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Abstract
In this thesis, we develop high order numerical methods for strong solution of Itô stochastic differential equations (SDEs). We first construct an approximate deterministic ODE with a random coefficient on each element using the Wong-Zakai approximation theorem. Since the resulting equation converges to the solution of the corresponding Stratonovich SDE, we apply a transformation to the drift term to obtain an ODE which converges to the solution of the original SDE. The corrected equation is then discretized using the standard continuous and discontinuous finite element methods for deterministic ODEs. Our methods are demonstrated to be strongly convergent, accurate, and computationally efficient. More precisely, numerical evidence demonstrate that our proposed continuous and discontinuous finite element methods, respectively, have strong convergence order of p/2 and p, when p-degree piecewise polynomials are used. Several linear and nonlinear test problems are presented to show the accuracy and effectiveness of the proposed method.
