Distance between two skew Brownian motions as a SDE with jumps and law of the hitting time

In this paper, we consider two skew Brownian motions, driven by the same Brownian motion, with different starting points and different skewness coefficients. We show that we can describe the evolution of the distance between the two processes with a stochastic differential equation. This S.D.E. possesses a jump component driven by the excursion process of one of the two skew Brownian motions. Using this representation, we show that the local time of two skew Brownian motions at their first hitting time is distributed as a simple function of a Beta random variable. This extends a result by Burdzy and Chen (2001), where the law of coalescence of two skew Brownian motions with the same skewness coefficient is computed.Comment: 27 page

Valuation of stock loans under exponential phase-type Lévy models.

Wong, Tat Wing.Thesis (M.Phil.)--Chinese University of Hong Kong, 2011.Includes bibliographical references (p. 53-55).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.1Chapter 2 --- Problem Formulation --- p.5Chapter 2.1 --- Phase-type distribution --- p.5Chapter 2.1.1 --- A generalization of the exponential distribution --- p.5Chapter 2.1.2 --- Properties of the phase-type distribution --- p.6Chapter 2.2 --- Phase-type jump diffusion model --- p.8Chapter 2.2.1 --- Jump diffusion model --- p.8Chapter 2.2.2 --- The stock price model --- p.9Chapter 2.3 --- Stock Loans --- p.10Chapter 3 --- General Properties of Stock Loans --- p.12Chapter 3.1 --- Preliminary results --- p.12Chapter 3.2 --- Characterization of the function V(x) --- p.15Chapter 4 --- ValuationChapter 4.1 --- Hyperexponential jumps --- p.25Chapter 4.1.1 --- Solution of the linear system --- p.29Chapter 4.1.2 --- Solution of the optimal exercise boundary --- p.30Chapter 4.2 --- Phase-type jumps --- p.33Chapter 4.3 --- The case for G'(1)≥ 0 --- p.36Chapter 5 --- Future Research Direction --- p.38Chapter 5.1 --- The fast mean-reverting stochastic volatility model --- p.38Chapter 5.2 --- Asymptotic expansion of stock loan --- p.39Chapter 5.2.1 --- The zeroth order term --- p.41Chapter 5.2.2 --- The first order term --- p.43Chapter 6 --- Conclusion --- p.52Bibliography --- p.5