Convergence in law to the multiple fractional integral

Abstract

We study the convergence in law in C 0([0,1]) , as ε→0, of the family of continuous processes { I η ε ( f)} ε>0 defined by the multiple integrals I η ε (f) t= ∫ 0 t ⋯ ∫ 0 t f(t 1,…,t n) dη ε(t 1)⋯ dη ε(t n); t∈[0,1], where f is a deterministic function and { η ε } ε>0 is a family of processes, with absolutely continuous paths, converging in law in C 0([0,1]) to the fractional Brownian motion with Hurst parameter H> 1 2 . When f is given by a multimeasure and for any family { η ε } with trajectories absolutely continuous whose derivatives are in L 2([0,1]), we prove that { I η ε ( f)} converges in law to the multiple fractional integral of f. This last integral is a multiple Stratonovich-type integral defined by Dasgupta and Kallianpur (Probab. Theory Relat. Fields 115 (1999) 505) on the space L 2( μ ̃ n) , where μ ̃ n is a measure on [0,1] n . Finally, we have shown that, for two natural families { η ε } converging in law in C 0([0,1]) to the fractional Brownian motion, the family { I η ε ( f)} converges in law to the multiple fractional integral for any f∈L 2( μ ̃ n) . In order to prove the convergence, we have shown that the integral introduced by Dasgupta and Kallianpur (1999a) can be seen as an integral in the sense of Solé and Utzet (Stochastics Stochastics Rep. 29(2) (1990) 203).