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Weak convergence of stochastic integrals driven by martingale measure

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  • Cho, Nhansook

Abstract

Let be the dual of Schwartz space, , {Mn} be a sequence of martingale measures and let F be some suitable function space such as , , p [greater-or-equal, slanted] 2 or . We find conditions under which (Xn, Mn) => (X, M) in the Skorohod topology in implies [integral operator] Xn(x, s)Mn(dx, ds) => [integral operator] X(x, s) M(dx, ds) in the Skorohod topology in . We use the idea of regularization to reduce to a metrizable subspace in order to apply the Skorohod representation theorem and then appropriate the randomized mapping constructed by Kurtz and Protter to get step functions approximating the integrands. Using this result, we prove weak convergence of certain double stochastic integrals studied by Walsh. Let , {[eta]n} be a sequence of Brownian density processes and {Wn} and {Zn} be two sequences of martingale measures generated by particle systems. We consider the weak convergence of [integral operator] [empty set](x, y)[eta]ns(dx)Wn(dx, dy) and [integral operator] [empty set](x, y)[eta]ns(dx)Zn(dx, dy).

Suggested Citation

  • Cho, Nhansook, 1995. "Weak convergence of stochastic integrals driven by martingale measure," Stochastic Processes and their Applications, Elsevier, vol. 59(1), pages 55-79, September.
    • Handle: RePEc:eee:spapps:v:59:y:1995:i:1:p:55-79
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    Cited by:

    1. Nhansook Cho & Youngmee Kwon, 2000. "Limit of Solutions of a SDE with a Large Drift Driven by a Poisson Random Measure," Journal of Theoretical Probability, Springer, vol. 13(2), pages 311-325, April.
    2. Ulrich Horst & Dorte Kreher, 2016. "A diffusion approximation for limit order book models," Papers 1608.01795, arXiv.org, revised Aug 2017.
    3. Horst, Ulrich & Kreher, Dörte, 2019. "A diffusion approximation for limit order book models," Stochastic Processes and their Applications, Elsevier, vol. 129(11), pages 4431-4479.

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