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 X t = x 0 + ∫ 0 t σ ( X s , M s ) d W 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 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 ξ λ , is distributed according to μ . We can view ( X t ∧ ξ λ ) 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) [21] 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.11 thor would like to thank Prof. Chris Rogers for helpful discussions.