A diffusion-type process with a given joint law for the terminal level and supremum at an independent exponential time
We construct a weak solution to the stochastic functional differential equation Xt=x0+∫0tσ(Xs,Ms)dWs, where Mt=sup0≤s≤tXs. Using the excursion theory, we then solve explicitly the following problem: for a natural class of joint density functions μ(y,b), we specify σ(.,.), so that X is a martingale, and the terminal level and supremum of X, when stopped at an independent exponential time ξλ, is distributed according to μ. We can view (Xt∧ξλ) 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)  using the excursion theory. This complements the recent work of Carr (2009)  and Cox et al. (2010) , who consider a standard one-dimensional diffusion evaluated at an independent exponential time.11The author would like to thank Prof. Chris Rogers for helpful discussions.
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Volume (Year): 121 (2011)
Issue (Month): 12 ()
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- Dmitry Davydov & Vadim Linetsky, 2001. "Pricing and Hedging Path-Dependent Options Under the CEV Process," Management Science, INFORMS, vol. 47(7), pages 949-965, July.
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