Martingale expansion in mixed normal limit

Nakahiro Yoshida

Stochastic Processes and their Applications, 2013, vol. 123, issue 3, 887-933

Abstract: The quasi-likelihood estimator and the Bayesian type estimator of the volatility parameter are in general asymptotically mixed normal. In case the limit is normal, the asymptotic expansion was derived by Yoshida [28] as an application of the martingale expansion. The expansion for the asymptotically mixed normal distribution is then indispensable to develop the higher-order approximation and inference for the volatility. The classical approaches in limit theorems, where the limit is a process with independent increments or a simple mixture, do not work. We present asymptotic expansion of a martingale with asymptotically mixed normal distribution. The expansion formula is expressed by the adjoint of a random symbol with coefficients described by the Malliavin calculus, differently from the standard invariance principle. Applications to a quadratic form of a diffusion process (“realized volatility”) are discussed.

Keywords: Asymptotic expansion; Martingale; Mixed normal distribution; Malliavin calculus; Random symbol; Double Itô integral; Quadratic form (search for similar items in EconPapers)
Date: 2013
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Citations: View citations in EconPapers (11)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:123:y:2013:i:3:p:887-933

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DOI: 10.1016/j.spa.2012.10.007

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