Weak convergence of stochastic integrals driven by martingale measure

Let S′(Rd) be the dual of Schwartz space, S(Rd), {Mn} be a sequence of martingale measures and let F be some suitable function space such as C0(Rd), Lp(Rd), p ⩾ 2 or Cm,y0(Rd). We find conditions under which (Xn, Mn) ⇒ (X, M) in the Skorohod topology in DF × S′(Rd)[0, ∞) implies ∫ Xn(x, s)Mn(dx, ds) ⇒ ∫ X(x, s) M(dx, ds) in the Skorohod topology in DS′(Rd)[0, ∞). We use the idea of regularization to reduce S′(Rd) to a metrizable subspace in order to apply the Skorohod representation theorem and then appropriate the randomized mapping constructed by Kurtz and Protter to get step functions approximating the integrands. this result, we prove weak convergence of certain double stochastic integrals studied by Walsh. Let ∅ ∈ S(R2d), {ηn} be a sequence of Brownian density processes and {Wn} and {Zn} be two sequences of martingale measures generated by particle systems. We consider the weak convergence of ∫ ∅(x, y)ηns(dx)Wn(dx, dy) and ∫ ∅(x, y)ηns(dx)Zn(dx, dy).