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• W2185173272 abstract "The rst passage time (FPT) problem for Brownian motion has been extensively studied in the literature. In particular, many incarnations of integral equations which link the density of the hitting time to the equation for the boundary itself have appeared. Most interestingly, Peskir (2002b) demonstrates that a master integral equation can be used to generate a countable number of new integrals via its dierentiation or integration. In this thesis, we generalize Peskir’s results and provide a more powerful unifying framework for generating integral equations through a new class of martingales. We obtain a continuum of new Volterra type equations and prove uniqueness for a subclass. The uniqueness result is then employed to demonstrate how certain functional transforms of the boundary aect the density function. Furthermore, we generalize a class of Fredholm integral equations and show its fundamental connection to the new class of Volterra equations. The Fredholm equations are then shown to provide a unied approach for computing the FPT distribution for linear, square root and quadratic boundaries. In addition, through the Fredholm equations, we analyze a polynomial expansion of the FPT density and employ a regularization method to solve for the coecients. Moreover, the Volterra and Fredholm equations help us to examine a modication of the" @default.
• W2185173272 created "2016-06-24" @default.
• W2185173272 creator A5086243574 @default.
• W2185173272 date "2010-03-03" @default.
• W2185173272 modified "2023-09-27" @default.
• W2185173272 title "First Passage Times: Integral Equations, Randomization and Analytical Approximations" @default.
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• W2185173272 hasPublicationYear "2010" @default.
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