Summary
We consider the space C[0, 1] together with its Borel σ-algebra A and a Wiener measure P. Let Ω denote a point in C[0, 1] and let x(Ω, t) denote the coordinate process. Then, {x(Ω, t), tε[0, 1]} is a Wiener process, and stochastic integrals of the form \(\int\limits_0^1 \varphi {\text{ }}(\omega ,t)dx(\omega ,t)\) can be defined for a suitable class of ϕ. In this paper we consider a sequence of Stieltjes integrals of the form
where {Ω n (Ω)} is a sequence of polygonal approximations to co. Conditions are found which ensure the quadratic-mean convergence of {I n }, and the limit is expressed as the sum of the stochastic integral \(\int\limits_0^1 \varphi {\text{ }}(\omega ,t)dx(\omega ,t)\) and a “correction term”.
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The research reported herein was supported in part by the U.S. Army Research Office, Durham under Grant DA-ARO-D-31-124-G776.
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Wong, E., Zakai, M. Riemann-Stieltjes approximations of stochastic integrals. Z. Wahrscheinlichkeitstheorie verw Gebiete 12, 87–97 (1969). https://doi.org/10.1007/BF00531642
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DOI: https://doi.org/10.1007/BF00531642