Contributions to Numerical Solution of Stochastic Differential Equations

Author: Anders Muszta; Chalmers Tekniska Högskola.; Chalmers University Of Technology.; [2005]

Keywords: MATEMATIK; MATHEMATICS; adaptive method; change of time; CIR model; CKLS model; convergence order; Euler method; heavy-tailed SDE; hyperbolic SDE; geometric integration; global Lipschitz condition; Levy process; Lie group method; local Lipschitz condition; local martingales; mean reversion; Milstein method; numerical method; semimartingale; stochastic differential equation; stochastic Taylor expansion; strong approximation; symplectic integration; volatility induced stationarity; waveform relaxation;

Abstract: This thesis consists of four papers:

Paper I is an overview of recent techniques in strong numerical solutions of stochastic differential equations, driven by Wiener processes, that have appeared the last then 10 years, or so.

Paper II studies theoretical and numerical aspects of stochastic differential equations with so called volatility induced stationarity. While being of great importance in contemporary applications, these equations are particularly difficult from a numerical point of view, to the extent that most or even all standard numerical procedures fail.

Paper III develops numerical procedures for stochastic differential equations driven by Levy processes. A general scheme for stochastic Taylor expansions is developed, together with an analysis of convergence properties.

Paper IV shows how to reduce the common global Lipschitz condition for numerical procedures for stochastic differential equations, to a local Lipschitz condition.

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