| Part I of this three-part thesis begins with a wavelet analysis of high frequency data of the Standard and Poor (S&P) 500 index between 1989 and 2000, and demonstrates a decrease in the degree of long range dependence as quantified by the Hurst parameter. The second chapter of Part I presents a micro-structure model that relates trading frequency to the Hurst parameter of the price. These findings provide a link between two well-known market phenomena: long range dependence in stock market returns and investor inertia. In chapter three, explicit arbitrage strategies are constructed for the models that arise from a market with inert investors. In an initial attempt to incorporate the feedback effect of the large traders and the strategic interaction between them, chapter four gives the Nash-equilibrium of a stochastic differential game with a non-Markovian differential.; Part II of the thesis studies quickest detection problems with a Poisson process. In these problems the rate of a Poisson process changes at an unobservable time, yet it is important to detect this change when it happens. The first chapter of Part II presents a solution to the Poisson disorder problem with an exponential penalty for the delay, which is especially relevant to financial applications since it gives a better account of the unrealized revenues due to the lost investment opportunities. In the second chapter, a complete solution of the standard problem is given which was formulated and partially solved by Davis. Efficient numerical algorithms for finding the optimal policy parameters are provided.; Finally, in Part III, the consistency problems associated with interest rate models, which were introduced by Bjork and Christensen, are studied. The consistency conditions for jump-diffusion models are derived, and it is shown that there exists no non-trivial jump diffusion model consistent with the Nelson-Siegel family, which is prominent in curve fitting practices employed by the most important central banks. The second chapter of Part III investigates whether an infinite dimensional forward rate model can be realized by a finite dimensional stochastic system. |
