Numerical treatment of stochastic differential equations: Diffusion and jump-diffusion processes with applications
| dc.contributor.author | Boukhelkhal, Ikram | |
| dc.date.accessioned | 2024-06-23T09:51:58Z | |
| dc.date.available | 2024-06-23T09:51:58Z | |
| dc.date.issued | 2024-06-13 | |
| dc.description.abstract | In thisdissertation,wedealwiththeproblemofsimulatingstochasticdifferentialequations driven byBrownianmotionorthegeneralL´evy processes.First,weestablishthebasic theory ofstochasticcalculusandintroducetheIt ˆo-Taylorexpansionforstochasticdifferen- tial equations(SDEs).Inaddition,wepresentvariousnumericalschemesderivedfrom the It ˆo-Taylorexpansion.ThesemethodsareusedtosolvethestochasticLorenzequa- tion, thestochasticDuffingequation,andtheMertonmodelequation.Inaddition,spec- tral techniquesareadaptedforthenumericalsolutionofnonlinearstochasticdifferential equations. Further,generalizedLagrangeinterpolationfunctionsareproposedforsolving various typesofSDEs,offeringsignificantperformanceimprovements. | en_US |
| dc.identifier.issn | MD/23 | |
| dc.identifier.uri | http://10.10.1.6:4000/handle/123456789/5035 | |
| dc.language.iso | en | en_US |
| dc.publisher | UNIVERSITY BBA | en_US |
| dc.subject | Stochastic differentialequation,Brownianmotion,jumpdiffusion,spectral method, numericalsolution,collocationmethod. i | en_US |
| dc.title | Numerical treatment of stochastic differential equations: Diffusion and jump-diffusion processes with applications | en_US |
| dc.type | Thesis | en_US |
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- Stochastic differentialequations(SDEs)governedbyBrownianmotionandbyajumpdif- fusion areimportanttoolsinawiderangeofapplications,includingbiology,chemistry, mechanics, economics,andphysics.Theyarebecomingmoreandmoreattractivedueto their applicationforsimulatingstochasticphenomenainvariousfields. These equationsareexplainedandinterpretedinthecontextoftheIt ˆo calculus.Unfortu- nately,thereisfrequentlynoanalyticalsolutiontotheseequations,andweareobligedtouse numerical approximations.Broadlyspeaking,therearetwobasicwaystoderivetheseap- proximations.Whenthesampletrajectoriesofthesolutionsneedtobeapproximated,mean squareconvergenceisemployed,andthemethodsthusderivedarecalledstrong.When we areinterestedonlyinthemomentsorotherfunctionalsofthesolution,whichinvolvea much weakerformofconvergence. The purposeofthisdissertationistoprovideabriefoverviewofthedifferentnumeri- cal methodsforsolvingstochasticdifferentialequationsandtoproposeanewmethodol- ogy thatimprovessomeexistingtechniques.Itcanbeseenthatthediscretizationstepsize plays animportantroleintheaccuracyforeachmethodthroughthesimulationofIt ˆo-Taylor schemes andinparticularbytheexaminationoftheeffectivenessofsomeschemesforthe approximationofthesolutionsofSDE.Whenthestepsizeiskeptverysmall,goodresults can beattained.Conversely,thecomputationalcomplexityisveryhighwhenweincrease the orderoftheschemes. Wehaveproposedtwonumericalapproachesthatcanbeusedforfindingapproximate solutions ofstochasticintegralequations.InterpolationbyLagrangepolynomialsandzeros of JacobipolynomialsareusedtoreducetheconsideredproblemofstochasticVolterrainte- gral equationstoanalgebraicsystemofequations.Approximatesolutionsofthestochastic 119 Chapter 3AnovelmethodtosolvenonlinearSIVIE Volterraintegralequationsarethenobtained.Atheoreticalinvestigationisalsocarriedout to confirmtheerrorandconvergenceanalysisofthesemethods.Thespectralconvergence rate forthedevelopedmethodisestablished.Inordertoprovethesuitabilityandaccuracy of ourmethodsseveralrelatednumericalexampleswithdifferentsimulationsofBrownian motion areincluded.Thenumericalresultsofthepresentedmethodsarealsocompared with theresultsofothernumericaltechniques. The secondnewtechniqueisbasedoncombiningJacobi-Gausscollocationpointsand generalized Lagrangefunctions.Theaccuracyandconsistencyofthenewtechniqueare evaluated andcomparedwithsometechniques.Inaddition,sufficientconditionsaregiven to ensurethattheestimationerrortendstozero.Thenewtechniqueshowssurprisingeffi- ciency overtheexistingtechniquesintermsofneededtime,computational,andapproxima- tion performance.Theaccuracyofthesolutionderivedbythenewtechniqueissignificantly higher thanthatoftheexistingmethods. Weareoptimisticthatitwillbepossibletogeneralizetheproposedmethodtoabroader class ofproblemswhilemaintainingtheefficiencyandaccuracyofthemethod.Extending our workrepresentsaninterestingtopicforfuturework,whichwecanidentifyasfollows • The abilitytoextendtheapproximationtohigherdimensions. • Our resultsleavethedooropenforfuturedevelopments,includingtheextensionof the currentresearchtostochasticdifferentialequationsdrivenbyotherstochasticpro- cesses. • Accordingtotheapproachpresentedinthisdissertation
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