The Annals of Probability

Martingale structure of Skorohod integral processes

Giovanni Peccati, Michèle Thieullen, and Ciprian A. Tudor

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Let the process {Yt,t∈[0,1]} have the form Yt=δ(u1[0,t]), where δ stands for a Skorohod integral with respect to Brownian motion and u is a measurable process that verifies some suitable regularity conditions. We use a recent result by Tudor to prove that Yt can be represented as the limit of linear combinations of processes that are products of forward and backward Brownian martingales. Such a result is a further step toward the connection between the theory of continuous-time (semi)martingales and that of anticipating stochastic integration. We establish an explicit link between our results and the classic characterization (owing to Duc and Nualart) of the chaotic decomposition of Skorohod integral processes. We also explore the case of Skorohod integral processes that are time-reversed Brownian martingales and provide an “anticipating” counterpart to the classic optional sampling theorem for Itô stochastic integrals.

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Ann. Probab., Volume 34, Number 3 (2006), 1217-1239.

First available in Project Euclid: 27 June 2006

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Primary: 60G15: Gaussian processes 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60G44: Martingales with continuous parameter 60H05: Stochastic integrals 60H07: Stochastic calculus of variations and the Malliavin calculus

Malliavin calculus anticipating stochastic integration martingale theory stopping times


Peccati, Giovanni; Thieullen, Michèle; Tudor, Ciprian A. Martingale structure of Skorohod integral processes. Ann. Probab. 34 (2006), no. 3, 1217--1239. doi:10.1214/009117905000000756.

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