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 , where M_t=sup_0≤s≤tX_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∧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) 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.¹