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Title: Stochastic differential inclusions
Author: Chen, Xiaoli
Awarding Body: University of Edinburgh
Current Institution: University of Edinburgh
Date of Award: 2006
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Stochastic differential inclusions (SDIs) on Rd have been investigated in this thesis, dx(t) Î a(t, x(t))dt +   (t, x (t)d where a is a maximal monotone mapping, b is a Lipschitz continuous function, and w is a Wiener process. The principal aim of this work is to present some new results on solvability and approximations of SDIs. Two methods are adapted to obtain our results: the method of minimization and the method of implicit approximation. We interpret the method of monotonicity as a method of constructing minimizers to certain convex functions. Under the monotonicity condition and the usual linear growth condition, the solutions are characterized as the minimizers of convex functionals, and are constructed via implicit approximations. Implicit numerical scheme is given and the result on the rate of convergence is also presented. The ideas of our work are inspired by N.V. Krylov, where stochastic differential equations (SDEs0 in Rd are solved by minimizing convex functions via Euler approximations. Furthermore, since the linear growth condition is too strong, an approach is proposed for truncating maximal monotone functions to get bounded maximal monotone functions. It is a technical challenge in this thesis. Thus the existence of solutions to SDIs is proved under essentially weaker growth condition than the linear growth. For a special case of SDEs, a few of recent results from [5] are generalized. Some existing results of the convergence by implicit numerical schemes are proved under the locally Lipschitz condition. We will show that under certain weaker conditions, if the drift coefficient satisfies one-sided Lipschitz and the diffusion coefficient is Lipschitz continuous, implicit approximations applied to SDEs, converge almost surely to the solution of SDEs. The rate of convergence we get is ¼.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available