A diffusion-type process with a given joint law for the terminal level and supremum at an independent exponential time

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:121

Article

Forde, Martin

关键词: One-dimensional diffusion processes;    Excursion theory;    Skorokhod embeddings;    Stochastic functional differential equations;    Barrier options;   

DOI  :  10.1016/j.spa.2011.07.009

来源: Elsevier

【 摘 要 】

We construct a weak solution to the stochastic functional differential equation X-t =x(0) + integral(t)(0) sigma(X-s, M-s) dW(s), where M-t = sup(0 <= s <= t) X-s. Using the excursion theory, we then solve explicitly the following problem: for a natural class of joint density functions mu(y, b), we specify sigma (., .), so that X is a martingale, and the terminal level and supremum of X, when stopped at an independent exponential time xi(lambda) is distributed according to mu. We can view (X-t boolean AND xi lambda) as an alternate solution to the problem of finding a continuous local martingale with a given joint law for the maximum and the drawdown, which was originally solved by Rogers (1993) [211 using the excursion theory. This complements the recent work of Carr (2009) [5] and Cox et al. (2010) [7], who consider a standard one-dimensional diffusion evaluated at an independent exponential time. I (C) 2011 Elsevier B.V. All rights reserved.

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