This dissertation studies the pricing of Outside barrier call options, when their activation starts at a hitting time. The pricing of Outside barrier options when their activation starts at time zero, and the pricing of standard barrier options when their activation starts at a hitting time of a pre speci ed barrier level, have been studied previously (see [21], [24]). The new work that this dissertation will do is to price Outside barrier call options, where they will be activated when the triggering asset crosses or hits a pre speci ed barrier level, and also the pricing of Outside barrier call options where they will be activated when the triggering asset crosses or hits a sequence of two pre specifed barrier levels. Closed form solutions are derived using Girsanov's theorem and the re ection principle. Existing results are derived from the new results, and properties of the new results are illustrated numerically and discussed. / Mathematical Sciences / M. Sc. (Applied Mathematics)

We consider the problem of testing whether r (>=2) samples are drawn from the same continuous distribution F(x). The test statistic we will study in some detail is defined as the maximum of the circular differences of the empirical distribution functions, a generalization of the classical 2-sample Kolmogorov-Smirnov test to r (>=2) independent samples. For the case of equal sample sizes we derive the exact null distribution by counting lattice paths confined to stay in the scaled alcove $\mathcal{A}_r$ of the affine Weyl group $A_{r-1}$. This is done using a generalization of the classical reflection principle. By a standard diffusion scaling we derive also the asymptotic distribution of the test statistic in terms of a multivariate Dirichlet series. When the sample sizes are not equal the reflection principle no longer works, but we are able to establish a weak convergence result even in this case showing that by a proper rescaling a test statistic based on a linear transformation of the circular differences of the empirical distribution functions has the same asymptotic distribution as the test statistic in the case of equal sample sizes. / Series: Research Report Series / Department of Statistics and Mathematics

This dissertation studies the pricing of Outside barrier call options, when their activation starts at a hitting time. The pricing of Outside barrier options when their activation starts at time zero, and the pricing of standard barrier options when their activation starts at a hitting time of a pre speci ed barrier level, have been studied previously (see [21], [24]). The new work that this dissertation will do is to price Outside barrier call options, where they will be activated when the triggering asset crosses or hits a pre speci ed barrier level, and also the pricing of Outside barrier call options where they will be activated when the triggering asset crosses or hits a sequence of two pre specifed barrier levels. Closed form solutions are derived using Girsanov's theorem and the re ection principle. Existing results are derived from the new results, and properties of the new results are illustrated numerically and discussed. / Mathematical Sciences / M. Sc. (Applied Mathematics)

Neste trabalho de tese discutimos resultados recentes sobre a regularidade e propriedades geomÃtricas de soluÃÃes variacionais de problemas de fronteira livre de duas fases regidos por equaÃÃes elÃpticas nÃo lineares degeneradas/singulares. Discutimos tambÃm resultados do tipo PhragmÃm-Lindelof para tais equaÃÃes classificando essas soluÃÃes em semi-espaÃos. / Neste trabalho de tese discutimos resultados recentes sobre a regularidade e propriedades geomÃtricas de soluÃÃes variacionais de problemas de fronteira livre de duas fases regidos por equaÃÃes elÃpticas nÃo lineares degeneradas/singulares. Discutimos tambÃm resultados do tipo PhragmÃm-Lindelof para tais equaÃÃes classificando essas soluÃÃes em semi-espaÃos. / In this work of thesis we discuss recents results on the regularity and geometric properties of variational solutions of two phase free boundary problems governed by singular/degenerate nonlinear elliptic equations. We also discuss PhragmÃn-Lindelof type results for such equations classifying those solutions in half spaces. / In this work of thesis we discuss recents results on the regularity and geometric properties of variational solutions of two phase free boundary problems governed by singular/degenerate nonlinear elliptic equations. We also discuss PhragmÃn-Lindelof type results for such equations classifying those solutions in half spaces.