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) [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.11The author would like to thank Prof. Chris Rogers for helpful discussions.