Existence of Law Density of a Pair (id/i-dimensional Diffusion iX/i, First Component Running Maximum iM/i) with Coefficients Depending on i(X, M)/i, d 1

The purpose of this paper is to consider a stochastic differential equation guiding X a continuous d-dimensional diffusion process, the coefficients of which depending on the pair i (X, M) /i, where M is the running supremum of the first component iX/isup1/sup. As an application of our work, we could think of a firm the activity of which is characterized by a set of processes (iX/isup1/sup, ···, iX/isupd/sup). But one of them, for instance iX/isup1/sup, could be linked to an alarm. Such a (id/i + 1) - dimensional process (X, M) could present a crucial interest in this case where M could be an alarm. Indeed, the possibility of an alarm at time t namely the event ∃s ≤ t: Xsup1/supsubs/sub u is identical to the event Msubt/sub iu/i when the specific (and dangerous) threshold iu/i is exceeded. This means that the law of M is closely linked to the law of the hitting time when iX/isup1/sup reaches such a dangerous level iu/i. Here is proved that, for all positive real number t; the law of the (id/i+1) -dimensional random vector (iX/isubt/sub, iM/isubt/sub) admits a density with respect to the Lebesgue measure. The solution of such a stochastic differential equation is built using a recursion method. The existence of the density of the law of (iX/i, iM/i) is based on Malliavin calculus. This density is solution of a partial differential equation in a weak sense. Moreover, such a recursive construction will allow to build simulated solutions. So finally such a tool could allow us to build an alarm system to detect the hitting time when the alarm could occur.