Boukhelkhal, Ikram2024-06-232024-06-232024-06-13MD/23http://10.10.1.6:4000/handle/123456789/5035In 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.enStochastic differentialequation,Brownianmotion,jumpdiffusion,spectral method, numericalsolution,collocationmethod. iNumerical treatment of stochastic differential equations: Diffusion and jump-diffusion processes with applicationsThesis