Strong Approximation of Stochastic Mechanical Systems with Holonomic Constraints

The main topic of this thesis is the solvability and numerical approximation of stochastic mechanical systems with nonlinear holonomic constraints. Such systems are described by second order stochastic differential-algebraic equations involving an implicitly given Lagrange multiplier process. By using an explicit representation of the Lagrange multiplier, one typically derives an underlying stochastic ordinary differential equation, such that the corresponding drift coefficient is not globally one-sided Lipschitz continuous. We show under mild assumptions on the underlying coefficients the solvability of the stochastic differential-algebraic equations by using a Gronwall-type argument depended on the geometric structure of the mechanical system. Concerning the numerical approximation we focus on half-explicit partially truncated Euler schemes. Those schemes fulfill the constraints at each discretization point and due a splitting of the discretized system we have P-almost sure solvability of the involved non-linear equations. In particular we establish the path-wise uniform L_p-convergence for the half-explicit increment-truncated Euler scheme as well as for am implementable modification thereof using Newton's algorithm to compute the solutions of the involved non-linear systems. The proof is based on a suitable decomposition of the discrete Lagrange multipliers and on norm estimates for the single components, enabling the verification of consistency, semi-stability and moment growth properties of the scheme. Given these properties suitable theorem of the theory of non-standard stochastic differential equations are guaranteeing the convergence.