Table Of ContentSpringer Proceedings in Mathematics & Statistics
Anatoliy Malyarenko
Ying Ni
Milica Rančić
Sergei Silvestrov Editors
Stochastic
Processes,
Statistical Methods,
and Engineering
Mathematics
SPAS 2019, Västerås, Sweden,
September 30–October 2
Springer Proceedings in Mathematics &
Statistics
Volume 408
This book series features volumes composed of selected contributions from
workshops and conferences in all areas of current research in mathematics and
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mostexcitingareasofmathematicalandstatisticalresearchtoday.
· · ·
Anatoliy Malyarenko Ying Ni Milica Rancˇic´
Sergei Silvestrov
Editors
Stochastic Processes,
Statistical Methods,
and Engineering
Mathematics
SPAS 2019, Västerås, Sweden,
September 30–October 2
Editors
AnatoliyMalyarenko YingNi
DivisionofMathematicsandPhysics DivisionofMathematicsandPhysics
MälardalenUniversity MälardalenUniversity
Västerås,Sweden Västerås,Sweden
MilicaRancˇic´ SergeiSilvestrov
DivisionofMathematicsandPhysics DivisionofMathematicsandPhysics
MälardalenUniversity MälardalenUniversity
Västerås,Sweden Västerås,Sweden
ISSN 2194-1009 ISSN 2194-1017 (electronic)
SpringerProceedingsinMathematics&Statistics
ISBN 978-3-031-17819-1 ISBN 978-3-031-17820-7 (eBook)
https://doi.org/10.1007/978-3-031-17820-7
MathematicsSubjectClassification:62P05,60H10,60F17,60K15,62M10,65C30,65K10
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Preface
This volume originated on the basis of selected contributions presented at the
international conference “Stochastic Processes and Algebraic Structures—From
TheoryTowardsApplications”(SPAS2019).Agroupofmathematicians,researchers
from related areas, and practitioners from industry, who contribute to the areas of
StochasticProcesses,StatisticalMethods,EngineeringMathematics,andAlgebraic
Structures,participatedintheconference,whichhasbeenorganizedbytheDivision
ofMathematicsandPhysicsattheMälardalenUniversityinVästerås,Sweden,and
hasbeenheldonSeptember30–October2,2019.
ThescopeofthevolumeisStochasticProcesses,StatisticalMethods,andEngi-
neeringMathematics.TheaccompanyingvolumecontainscontributionstoAlgebraic
Structures.
The purpose of the book is to highlight the latest advances in the above areas
and focus on the mathematical structures, models, concepts, problems, computa-
tionalmethods,andalgorithmsthatareimportantforapplicationsinvariousfields
ofscienceandsociety.
Thevolumeisdividedintothreeparts,accordingtotheareasofthebook’sscope
describedabove.
Part I, called “Stochastic Processes and Analysis”, contains 21 Chapters.
Chapter 1, written by Mohammed Albuhayri, Christopher Engström, Anatoliy
Malyarenko, Ying Ni, and Sergei Silvestrov, is concerned with the double-mean-
revertingmodelproposedbyJimGatheral.Theasymptoticexpansionoftheimplied
volatilityofaEuropeancalloptionwrittenonastockintheabovemodelisimproved
uptoorder3.
Chapter2,writtenbyMohammadJamsherAliandKalevPärna,studiestheruin
probabilityofthesumoftwoclassicalriskprocessesundertheassumptionthatthe
claimsizedistributionsareofphasetypeandthatthetwoPoissonprocessesofclaim
arrivalsarecorrelated.Anexactformulafortheruinprobabilityofthemergedrisk
processisconstructed.
Chapter3byTinNweAyeandLinusCarlssonstudiesanaquaticecologicalsystem
containingonefishspeciesandanunderlyingresource.Thismodelisinvestigatedin
boththedeterministicandthestochasticsettings.Theauthorsprovideestimatesfor
v
vi Preface
theexpectedoutcomeofpopulationproperties,measuresofdispersion,probability
ofextinction,andtherecoverypotentialofaspecies,amongothers.
Chapter4byDomingosDjinja,SergeiSilvestrov,andAlexBehakaniraTumwe-
sigyeisdevotedtopairsofintegraloperatorsonthespacesL overarbitrarymeasure
p
spaces.Theauthorsobtainconditionsonthekernelsofintegraloperatorstosatisfy
general covariance type commutation relations associated to polynomial actions
important in applications of noncommutative analysis, noncommutative geometry,
andoperatoralgebramethodsinstochasticprocesses,harmonicanalysis,quantum
physics,andengineering.
Chapter5byVitaliyGolomoziystudiesthefirstsimultaneoushittingoftheatom
bytwodiscrete-time,inhomogeneousMarkovchainswithvaluesinageneralphase
space.Bothconditionsfortheexistenceandcomputableboundsforthehittingtime’s
exponentialmomentareestablished.
Chapter6,writtenbyLuJin,MarkoDimitrov,andYingNi,considersthepricing
ofAmericanoptionswhentheunderlyingassetisgovernedbytheMarkovregime-
switchingprocess.Theexistenceofamonotonicoptimalexercisepolicywithrespect
totheholdingtime,assetprice,andeconomicconditionsisproved.
Chapter 7 by Ya. M. Khusanbaev and Kh. E. Kudratov gives the upper bounds
formomentsandcentralmomentsofbranchingprocessesinavaryingenvironment
startingwitharandomnumberofparticles.
InChap.8,writtenbyAbderrahimKitouniandFatihaMessaci,afunctionallaw
of the iterated logarithm for the increment functions of empirical processes with
twicecensoreddataisestablished.Stronglawsforkernelestimatorsofthedensity
andthefailurerateofthelifetimearealsoderived.
Chapter 9 by Pavlos Kolias and Alexandra Papadopoulou describes a DNA
sequence by a semi-Markov chain with discrete state space consisting of the four
nucleotides. Both strong and weak d-periodic and quasiperiodic behaviour of this
modelischaracterizedbyequationsinclosedanalyticform.Therelatedprobabilities
andthecorrespondingindexesareprovided.
In Chap. 10, Sergey Krasnitskiy, Oleksandr Kurchenko, and Olga Syniavska
proveaBaxter-typetheoremforGaussiangeneralizedrandomprocesseswithinde-
pendent values. Sufficient conditions for the singularity of probability measures
correspondingtosuchprocessesarealsogiven.
InChap.11,EugeneLebedev,VadymPonomarov,andHannaLivinskainvestigate
abivariateMarkovprocess.Thestatespaceoftheprocessisalatticesemi-strip.Such
amodeldescribestheservicepolicyofamulti-serverretrialqueueinwhichtherate
of repeated flow does not depend on the number of sources of repeated calls. The
ergodicityconditionsandavector-matrixrepresentationofsteadystatedistribution
areobtained.
InChap.12,AnatoliyMalyarenkoandHosseinNohrouzianusecubatureformulae
ofdegrees5and7onWienerspacetopriceEuropeanoptionsintheclassicalBlack-
Scholesmodel.Theobtainednumericalresultsarecomparedwiththewell-known
Nobelprize-awarded closedformsolution,andseveral importantproperties ofthe
cubaturemethodsareestablished.
Preface vii
In Chap. 13, Yuliya Mishura, Georgiy Shevchenko, and Sergiy Shklyar study
Volterra processes. The celebrated fractional Brownian motion appears here as a
particularcase.Thesmoothnesspropertiesoftheaboveprocesses,includingconti-
nuityandHölderproperty,areestablished.Theproblemofinverserepresentationof
thestandardBrownianmotionviaaVolterraprocessisinvestigated.
InChap.14,GiuliaDiNunno,YuliyaMishura,andKostiantynRalchenkostudy
the existence and uniqueness of solutions to stochastic differential equations with
VolterraprocessesdrivenbyLévynoise.Specialattentionisgiventotwokindsof
Volterra-Gaussian processes that generalize the compact interval representation of
fractionalBrownianmotiontostochasticequationswithsuchprocesses.
InChaps.15–17,TalatNazirandSergeiSilvestrovinvestigatetheexistenceand
otherpropertiesoffixedpoints,jointfixedpoints,andperiodicpointsformappings
andpairsofmappingsonmultiplicativemetricspacessatisfyingvariousgeneralized
contractionandcyclicconditions.
InChap.18,GodwinAmechiOkeke,MujahidAbbas,andSergeiSilvestrovintro-
ducearandomversionofsomeknownfastfixedpointiterativeprocesses,approxi-
matetherandomfixedpointofageneralizedrandomoperatorusingtheserandom
iterativeprocesses,andprovetheBochnerintegrabilityoftherandomfixedpoints
forthiskindofgeneralizedrandomoperatorsandthealmostsureT-stabilityofthe
aboverandomiterativeprocesses.
In Chap. 19, Salvador Cruz Rambaud presents mathematical results for the
absence of asset price bubbles by using, as an algebraic tool, a state-price deflator
across an infinite time horizon. A financial market with both uncertainty, a
finite number of corporate securities, and a countable number of trading dates is
investigatedthere.
José L. da Silva, Custódia Drumond, and Ludwig Streit give an account of the
formfactorsofpathsforacertainclassofnon-GaussianprocessesinChap.20.A
closedanalyticformfortheformfactorsandtheDebyefunctionareobtained.The
relation between the mean square end-to-end length and the radius of gyration is
explicitlyderived.
In Chap. 21, Dmitrii Silvestrov obtains necessary and sufficient conditions for
convergence in distribution and in Skorokhod J-topology for counting processes
generatedbyflowsofrareeventsforperturbedsemi-Markovprocesses.
PartII,called“StatisticalMethods”,contains7chapters.Itbeginswiththechapter
by Guglielmo D’Amico, Bice Di Basilio, Filippo Petroni, and Fulvio Gismondi
(Chap.22),inwhichtwodrawdown-basedriskmeasuresformanagingmarketcrisis
areconsidered.Thesetwomeasuresarethenanalysedusinghigh-frequencymarket
dataandsyntheticdatageneratedbyARMA,GARCH,andEGARCHmodels.
Chapter23byVladimirAnisimovandMatthewAustindealswithpatientenrol-
ment modelling and forecasting. Here, modelling of enrolment on different levels
and with restrictions are discussed, and new analytical techniques are proposed to
findthesolutiontothecorrespondingoptimizationproblem.
Chapter 24 by Collins Anguzu, Christopher Engström, Henry Kasumba, John
MageroMango,andSergeiSilvestrovdealswithcentralitymeasuresingraphtheory.
viii Preface
Algorithms are developed for recalculating two centrality measures, namely alpha
andeigenvectorcentralitymeasures,usinggraphpartitioningtechniques.
In Chap. 25 by Yuriy Kozachenko and Iryna Rozora, estimation of the impulse
response function of a time-invariant continuous linear system with a real-valued
impulse response function was considered. Statistical properties of this impulse
responsefunctionandcriteriononitsshapearegiven.
Chapters26and27,byAsaphKeikaraMuhumuza,KarlLundengård, Anatoliy
Malyarenko, Sergei Silvestrov, John Magero Mango, and Godwin Kakuba, are
devotedtousefulapplicationsofextremepointsforVandermondedeterminants.In
Chap.26,theextremepoints,optimizedonvarioussurfaces,areusedtoconductthe
risk-minimizationtaskinassetpricingandoptimalportfolioselection.InChap.27,
the extreme points maximize the Wishart probability distribution based on the
boundaryofthesymmetricconesinJordanalgebra.
Finally,Part2endswithChap.28byNataliyaShchestyukandSerhiiTyshchenko
proposing a new approach to option pricing. A concept of investor optimal price
is defined as the optimal decision of a investor maximizing expected profit. This
investor optimal pricing, integrated with risk management, is then conducted by
stochasticoptimization.
Part III, called “Engineering Mathematics”, contains 12 chapters. Chapter 29
by Magnus Ögren deals with the one-dimensional Stefan problem with a general
time-dependentboundaryconditionatthefixedboundary.Applyingdiscreterandom
walks,stochasticsolutionshavebeenobtainedandconfirmedagainstanalyticalor
numericalsolutions(FDM).
InChap.30,DoghonayArjmandexploresthepossibilityofintegratingperfectly
matched layers to the local wave equation. In particular, questions in relation to
accuracyandreducedcomputationalcostsareaddressed.Numericalsimulationsare
providedinasimplifiedone-dimensionalsettingtoillustratetheideas.
The objective of the work done by Imran M. Chandarki and Brijbhan Singh in
Chap.31wastorevisittheproblempertainingtoaverticallyflowingfluidpasseda
modelofathinverticalfininasaturatedporousmedium.Thegoverningequations
havebeensimplifiedusingthesimilaritytransformationtoyieldordinarydifferential
equations.Theseequationshavebeensolvedbyhomotopyanalysismethod(HAM).
InChap.32,Vucˇkovic´ etal.presentmodellingofapermanentmagnetshapedas
truncated cone and positioned in the vicinity of a body of finite dimensions made
of soft magnetic material. The force calculation between the permanent magnet
andsoftmagneticcylinderisperformedusingthehybridboundaryelementmethod
alongwithsemi-analyticalapproachbasedonfictitiousmagnetizationchargesand
discretizationtechnique.
Chapters33and32tackleproblemsrelatedtothepopulationdynamics.Specif-
ically, in Chap. 33, Loy Nankinga and Linus Carlsson deal with interactions of a
consumer-resource system with harvesting in which African Catfish consumes the
foodresource.ThedynamicsofthefoodresourceandtheAfricanCatfishresultin
asystemofordinarydifferentialequationscalledastage-structuredfishpopulation
model.Analysisofeightharvestingscenariosrevealedthatharvestinglargejuveniles
Preface ix
andsmalladultsunderequalharvestingratesgivesthehighestmaximumsustainable
yieldcomparedtootherharvestingscenarios.
In Chap. 34, Sam Canpwonyi and Linus Carlsson studied dynamics of forage
resourceandlivestockpopulationinagrasslandecosystemdescribingitbycoupled
ordinary differential equations. By solving this system, one is able to predict the
density-dependentpropertiesofthepopulationsincethesystemprovidesasomewhat
close-to-realitydescriptionofthenaturalandtraditionalgrazingsystem.
In Chaps. 35–38 by Prashant G. Metri and coauthors, numerical and analytical
methodsareappliedtoinvestigationofsolutionsofboundaryandinitialvalueprob-
lemsforsystemsofpartialdifferentialequationinfluidmechanicsandelectromag-
netism applications, including magnetohydrodynamic Casson nanofluid flow over
anonlinearstretchingsheetwithvelocityslipandconvectiveboundaryconditions,
mathematicalandcomputational analysisofMHDviscoelasticfluidflowandheat
transferoverstretchingsurfaceembeddedinasaturatedporousmedium,numerical
solutionofboundarylayerflowproblemofaMaxwellfluidpastaporousstretching
surface,andeffectofelectromagneticfieldonmixedconvectionoftwoimmiscible
conductingfluidsinaverticalchannel.
Finally, Chaps. 39 and 40 cover new stochastic digital measurement method
(SDMM)anditsroleindesigninglow-costdigitalhighprecisionpowergridelec-
tricalenergymeters.InChap.39,authoredbyVujicˇic´etal,mathematicalproperties
oftheSDMMaregiven.Practicalandusefulformulasarederivedthatconnectthe
measurementparameterswiththeprecision.Hardwareofatwo-bitSDMMissimple,
sosourcesofsystematicerrorscaneasilybeidentifiedanderrorcorrected.Inaddition,
simplehardwareenableslarge-scaleparallelizationofmeasurementsandprocessing.
UsingthemultibitSMI(StochasticMeasurementInstruments)mathematicalmodel
developedin39and40,aworkinghardwareprototypeofa4-bitSDEEM(Stochastic
DigitalElectricalEnergyMeter)wasbuiltandrigorouslytestedbyMarjanUrekar
andJelenaDjordjevic´Kozarov.Thisconfirmedthevalidityofthetheoreticalmodel
andthatSDEEMisanidealsolutionforaSmartMeterinSmartGridinIndustry4.0
applications,duetoitshighprecisionandaccuracy,highreliability,digitalcontrols
andeaseofinterfacingwithIoTandIIoT,simplehardware,andlowcost.
The volume is intended for researchers, graduate and Ph.D. students, and prac-
titionersintheareasofMathematics,Statistics,Finance,andEngineering,whoare
interestedinasourceforinspiration,cutting-edgeresearch,andapplications.
Thisbookcomprisesselectedrefereedcontributionsfromseverallargeresearch
communities in modern stochastic processes, probability theory, statistics, anal-
ysis,computationalmathematics,engineeringmathematics,andtheirinterplayand
applications. The book will be a useful source of inspiration for a broad spec-
trumofresearchersandresearchstudentsinthefieldofMathematicsandApplied
Mathematics,aswellasinthespecificareasofapplicationsconsideredinthebook.
This collective book project has been realized thanks to the strategic support
offeredbyMälardalenUniversityfortheresearchandresearcheducationinMath-
ematicswhichisconductedbytheresearchenvironmentMathematicsandApplied
Mathematics (MAM) in the established research specialization of Educational
SciencesandMathematicsattheSchoolofEducation,CultureandCommunicationat
