In recent years symplectic geometry and symplectic topology have grown to large subbranches in mathematics and had a great impact on other areas in mathematics. When interested in geometry, a geometer always considers geometric structures that arise on immersed submanifolds. In symplectic geometry there is a distinguished class of immersions, known as Lagrangian submanifolds . In particular, minimal Lagrangian submanifolds, called special Lagrangians, are very important in mirror symmetry. Lagrangian mean curvature flow is an important example of Lagrangian deformation. From which we can get the special Lagrangian submanifolds. In recent years, there have been many papers about this subject and the result by K.Smoczyk and Mu-Tao Wang [WS] is very important and beautiful. Our main purpose in this article is to give a new proof for the main result in [WS] from the viewpoint of fully nonlinear partial differential equations.

Generated by stochastic recursions, perpetuities encompass a vast range of discretetime financial behaviours. When focusing on the dramatic changes occurring in such processes, the analysis of threshold exceedances provides an extensive description of their underlying mechanisms. Asymptotically, an exceedance point process tends to a compound Poisson measure, highlighting a tendency to cluster. Now, the parameters of this limit law are known, but complex. Here, an empirical approach is adopted, and a class of explicit compound Poisson models developed, with a bound on the error, for the exceedance point process of a finite, multidimensional perpetuity. In a financial regulatory context, this provides a new way of examining the Value-at-Risk criterion for securities.

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Thesis (Ph.D.)--Auburn University, 2006. / Abstract. Vita. Includes bibliographic references (ℓ.28-29)

Proefschrift Universiteit van Amsterdam. / Met lit. opg. - Met samenvatting in het Nederlands.

Licentiatavhandling Uppsala, Univ : 2005.

Thesis (Ph. D.)--School of Mathematics and Statistics, Faculty of Science, University of Sydney, 2005. / Bibliography: p. 89-91.

Thesis (Ph. D.)--Oregon State University, 2009. / Printout. Includes bibliographical references (leaves 50-53). Also available on the World Wide Web.

Thesis (Ph.D.)--University of Western Australia, 2007.

Thesis (Ph. D.)--University of Sydney, 2005. / Title from title screen (viewed 22 May 2008). Submitted in fulfilment of the requirements for the degree of Doctor of Philosophy to the School of Mathematics and Statistics, Faculty of Science. Includes bibliographical references. Also available in print form.