Weak convergence to the multiple Stratonovich integral

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Abstract

We have considered the problem of the weak convergence, as ε tends to zero, of the multiple integral processes {(Latin small letter esh)t0 ⋯ (Latin small letter esh)t0 f(t1, . . . , tn)dηε(t1) ⋯ dηε(tn),t ∈ [0, T]} in the space script C sign0([0,T]), where f ∈ L2([0, T]n) is a given function, and {ηε(t)}ε>0 is a family of stochastic processes with absolutely continuous paths that converges weakly to the Brownian motion. In view of the known results when n≥2 and f(t1, . . . , tn) = 1{t1 <t2 <⋯<tn}, we cannot expect that these multiple integrals converge to the multiple Itô-Wiener integral of f, because the quadratic variations of the ηε are null. We have obtained the existence of the limit for any {ηε}, when f is given by a multimeasure, and under some conditions on {ηε} when f is a continuous function and when f(t1, . . . , tn) = f1(t1) ⋯ fn(tn)1{t1 <t2 < ⋯ <tn}, with fi ∈ L2([0, T]) for any i = 1, . . . , n. In all these cases the limit process is the multiple Stratonovich integral of the function f. © 2000 Elsevier Science B.V. All rights reserved.
Original languageEnglish
Pages (from-to)277-300
JournalStochastic Processes and their Applications
Volume90
Issue number2
DOIs
Publication statusPublished - 1 Jan 2000

Keywords

  • Donsker approximations
  • Multimeasure
  • Multiple stratonovich integral
  • Weak convergence

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