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Absolute continuity of the laws, existence and uniqueness of solutions of some SDEs and SPDEs
[Thesis]. Manchester, UK: The University of Manchester; 2014.
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Abstract
This thesis consists of four parts. In the first part we recall some background theory that will be used throughout the thesis. In the second part, we studied the absolute continuity of the laws of the solutions of some perturbed stochastic differential equaitons(SDEs) and perturbed reflected SDEs using Malliavin calculus. Because the extra terms in the perturbed SDEs involve the maximum of the solution itself, the Malliavin differentiability of the solutions becomes very delicate. In the third part, we studied the absolute continuity of the laws of the solutions of the parabolic stochastic partial differential equations(SPDEs) with two reflecting walls using Malliavin calculus. Our study is based on Yang and Zhang \cite{YZ1}, in which the existence and uniqueness of the solutions of such SPDEs was established. In the fourth part, we gave the existence and uniqueness of the solutions of the elliptic SPDEs with two reflecting walls and general diffusion coefficients.
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Keyword(s)
Malliavin differentiability; Absolute continuity; Comparison Theorem; Stochastic differential equations; Stochastic partial differential equations; Diffusion processes; Peturbed diffusion processes; Reflecting walls;