Weak Convergence of Stochastic Integrals and Differential Equations∗

Let W denote a standard Wiener process with W0 = 0. For a variety of reasons, it is desirable to have a notion of an integral ∫ 1 0 HsdWs, where H is a stochastic process; or more generally an indefinite integral ∫ t 0 HsdWs, 0 ≤ t < ∞. If H is a process with continuous paths, an obvious way to define a stochastic integral is by a limit of sums: let πn[0, t] be a sequence of partitions of [0, t], with mesh (πn) = supi(ti+1 − ti), where 0 = t0 < t1 < · · · < tn = t are the successive points of the partition. Then one could define ∫ t