual lives and inactivity times at quantiles. Second, we introduce and

study a new stochastic order that occupies an intermediate position between two of these

transform orderings, namely, the convex order and the star order. As an application of

these results, we provide new relationships among some classes of life distributions, in-

cluding characterizations of the IHR and DHR classes in terms of aging properties of their

residual lives at any time.

References

1. Ahmad I.A. (2005) Further results involving the MIT order and the IMIT class. Prob.

Eng. Inf. Sci. 19, 377–395.

2. Barlow R.E., Proschan F. (1975) Statistical theory of reliability and life testing. Holt,

Reinhart and Winston, Inc.

3. Belzunce F., Candel J., RuizFeller J.M. (1996) Dispersive orderings and characterizations

of ageing classes. Statist. Prob. Lett. 28, 321–327.

4. Belzunce F. (1999) On a characterization of right spread order by the increasing convex

order. Statist. Prob. Lett. 45, 103–110.

5. Belzunce F., Hu T., Khaledi B. (2003) Dispersion-type variability orders. Prob. Eng. Inf.

Sci. 17, 305–334.

6. Gilchrist W.G. (2000) Statistical Modelling with Quantile Functions. Boca Raton, Florida,

USA: Chapman and Hall/CRC Press.

7. Kayid M. and Ahmad I.A. (2004) On the mean inactivity time ordering with reliability

applications. Prob. Eng. Inf. Sci. 18, 395–409.

8. Kayid M., Izadkhah S. (2014) Mean inactivity time function, associated orderings and

classes of life distributions. IEEE Trans. Rel. 63, 593–602.

9. Kochar S.C., Wiens D. (1987) Partial orderings of life distributions with respect to their

aging properties. Nav. Res. Logist. 34, 823–829.

10. Mu˜noz–P´erez J. (1990) Dispersive ordering by the spread function. Statist. Prob. Lett.

10, 407–410.

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Workshop SMART 2016 Book of Abstracts

L´evy processes with Poisson and Gamma times

Luisa Beghin

Dipartimento di Scienze Statistiche, Sapienza University of Rome, Italy

We consider L´evy processes driven by an independent random time, which will be rep-

resented by a Poisson or a Gamma process, endowed with a drift. Our attention will

be addressed to the semigroups and the inﬁnitesimal generators of these processes. In

particular, when the leading process is an α-stable, the governing equation is expressed in

terms of new pseudo-diﬀerential operators involving the Riesz-Feller fractional derivative

of order α∈(0,2]. The special case α= 2 is particularly interesting because it concerns

the Brownian motion with randomly intermitting times.

Thus we deal with extensions of the space-fractional diﬀusion equation, which is satisﬁed

by the density of a stable process: the ﬁrst case considered here is obtained by adding a

fractional exponential diﬀerential operator. We prove that this produces a random addi-

tional term in the time-argument of the corresponding stable process, which is represented

by the so-called Poisson process with drift. Analogously, if we add, to the space-fractional

diﬀusion equation, a logarithmic diﬀerential operator involving the Riesz-derivative, we

obtain, as a solution, the transition semigroup of a stable process subordinated by an

independent gamma subordinator with drift. Finally, we show that a non-linear exten-

sion of the space-fractional diﬀusion equation is satisﬁed by the transition density of the

process obtained by time-changing the stable process with an independent linear birth

process with drift.

References

1. Beghin L. (2014) Geometric stable processes and fractional diﬀerential equation related

to them, Electron. Commun. Probab.,19, no. 13, 1–14.

2. Beghin L. (2015) Fractional gamma and gamma-subordinated processes, Stoch. Anal.

Appl., 33, no. 5, 903–926.

3. Beghin L. (2016) Fractional diﬀusion-type equations with exponential and logarithmic

diﬀerential operators, arXiv :1601.01476.

4. Beghin L., D’Ovidio M. (2014) Fractional Poisson process with random drift, Electron. J.

Probab., 19, no. 122, 1–26.

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Workshop SMART 2016 Book of Abstracts

Approximation of Markov Chains by hybrid switching jump

diﬀusion processes

Enrico Bibbona and Roberta Sirovich

Department of Mathematics, University of Torino, Italy

There is a large body literature concerning diﬀusions and other approximations for large

state space continuous time Markov Chains. In two recent papers [1,2] we proposed an

hybrid switching jump diﬀusion approximation that aims at being an extension of the

diﬀusion approximation proposed in [3]. In particular we succeed in approximating the

discrete model even when the process reaches the boundaries of the state space with non-

negligible probability. We study the error of the approximation of the Markov Chain

theoretically and the results are illustrated on few examples drawn from biological system

modelling.

References

1. Beccuti M., Bibbona E., Horvath A., Sirovich R., Angius A., Balbo G. (2014) Analysis of

Petri Net models through Stochastic Diﬀerential Equations. Proceedings of the Interna-

tional Conference on application and theory of Petri nets and other models of concurrency

(ICATPN14), Springer LNCS vol. 8489, pp. 273–293.

2. Angius A., Balbo G., Beccuti M., Bibbona E., Horvath A., Sirovich R. (2015) Approximate

analysis of biological systems by hybrid switching jump diﬀusion. Theoretical Computer

Science 587, 49–72.

3. Kurtz T.G. (1976) Limit theorems and diﬀusion approximations for density dependent

Markov Chains. Stochastic Systems: modeling, Identiﬁcation and Optimization I, 67–78.

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Some notes on stochastic diﬀusion equations for neurons

Stefano Bonaccorsi

Department of Mathematics, University of Trento, Italy

Neurons continuously perform computations while converting synaptic inputs into action

potential output. In some recent papers, we investigates how the spatial extension of the

dendritic tree inﬂuences the responses of a single neuron or a whole system of neurons.

In this talk, we shall brieﬂy discuss some models of synaptic activities, in order to get a

physically meaningful and mathematically treatable model.

Then we review some techniques of solving evolution systems of equations with inhomoge-

neous boundary conditions and we apply these results to neurons and systems of neurons.

References

1. Bonaccorsi S, Mugnolo D. (2010) Existence of strong solutions for neuronal network dy-

namics driven by fractional Brownian motions. Stochastics and Dynamics 10(3), 441–464.

2. Bonaccorsi S., Ziglio G. (2014) A variational approach to stochastic nonlinear diﬀusion

problems with dynamical boundary conditions. Stochastics 86(2), 218–233.

3. Bonaccorsi S., Ziglio G. (2006) A semigroup approach to stochastic dynamical boundary

value problems. IFIP International Federation for Information Processing 202, 55–65.

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Workshop SMART 2016 Book of Abstracts

Mathematical modeling of tumor-driven angiogenesis

Vincenzo Capasso

ADAMSS (Advanced Applied Mathematical and Statistical Sciences)

Universit´a degli Studi di Milano, Italy

In the mathematical modelling of tumor-driven angiogenesis, the strong coupling between

the kinetic parameters of the relevant stochastic branching-and-growth of the capillary

network, and the family of interacting underlying ﬁelds is a major source of complexity

from both the analytical and computational point of view. Our main goal is thus to ad-

dress the mathematical problem of reduction of the complexity of such systems by taking

advantage of its intrinsic multiscale structure; the (stochastic) dynamics of cells will be

described at their natural scale (the microscale), while the (deterministic) dynamics of

the underlying ﬁelds will be described at a larger scale (the macroscale). In this pre-

sentation, starting from a conceptual stochastic model including branching, elongation,

and anastomosis of vessels, we discuss the possible derivation of a deterministic mean

ﬁeld approximation of the vessel densities, leading to deterministic nonlinear partial dif-

ferential equations for the underlying ﬁelds. The propagation of chaos assumption on a

single replica is criticized, as opposed to the usual law of large numbers applied to a large

number of replicas. Outcomes of relevant numerical simulations will be presented.

References

1. Capasso V., Morale, D. (2009) Stochastic modelling of tumour-induced angiogenesis. J.

Math. Biol. 58, 219–33.

2. Capasso V., Morale D., Facchetti, G. (2012) The role of stochasticity for a model of retinal

angiogenesis. IMA J. Appl. Math. 77, 729–747.

3. Bonilla L.L., Capasso V, Alvaro M., and Carretero M. (2014) Hybrid modeling of tumor-

induced angiogenesis. Phys. Rev. E 90, 062716.

4. Terragni F., Carretero M., Capasso V., Bonilla L.L. (2015) Stochastic model of tumor-

induced angiogenesis: ensemble averages and deterministic equations. Submitted.

5. Bonilla L.L., Capasso V., Alvaro M., Carretero M., Terragni F. (2015) On the mathemat-

ical modelling of tumor-induced angiogenesis. Submitted.

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Workshop SMART 2016 Book of Abstracts

The F¨ollmer-Schweizer decomposition under incomplete

information and ﬁnancial applications

Claudia Cecia, Katia Colaneriaand Alessandra Cretarolab

aDepartment of Economics, University “G. D’Annunzio” of Chieti-Pescara, Viale Pindaro, 42,

I-65127 Pescara, Italy

bDepartment of Mathematics and Computer Science, University of Perugia, Via Vanvitelli, 1,

I-06123 Perugia, Italy

The aim is to derive and characterize the F¨ollmer -Schweizer decomposition of a square

random variable with respect to a given semimartingale under restricted information.

The F¨ollmer-Schweizer decomposition has a relevant application in ﬁnance. Indeed, un-

der suitable assumptions, the integrand in the decomposition of the square integrable

random variable representing the discounted payoﬀ of a given European type contingent

claim, provides the locally risk-minimizing hedging strategy in an incomplete ﬁnancial

market driven by semimartingales. In a partial information framework, where agents

have a limitative knowledge on the market, the F¨ollmer-Schweizer decomposition under

restricted information plays an analogous role. We discuss both the cases where the ran-

dom variable is observable or not observable and for partially observed Markovian models

we characterize the integrand of the F¨ollmer -Schweizer decomposition by means of ﬁlter-

ing problems.

References

1. Ceci C., Colaneri K., Cretarola A. (2015) Local risk-minimization under restricted infor-

mation on asset prices. Electronic Journal of Probability 20(96) 1–30

http://arxiv.org/abs/1312.4385

2. Ceci C., Colaneri K., Cretarola A. (2015) The F¨ollmer-Schweizer decomposition under

incomplete information. Submitted

http://arxiv.org/abs/1511.05465

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Workshop SMART 2016 Book of Abstracts

On a fractional growth process with two kinds of jumps

Antonio Di Crescenzo, Barbara Martinucci and Alessandra Meoli

Dipartimento di Matematica, University of Salerno, Italy

Fractional order diﬀerential equations play a relevant role in the modeling of memory-

dependent phenomena, since the deﬁnition of fractional derivative involves an integration

which is a non-local operator. The need of dealing with processes exhibiting power-

law behavior and multiple occurrences of events stimulates us to investigate fractional

Poisson-type processes subject to multiple kinds of jumps, especially in view of their po-

tential applications in several ﬁelds of science and engineering. Therefore, we consider

a suitable fractional jump process describing growth phenomena, that may be viewed as

a counting process characterized by 2 kinds of jumps with size 1 and 2. We obtain the

probability generating function and the probability law of the process, expressed in terms

of the generalized Mittag-Leﬄer function. The mean, variance, and squared coeﬃcient of

variation are also provided.

References

1. Di Crescenzo A., Martinucci B., Meoli A. (2015) Fractional growth process with two kinds

of jumps. In: R. Moreno-Diaz et al. (Eds.) Computer Aided Systems Theory – EURO-

CAST 2015, Lecture Notes in Computer Science 9520, 158–165, Springer International

Publishing Switzerland.

2. Beghin L., Macci C. (2014) Fractional discrete processes: compound and mixed Poisson

representations. J. Appl. Prob. 51, 19–36.

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Workshop SMART 2016 Book of Abstracts

Multivariate Gini-type index

Antonio Di Crescenzoaand Motahareh Parsab

aUniversit`a degli Studi di Salerno, Italy

bFerdowsi University of Mashhad, Iran

Recently, Gini-type index (GT) has been introduced (Kaminsky and Krivtsov, 2010) to

assess the degree of ageing or rejuvenating of the repairable or non-repairable systems.

The GT for a system working in [0, t] is introduced as

C(t) = 1 −Rt

0H(u)du

0.5t H(t),

in which H(t) represent the cumulative hazard rate function.

The extension of this index for higher dimensions does not appear to be clearly resolved.

Shaked and Shanthikumar (1987), deﬁned the multivariate conditional hazard rate func-

tions, and also, described the total accumulated hazards for various components by their

failure time.

Here, we introduce possible extensions for the GT in the multivariate case according to

Shaked and Shanthikumar (1987) deﬁnitions. Besides, the properties of these deﬁnitions

are considered for some multivariate distributions representing the lifetime of dependent

components working in the same system.

References

1. Kaminsky M.P., Krivtsov V.V. (2010) A Gini-type index for ageing/rejuvenating ob-

jects. In: Mathematical and Statistical Models and Methods in Reliability: Applications

to Medicine, Finance, and Quality Control (Eds. Rykov V.V., Balakrishnan N., Nikulin

M.S.) Statistics for industry and Technology, Birkh¨auser, pp. 133–140.

2. Shaked M., Shanthikumar G. (1987) The multivariate hazard construction. Stoch. Proc.

Appl. 24, 241–258.

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Workshop SMART 2016 Book of Abstracts

A new model of population growth

Antonio Di Crescenzo and Serena Spina

Dipartimento di Matematica, Universit`a degli Studi di Salerno

Via Giovanni Paolo II, 132, I-84084, Fisciano (SA), Italy

We introduce a new model of population growth, which is able to describe growth phe-

nomena sharing some characteristics of the Gompertz and Korf laws:

N(t) = yexp (α

βh1−(1 + t)−βi), t > 0, N(0) = y > 0, α, β > 0.

The proposed model has the same carrying capacity of the previous ones, but is able

to capture diﬀerent evolutionary dynamics. We investigate the main properties of N(t),

with reference to the correction function, the relative growth rate, the inﬂection point, the

maximum speciﬁc growth rate, the lag time and the threshold crossing problem. Some

comparison with Gompertz and Korf models are also studied. We discuss some data an-

alytic examples that pinpoint certain features of the proposed model.

We further deﬁne and study the corresponding time non-homogeneous birth-death process

X(t), whose mean satisﬁes the equation of N(t). We study the transition probabilities,

the mean, the variance and the population extinction probability of X(t). Our study is

supported by a scrutinized analysis of the model behaviour for diﬀerent choices of the

involved parameters.

References

1. Lindsey J.K. (2004) Statistical Analysis of Stochastic Processes in Time. Cambrige Uni-

versity Press, New York.

2. Parthasarathy P.R., Krishna Kumar B. (1991) A birth and death process with logistic

mean population. Comm. Stat. Theory Meth. 20, 621–629.

3. Ricciardi L.M. (1986) Stochastic Population Theory: Birth and Death Processes, In:

Mathematical Ecology, 155–190. Biomathematics 17, Springer, Berlin.

4. Tan W.Y., Piantadosi S. (1991) A stochastic growth processes with application to stochas-

tic logistic growth. Statistica Sinica 1, 527–540.

5. Tsoularis A., Wallace J. (2012) Analysis of logistic growth models. Math. Biosci. 179,

21–55.

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Workshop SMART 2016 Book of Abstracts

Stochastic variability of synaptic responses

Vito Di Maio and Silvia Santillo

Istituto di Scienze Applicate e Sistemi intelligenti del CNR, Pozzuoli (NA), Italy

Excitatory synaptic response shows a large variability which is inﬂuenced by several fac-

tors both of pre and post-synaptic origin. The average number of synapses on a single

hippocampal CA1 pyramidal neuron is ∼3×104[2]. Each synapse on the dendritic tree is

then inﬂuenced by the electrical activity of other synapses. We have simulated the single

synaptic event in terms of a diﬀerence of two exponential as follows:

V(t) = −k(e−t

τ1−e−t

τ2) (1)

where kmodulates the Excitatory Post Synaptic Potential (EPSP) peak amplitude and τ1

and τ2are the rising and decay time constants of the EPSP. Although k,τ1and τ2can be

all random variables, we have used their mean values for a typical EPSP in order to study

the eﬀect of synaptic population and of electrical activity of the cell on the stochastic

variability of the single synaptic event [1].

We have tested a sinusoid wave which can well simulate the eﬀect of a ﬁltered retrograde

spike

Vs(t) = V(t) + αsin(ωt +φ) (2)

where αis the wave amplitude which depends on the distance of the synapse with respect

to the soma, ωt is the frequency and φis the phase of the sinusoid wave. We have

analyzed the variability of the EPSP as function of the amplitude, frequency and phase

of the background wave.

In addition we have tested the eﬀect of a gaussian noise G(µ, σ) both with respect to a

ﬁxed level of resting potential

Vw(t) = V(t) + G(µ, σ) (3)

and in a cooperation with the sinusoid wave

Vw,s(t) = V(t) + αsin(ωt +φ) + G(µ, σ) (4)

The variability of the EPSPs has been analyzed as function of the stochastic processes

occurring outside the synapse depending on the frequency and phase of the noise.

References

1. Di Maio V., L´ansk´y P., Rodriguez R. (2004) Diﬀerent type of noise in leaky integrate-

and-ﬁre model of neuronal dynamics with discrete periodical input. General Physiology

and Biophysics 23, 21–38.

2. Ventriglia F., Di Maio V. (2006) Multisynaptic activity in a pyramidal neuron model and

neural code. Biosystems 86, 18–26.

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Sample estimations and numerical evaluations of ﬁring neural

activity

Giuseppe D’Onofrioa, Enrica Pirozziaand Marcelo O. Magnascob

aDipartimento di Matematica e Applicazioni, Universit`a di Napoli Federico II, Italy

bLaboratory of Mathematical Physics, The Rockefeller University, New York, USA

The ﬁring activity of stimulated neurons is aﬀected by temporal structure of the injected

current. In [1] Taillefumier and Magnasco focus on the following stochastic leaky integrate-

and-ﬁre (sLIF) model for a single neuron

dX(t) = −αX(t)dt +dC (t) + σdW (t), α > 0, X(t0) = r < l, t ≥t0(5)

in the presence of a constant threshold land a H¨older-continuous load function C(t) such

that dC(t) = I(t)dt.I(t) is a frozen noise conveying random current pulses of varying

amplitudes and a constant current input. The authors generate numerically a train of

spikes of the Ornstein-Uhlenbeck X(t) as successive ﬁrst passage times (FPTs) to an ef-

fective boundary L(t) = l−Zt

t0

e−α(t−s)dC(s).The determination of FPT density of these

successive spikes with a particular reset condition is called recurring-passage problem [1].

In order to address the model (5), we propose a stochastic model for the prediction of the

successive spikes that constitute the train by means of FPTs of a succession of Gauss-

Markov processes with a particular mean function involving the FPT already occurred

([2],[3]). In [1] the H¨older exponent Hof the input signal is connected with the regime of

the ﬁring activity and the authors show evidences of a phase transition of the probability

distribution of observing a given instantaneous ﬁring rate. The computed cumulative

distributions of the spiking times for diﬀerent exponents Hcoincide even across the phase

transition. This kinematic property of the recurring problem turns out to be not depen-

dent on the shape of the boundary. We verify if it is valid for the FPT through a constant

boundary using the method of moments, the results of [2], [3] and [4].

References

1. Taillefumier T., Magnasco M.O. (2014) A transition to sharp timing in stochastic leaky

integrate-and-ﬁre neurons driven by frozen noisy input. Neural Comput. 26(5), 819–859.

2. D’Onofrio G., Pirozzi E. (2016) Successive spike times predicted by a stochastic neuronal

model with a variable input signal Math. Biosci. Engin. 13(3), to appear.

3. D’Onofrio G., Pirozzi E., Magnasco M.O. (2015) Towards stochastic modeling of neuronal

interspike intervals including a time-varying input signal, Lecture Notes in Computer Sci-

ence - EUROCAST 2015,9520, 166–173.

4. Ricciardi L.M., Sato S. (1988) First-passage-time density and moments of the Ornstein-

Uhlenbeck process. J. Appl. Prob. 25(1), 43–57.

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Workshop SMART 2016 Book of Abstracts

Revisiting some macroscopic limit results for particle systems

under the view of modelling tumor growth

Franco Flandoli

Dipartimento di Matematica, Universit`a di Pisa, Italy

When considering an aggregate of tumor cells, some of the main phenomena which are

observed are proliferation, interaction between cells and environment, transport, diﬀu-

sion, genetic changes. Some of these features can be described by stochastic diﬀerential

equations and jump processes. When the number of cells is very large, a macroscopic

description by means of partial and ordinary diﬀerential equations becomes relevant, in

particular for simulation purposes. We summarize some rigorous and heuristic results

in this direction. In particular, we discuss the question of proliferation and change of

species (see for instance 1, 3) and the question of interaction (see for instance 2, 4). The

limit PDE for proliferation is relatively clear, although the rigorous justiﬁcation is only

partial. In the case of interactions like volume-constraint and membrane adhesion, the

limit PDE is on the contrary non clear yet, so we discuss some approximate models and

some conjectures supported by numerical simulations.

References

1. Flandoli F., Leimbach M., Olivera C., The FKPP equation as a macroscopic limit of

proliferating particles. In preparation.

2. Oelschl¨ager K. (1985) A law of large numbers for moderately interacting diﬀusion pro-

cesses. Zeitschrift fur Wahrsch. Verwandte Gebiete 69, 279–322.

3. Stevens A. (2000) The derivation of chemotaxis equations as limit dynamics of moderately

interacting stochastic many-particle systems. SIAM J. Appl. Math. 61, no. 1, 183–212.

4. Varadhan S.R.S. (1991) Scaling limit for interacting diﬀusions. Comm. Math. Phys. 135,

313–353.

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Workshop SMART 2016 Book of Abstracts

Some examples of the interplay between Probability and Number

Theory

Rita Giuliano

Department of Mathematics, University of Pisa, Italy

Probability theory have been often used to study problems in number theory: for in-

stance the so–called “probabilistic method” is a powerful tool which traces back to P.

Erd¨os. More recently also the large deviations theory has been applied in number theo-

retical settings; we provide a list of references here below (papers from [5] to [10]). As the

authors of [5] say, “it might have the potential to become a useful tool in analytic number

theory”. In my talk I shall illustrate the main results of the papers [1], [2], [3], [4], which

are in the same circle of ideas. All these papers are in collaboration with Claudio Macci.

References

1. Giuliano R., Macci C. (2015) Asymptotic results for weighted means of random variables

which converge to a Dickman distribution, and some number theoretical applications.

ESAIM Probab. Stat. 19, 395–413.

2. Giuliano R., Macci C., Asymptotic results for a class of triangular arrays of multivariate

random variables with Bernoulli distributed components. Submitted.

3. Giuliano R., Macci C., Convergence results in the generalized Cramer’s model for prime

numbers. In preparation.

4. Giuliano R., Macci C., Convergence results for logarithmic means in the generalized

Cramer’s model for prime numbers. In preparation.

5. Mehrdad B., Zhu L. (2013) Moderate and Large Deviations for the Erd˝os-Kac Theorem.

http://arxiv.org/abs/1311.6180.pdf.

6. Lulu Fang (2015) Large and moderate deviation principles for alternating Engel expan-

sions. Journal of Number Theory 156, 263–276.

7. Wei Hu (2015) Moderate deviation principles for Engel’s, Sylvester’s series and Cantor’s

products. Stat. Prob. Lett. 96, 247–254.

8. Zhu L. (2014) On the large deviations for Engel’s, Sylvester’s series and Cantor’s products.

Electron. Comm. Probab. 19(2), 1–9.

9. Mehrdad B., Zhu L. (2013) Limit theorems for empirical density of greatest common

divisors. http://arXiv:1310.7260.pdf.

10. Fern´andez J.L., Fern´andez P. (2013) Asymptotic normality and greatest common divisors.

http://arxiv.org/pdf/1302.2357.pdf.

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Workshop SMART 2016 Book of Abstracts

Asymptotic results for runs and empirical cumulative entropies

Rita Giulianoa, Claudio Macciband Barbara Pacchiarottib

aDepartment of Mathematics, University of Pisa, Italy

bDepartment of Mathematics, University of Rome Tor Vergata, Italy

We prove large and moderate deviations results (see e.g. [1]) for two sequences of estima-

tors based on the order statistics and, more precisely, on spacings. In the ﬁrst case we

deal with runs, and we have sums of independent Bernoulli distributed random variables.

In the second case we deal with empirical cumulative entropies, and we have linear com-

binations of independent exponentially distributed random variables.

References

1. Dembo A., Zeitouni O. (1998) Large deviations techniques and applications. Second edi-

tion. Springer, New York.

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Workshop SMART 2016 Book of Abstracts

Size biased couplings and the spectral gap for random regular

graphs

Larry Goldstein

Department of Mathematics, University of Southern California, Los Angeles, USA

The second largest absolute eigenvalue λof a graph’s adjacency matrix is closely related

to its expansion properties. Bounding λfor Gn,d, a uniformly chosen graph from the

collection of all d-regular graphs on nvertices, has been a major topic of research over the

last thirty years. A conjecture by Vu states that as n, and optionally d, tend to inﬁnity,

the second eigenvalue λis bounded by C√dwith probability tending to 1, regardless of

the growth of d. This bound was formerly known to hold only if d=o(n1/2). We prove

that it holds as long as d=O(n2/3). We use the Kahn-Szemer`edi approach, which is

based on showing concentration for Rayleigh quotients of the graph’s adjacency matrix.

To prove that the required concentration estimates hold, we extend previous work for

showing concentration via size biased couplings to cases where the coupling may be un-

bounded. This is joint work with Nicholas Cook and Tobias Johnson.

References

1. Cook N., Goldstein L., Johnson T. (2015) Size biased couplings and the spectral gap for

random regular graphs. http://arxiv.org/abs/1510.06013

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Workshop SMART 2016 Book of Abstracts

Stochastic single vehicle routing problem with ordered customers

Epaminondas G. Kyriakidisa, Theodosis D. Dimitrakosb

and Constantinos C. Karamatsoukisc

aDepartment of Statistics, Athens University of Economics and Business, Patission 76, 10434,

Athens, Greece

bDepartment of Mathematics, University of the Aegean, Karlovassi, 83200, Samos, Greece

cHellenic Army Academy, Department of Mathematics and Engineering Sciences, 16673, Vari,

Greece

We consider a set of nodes V={0,...,N}with node 0 denoting the depot and nodes

1,...,N corresponding to customers. There are two similar but not identical products to

be delivered to the customers. We refer to these products as product 1 and product 2.

The items of both products are of the same size. It is assumed that the depot contains

enough items of both products to satisfy the demands of all customers. The customers

are serviced in the order 1,...,N by a vehicle, which may carry any quantity of product

i∈ {1,2}provided that its total capacity Qis not exceeded. The vehicle starts its route

from the depot with a total load of Qitems of both products and after servicing all

customers it returns to the depot. We denote by cj,j +1,j= 1,...,N −1, the travel cost

between customers jand j+ 1, and by cj0,c0j,j= 1,...,N, the travel cost between

customer jand the depot and the cost between the depot and customer j, respectively.

These costs can be considered as the costs of the gasoline that the vehicle needs to

cover the distances between consecutive customers or the distances between customers

and the depot. We naturally assume that these costs are symmetric and satisfy the

triangle inequality, i.e. ci0=c0i, i = 1,...,N and ci,i+1 ≤ci0+c0,i+1 , i = 1,...,N −1.

Each customer j∈ {1, . . . , N}demands either product 1 or product 2. We assume that

customer j∈ {1,...,N}prefers product 1 with known probability pjand product 2

with probability 1 −pj. The actual preference of each customer becomes known when

the vehicle visits the customer. The number of items that each customer demands is a

discrete random variable ξjwith known distribution. The actual demand of each customer

cannot exceed the vehicle capacity, i.e. max1≤j≤Nξj≤Q. If the vehicle contains a smaller

quantity of the product that a customer prefers, it is permissible (but not compulsory)

to deliver items of the other product. It is assumed that there is a penalty cost πj,

j= 1,...,N, if the vehicle delivers one item of the product that is not preferred by

customer j. The vehicle is allowed during its route to return to the depot to restock with

items of both products. The objective is to ﬁnd the routing strategy that minimizes the

expected total cost among all possible routing strategies. It is possible to ﬁnd the optimal

routing strategy using a suitable dynamic programming algorithm. It is also possible to

prove that the optimal routing strategy has a speciﬁc threshold-type structure.

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Workshop SMART 2016 Book of Abstracts

The impact of degree variability on connectivity properties of

large networks

Lasse Leskel¨a

Department of Mathematics and Systems Analysis, Aalto University, Finland

The goal of this work is to study how increased variability in the degree distribution im-

pacts the global connectivity properties of a large network. We approach this question

by modeling the network as a uniform random graph with a given degree sequence. We

analyze the eﬀect of the degree variability on the approximate size of the largest connected

component using stochastic ordering techniques. A counterexample shows that a higher

degree variability may lead to a larger connected component, contrary to basic intuition

about branching processes. When certain extremal cases are ruled out, the higher degree

variability is shown to decrease the limiting approximate size of the largest connected

component. (Based on joint work with Hoa Ngo, Aalto University).

References

1. Leskel¨a L., Ngo H. (2015) The impact of degree variability on connectivity properties

of large networks. Proc. 12th Workshop on Algorithms and Models for the Web Graph

(WAW). Preprint: arXiv:1508.03379

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Workshop SMART 2016 Book of Abstracts

Two stochastic dominance criteria based on tail comparisons

Julio Muleroa, Miguel Angel Sordob, Marilia C. de Souzaband Alfonso Su´arez-Llorensb

aFacultad de Matem´aticas, Universidad de Alicante, Spain

bDpto. Estad´ıstica e I.O., Universidad de C´adiz, Spain

Actuarial risks and ﬁnancial asset returns are typically heavy tailed. In this paper, we

introduce two stochastic dominance criteria, called the right tail order and the left tail

order, to compare stochastically these variables. The criteria are based on comparisons of

expected utilities, for two classes of utility functions that give more weight to the right or

the left tail (depending on the context) of the distributions. We study their properties,

applications and connections with other classical criteria, including the increasing convex

and the second order stochastic dominance. Finally, we rank some parametric families

of distributions and provide empirical evidence of the new stochastic dominance criteria

with an example using real data.

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Workshop SMART 2016 Book of Abstracts

Comparisons of residual lifetimes of coherent systems under

dependence

Jorge Navarro

Facultad de Matem´aticas, Universidad de Murcia, Spain

Consider a coherent system with a ﬁxed dependence structure (copula) between its com-

ponent lifetimes. Then we compare, by using diﬀerent criteria (stochastic orders), the

residual lifetime at time tof the system under two diﬀerent assumptions: when, at time

t, we know that all the components are working or when, at time t, when we just know

that the system is working. The comparison results obtained are distribution-free, that

is, they do not depend on the distributions of the component lifetimes. Some illustrative

examples are included. They prove that, in some cases, these residual system lifetimes

are not ordered. Even more, surprisingly, sometimes the second residual lifetime is better

than the ﬁrst one when the component lifetimes satisfy a given dependence model.

This work is supported by Ministerio de Econom´ıa y Competitividad of Spain under grant

MTM2012-34023-FEDER

References

1. Navarro J. (2015) Comparisons of the residual lifetimes of coherent systems under diﬀerent

assumptions. Submitted.

2. Navarro J., Durante F. (2015) Copula–based representations for the reliability of the

residual lifetimes of coherent systems with dependent components. Submitted.

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Workshop SMART 2016 Book of Abstracts

Gradient ﬂow to the optimal transport plan

Giovanni Pistone

de Castro Statistics, Collegio Carlo Alberto, Via Real Collegio 30, 10024 Moncalieri, Italy

This talk is based on work in progress joint with Luigi Malag`o (Shinshu University, Japan)

and Luigi Montrucchio (Universit`a di Torino and Collegio Carlo Alberto). A research

announcement has been done at an ICMS workshop, Edinburgh Sep 21–25, 2015.

1. Statistical bundle The statistical bundle is an extension of the exponential manifold

introduced by G. Pistone and C. Sempi (1995). In particular, we consider a ﬁnite state

space Ω and its positive probability functions, i.e. the points in the interior of the proba-

bility simplex, γ∈∆◦(Ω). For each such a γ, we consider the vector space Bγof random

variables Vwith EγV= 0. Each one dimensional regular statistical model θ7→ γ(θ) has

Fisher score Dγ(θ) = dlog γ(θ)/dθ ∈Bγ(θ). It follows that Bγis an expression of the

tangent space at γ. The statistical bundle is the set of couples (γ, V ), V∈Bγ, see [2].

Given a regular function F: ∆◦(Ω), its statistical gradient is a section of the statistical

bundle, γ7→ F′(γ)∈Bγ, such that dF (γ(θ))/dθ =Eγ(θ)(F′(γ(θ))Dγ(θ)). It corresponds

to S.i. Amari’s natural gradient. The gradient ﬂow of Fis the solution of the equation

Dγ(θ) = −F′(γ(θ). The gradient ﬂow equation is expected to go towards a minimum

point of F. The most interesting cases arise when the curve γ(·) is restricted to belong

to some smooth sub-model, either exponential or mixture, see eg [1].

2. Gradient ﬂow to optimal transport Assume Ω = Ω1×Ω2and denote by Γ◦(µ1, µ2)

the set of positive probability functions with given marginals µ1, µ2. Given a cost function

c: Ω, we consider minimising F(γ) = Eγ(c) under the condition γ∈Γ◦(µ1, µ2). The mini-

mum value of this problem for c(x, y) = |x−y|is the Gini’s dissimilarity index, while for

c(x, y) = |x−y|2its square root is the Wasserstein 2-distance. The problem does not have

a minimum on positive probability function, but it does have a solution γ∗if we allow zero

probabilities. In this last case it is an instance of a linear programming problem whose

dual has been identiﬁed and studied by L. Kantorovich. There is a considerable body

research we cannot refer to here. We want to discuss an analytic approach, consisting in

computing the statistical gradient of the minimum expected cost problem and looking for

a gradient ﬂow trajectory going from the product of the marginals µ1⊗µ2toward the

solution γ∗. To this aim we compute the sub-bundle of scores when a curve is restricted

to have assigned marginals and show that such scores at each point are random variables

of the interaction type, i.e. they are γ-orthogonal to constants and to simple eﬀects.

References

1. Malag`o L., Pistone G. (2015) Natural Gradient Flow in the Mixture Geometry of a

Discrete Exponential Family. Entropy 17(6), 4215–4254.

2. Pistone G. (2013) Examples of Application of Nonparametric Information Geometry to

Statistical Physics. Entropy 15(10), 4042–4065.

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Workshop SMART 2016 Book of Abstracts

An unpredictable talk on prediction, the hot hand in basketball,

and some statistical principles

Yosef Rinott

Department of Statistics, The Hebrew University, Jerusalem, Israel

The old discussion of whether there exists a ‘hot hand’ phenomenon in basketball and

related questions arise from time to time with new analyses of the data. I will discuss

some new aspects, which raise some basic questions on statistical principles.

References

1. Rinott Y., Bar-Hillel M. (2015) Comments on a ‘Hot Hand’ Paper by Miller and Sanjurjo.

SSRN. http://ssrn.com/abstract=2642450

2. Miller J.B., Sanjurjo A. (2015) Surprised by the gambler’s and hot hand fallacies? A truth

in the Law of Large Numbers. SSRN. http://ssrn.com/abstract=2627354

3. Gilovich T., Vallone R., Tversky A. (1985) On the misperception of random sequences.

Cognitive Psychology 17, 295–314.

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Workshop SMART 2016 Book of Abstracts

First passage times of two-dimensional correlated diﬀusion

processes with application to neural modelling

Laura Sacerdote

Dept. of Mathematics “G. Peano”, University of Torino, Italy

A large literature concerns First Passage Time problems (FPT) for one dimensional dif-

fusion processes through constant or time dependent boundaries. Analytical, numerical

as well as simulation results are nowadays available and their application plays an impor-

tant role in many contexts ranging from physics to engineering, psychology or biology.

Motivations for such study came from the modelling of the ﬁring time of a single neuron.

In this framework, a diﬀusion process models the membrane potential evolution and we

identify the ﬁring time of the neuron with the times in which the process crosses a suitable

boundary.

Nowadays an increasing number of modeling instances requests to determine the joint

density of the FPTs of multivariate diﬀusion processes in presence of constant or time

dependent boundaries. The theoretical diﬃculties of the bivariate and multivariate case

are similar and here we limit ourselves to the two dimensional instance.

To deal with this problem we note that the joint density of two FPTs depends on the

joint density of the FPT of the ﬁrst crossing component and of the position of the second

crossing component before its crossing time [1]. First, we derive explicit expressions for

this and other quantities of interest in the case of a bivariate Wiener process. Then we

consider a bivariate diﬀusion process. We show that the densities of the position of the

second crossing component before its crossing time are solutions of a system of Volterra-

Fredholm integral equations of the ﬁrst kind. We propose a numerical algorithm to solve

it and we prove its convergence. Then we describe its use to evaluate the joint density of

the FPTs and we illustrate the application of the method through a set of examples on a

bivariate Ornstein-Uhlenbeck process.

These results are a ﬁrst tool for the study of a generalization of the Leaky Integrate and

Fire model for a single neuron to the case of a small neural network. The necessity to

model the joint behavior of two or more neurons suggested us the development of this

model. Its formulation implies to prove the weak convergence of marked point processes

generated by the crossings of multivariate jump process that tends to a diﬀusion [2]. We

ﬁnally brieﬂy discuss the use of available results on bivariate FPTs in this context and

we introduce some related open problems.

References

1. Sacerdote L., Tamborrino M., Zucca C. (2016) First passage times of two-dimensional

correlated processes: Analytical results for the Wiener process and a numerical method

for diﬀusion processes. Journal of Computational and Applied Mathematics 296, 275–292.

2. Tamborrino M., Sacerdote L., Jacobsen M. (2014) Weak convergence of marked point

processes generated by crossings of multivariate jump processes. Applications to neural

network modeling. Physica D: Nonlinear Phenomena 288, 45–52.

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Workshop SMART 2016 Book of Abstracts

Inference in the early phase of an epidemic

Gianpaolo Scalia Tombaaand Tom Brittonb

aDept. Mathematics, Univ. of Rome Tor Vergata, Italy

bDept. Mathematical Statistics, Univ. of Stockholm, Sweden

In several recent epidemic outbreaks (SARS, A(H1N1) ﬂu, Ebola,...), eﬃcient data col-

lection has allowed inference on important epidemic parameters such as R0, exponential

increase rate (r), generation time distribution characteristics, case fatality rate (CFR),

already in the early phase of spread (see e.g. [1]). However, statistical analysis of such

data poses interesting non-standard problems. These problems will be discussed and some

solution proposals presented.

References

1. WHO Ebola Response Team (2014) Ebola Virus disease in West Africa - The ﬁrst 9

months of the epidemic and forward projections. N Engl J Med 371, 1481-95.

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Workshop SMART 2016 Book of Abstracts

Some remarks on multivariate conditional hazard rates and

dependence modeling

Fabio L. Spizzichino

Department of Mathematics, University La Sapienza, Piazzale Aldo Moro, 5, 00185 Rome, Italy

In the absolutely continuous case, the joint distribution of nnon-negative random vari-

ables X1, ..., Xncan be characterized in terms of the so-called Multivariate Conditional

Hazard Rate (MCHR) functions. Such a characterization is alternative, but equivalent,

to the one based on the joint density function (or on other tools such as joint survival

function, marginal distributions and survival copula). These two types of characteriza-

tions imply however completely diﬀerent methods for building multivariate dependence

models and for describing stochastic dependence among the variables X1,...,Xn. The

method of the M.C.H.R. functions is, in fact, specially adapt to describe dynamic models

of dependence.

In this talk we will ﬁrst recall some basic deﬁnitions and provide some related comments.

On this basis, we shall later on establish a comparison between two special classes of

dependence models: those deﬁned in terms of conditional independence and those of the

type Load-Sharing (a property that can be directly deﬁned in terms of the M.C.H.R.

functions). Such a comparison can shed some light on topics of interest in both the two

ﬁelds of Reliability and Portfolio Credit Risk. In the ﬁnal part of the talk we will, in

particular, concentrate attention on the special cases when the afore-mentioned models

are exchangeable. We will be thus in a position to pointing out some relevant aspects

concerning concepts of positive dependence for vectors of exchangeable random variables.

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Workshop SMART 2016 Book of Abstracts

Notes

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Workshop SMART 2016 Book of Abstracts

Notes

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Workshop SMART 2016 Book of Abstracts

Notes

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Workshop SMART 2016 Book of Abstracts

Notes

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Workshop SMART 2016 Book of Abstracts

Notes

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Workshop SMART 2016 Book of Abstracts

PROGRAM

Thursday – January 21

8:15–8:45 bus from Salerno

9:00–9:15 registering

9:15–9:30 opening

Session 1 Chairperson: V. Capasso

9:30–9:50 Size biased couplings and the spectral gap for random regular graphs

L. Goldstein (pag. 17)

9:55–10:15 Revisiting some macroscopic limit results for particle systems under

the view of modelling tumor growth

F. Flandoli (pag. 14)

10:20–10:40 The F¨ollmer-Schweizer decomposition under incomplete information

and ﬁnancial applications

C. Ceci∗, K. Colaneri, A. Cretarola (pag. 8)

10:45–11:15 coffee break

Session 2 Chairperson: G. Scalia Tomba

11:15–11:35 Stochastic single vehicle routing problem with ordered customers

E.G. Kyriakidis∗, T.D. Dimitrakos, C.C. Karamatsoukis (pag. 18)

11:40–12:00 First passage times of two-dimensional correlated diﬀusion processes

with application to neural modelling

L. Sacerdote (pag. 24)

12:05–12:25 Gradient ﬂow to the optimal transport plan

G. Pistone (pag. 22)

12:30–12:55 On the inverse ﬁrst-passage-time problems for one-dimensional

diﬀusions

M. Abundo (pag. 1)

13:00–14:30 lunch break

Session 3 Chairperson: F. Flandoli

14:30–14:50 Comparisons of residual lifetimes of coherent systems under

dependence

J. Navarro (pag. 21)

14:55–15:15 New classes of priors based on stochastic orders and distortion

functions

J.P. Arias-Nicol´as, F. Ruggeri, A. Su´arez-Llorens∗(pag. 2)

15:20–15:40 Some remarks on multivariate conditional hazard rates and

dependence modeling

F.L. Spizzichino (pag. 26)

15:45–16:05 Some examples of the interplay between Probability and Number

Theory

R. Giuliano (pag. 15)

16:10–16:40 coffee break

16:40–17:30 Poster Session

17:30–18:00 bus to Salerno

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Workshop SMART 2016 Book of Abstracts

Poster Session

Thursday – January 21

•Comparing residual lives and inactivity times by transform stochastic orders

A. Arriaza∗, M.A. Sordo, A. Su´arez-Llorens (pag. 3)

•On a fractional growth process with two kinds of jumps

A. Di Crescenzo, B. Martinucci, A. Meoli∗(pag. 9)

•Multivariate Gini-type index

A. Di Crescenzo, M. Parsa∗(pag. 10)

•A new model of population growth

A. Di Crescenzo, S. Spina∗(pag. 11)

•Stochastic variability of synaptic responses

V. Di Maio∗, S. Santillo (pag. 12)

•Sample estimations and numerical evaluations of ﬁring neural activity

G. D’Onofrio∗, E. Pirozzi, M.O. Magnasco (pag. 13)

•Asymptotic results for runs and empirical cumulative entropies

R. Giuliano, C. Macci∗, B. Pacchiarotti (pag. 16)

•Two stochastic dominance criteria based on tail comparisons

J. Mulero, M.A. Sordo, M.C. de Souza∗, A. Su´arez-Llorens (pag. 20)

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Workshop SMART 2016 Book of Abstracts

PROGRAM

Friday – January 22

8:20–9:00 bus from Salerno

Session 4 Chairperson: F.L. Spizzichino

9:30–9:50 An unpredictable talk on prediction, the hot hand in basketball,

and some statistical principles

Y. Rinott (pag. 23)

9:55–10:15 Mathematical modeling of tumor-driven angiogenesis

V. Capasso (pag. 7)

10:20–10:40 L´evy processes with Poisson and Gamma times

L. Beghin (pag. 4)

10:45–11:15 coffee break

Session 5 Chairperson: L. Sacerdote

11:15–11:35 The impact of degree variability on connectivity properties of

large networks

L. Leskel¨a (pag. 19)

11:40–12:00 Some notes on stochastic diﬀusion equations for neurons

S. Bonaccorsi (pag. 6)

12:05–12:25 Approximation of Markov Chains by hybrid switching jump

diﬀusion processes

E. Bibbona, R. Sirovich∗(pag. 5)

12:30–12:55 Inference in the early phase of an epidemic

G. Scalia Tomba∗, T. Britton (pag. 25)

13:00–14:00 lunch break

14:00–14:15 closing

14:15–14:45 bus to Salerno

34

‎Generalized Gini-type Index

Abstract

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