"Numerical methods for the approximation of strong solutions of stochastic differential equations of jump type"
Numerical methods for the approximation of strong solutions of stochastic differential equations of jump type
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Random dynamical systems in financial applications, bioeconomics and other areas of research are often modelled by stochastic ordinary differential equations of jump type (JSODEs). Only a limited class of JSODEs admits an explicit solution and consequently methods for the numerical approximation of solutions become very important. In this thesis we begin with the construction of stochastic integrals w.r.t. random measures and afterwards introduce the JSODEs and stochastic calculus for their treatment. Subsequently we present existence and uniqueness results for JSODEs. Further we discuss the basic concepts of stochastic numerics particularly for semi?implicit one?step methods for the strong approximation of solutions to JSODEs. A convergence proof of such methods under certain conditions on the increment function is presented. As examples of such methods we consider stochastic - Maruyama methods and additionally propose classes of derivative?free strong order 0.5 stochastic Runge-Kutta (SRK) schemes and prove their
convergence. Particularly the error behaviour of these methods is analysed for problems with small noise. Finally, numerical experiments on the accuracy of several methods are presented.