Calcul des variations stochastique pour les martingales

Let (Mt)t∈IR+ be a real locally integrable martingale starting from 0, with jumps of height one on a probability space (Ω, (Ft),F∞, P ), with the decomposition M = M c + X − ν, where (M c t )t∈IR+ is the continuous part of (Mt)t∈IR+ , (Xt)t∈IR+ is a point process on IR+ independent of (M c t )t∈IR+ , and (νt)t∈IR+ is the compensator of (Xt)t∈IR+ . Let β =< M c > denote the quadratic variation of (M c t )t∈IR+ . There is a Wiener process (Bt)t∈IR+ and a Poisson process (Nt)t∈IR+ such that M c = B ◦ β and X = N ◦ ν. Let (F t ) and (F t ) denote the filtrations generated by (Bt)t∈IR+ and (Nt)t∈IR+ . We assume that the trajectories of (βt)t∈IR+ and (νt)t∈IR+ are continuous and strictly increasing, and that (β−1 t )t∈IR+ , resp. (ν −1 t )t∈IR+ , is (F t ), resp. (F t )-adapted. Let (hk)k∈IN be an orthonormal basis of L(IR+, dt), let ξk = ∫∞ 0 hk(t)dBt and denote by τk the time between the jumps k and k + 1 of (Nt)t∈IR+ , k ∈ IN. Let P be a dense set of smooth cylindrical functionals in L(Ω), and denote by D̂, resp. D̃, the gradient operator on the Wiener, resp. Poisson space, cf. [1], [2], [3]. We define a closable gradient operator D and a random CameronMartin space H as DF = ((D̂F ) ◦ β, (D̃F ) ◦ ν), F ∈ P ,