Product of two multiple stochastic integrals with respect to a normal martingale
Let \(I_n(g)\) and \(I_m(f)\) be two multiple stochastic integrals with respect to a normal martingale \(M\), where \(g\) and \(f\) are symmetric, square integrable functions on \(R^n_+\) and \(R^m_+\), respectively. The product \(I_n(g)I_m(f)\) is decomposed as a sum of \(n\land m\) terms \(H_k\) where \(H_k\) is the integral over \(R^k_+\) of the function \(t\to I_{n+m-2k}(h_k(t,.))\), with respect to the \(k\)-tensor product of \(d[M,M]\), \(h_k\) being an explicit function depending only on \(g\) and \(f\). This extends some earlier results achieved for the cases when \(M\) is a Brownian motion or a compensated Poisson process.
scientific article; zbMATH DE number 1409163
scientific article