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≤tXsMt=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)μ(y,b), we specify σ(.,.)σ(.,.), so that XX is a martingale, and the terminal level and supremum of XX, when stopped at an independent exponential time ξλξλ, is distributed according to μμ. We can view (Xt∧ξλ)(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) [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