Product of two multiple stochastic integrals with respect to a normal martingale

Let M be a normal martingale (i.e. 〈M, M〉 (t) = t), we decompose the product of two multiple stochastic integrals (with respect to M) In(f)Im(g) as a sum of n ∧ m terms Hk. Hk is equal to the integral over Rk+ of the function t → In+m−2k(hk(t,.)), with respect to the k-tensor product of d[M,M]., hk being an explicit function depending only on f and g. Our formula generalizes the well-known result concerning Brownian motion and compensated Poisson process and allows us to improve some results of Emery related to the chaos representation property of solution of the structure equation.