In Chapter 2,I consider a queuing system in which there are two exponential servers, each having his own queue, and arriving customers will join the two queues separately, and analyze the two-dimensional Markov process representing the numbers of jobs in the two stations. A functional equation for the generating function of the stationary distribution of this two-dimensional process is derived and solved through the theory of Riemann-Hilbert boundary value problems. In Chapter 3, I consider the system with two parallel queues, in which arrivals join the shorter queue.I describe the problem of coupling and show that the generating function Fcan be continued as a meromorphic function to the whole complex plane.

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Sset future prices can be forecasted with filtering historical simulation by drawing from standardized residual density. The problem of inappropriate risk neutral assumption could be resolved by the calibration from density transformation. The transformation is to identify the state price density per unit probability through the parameter re-estimation by minimizing the mean squared error of market and model prices.

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The trigonometric uniform polynomial curve constructed in this paper has two advantages. One is that the curve can be continuous and the shape can be adjusted with the fixed tangents. The other is that only adding one new knot, the curve can pass through the tangent, which reduces excess points.

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