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Variation of iterated Brownian motion

Krzysztof Burdzy

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Abstract

In this paper, we study higher order variations of iterated Brownian motion (IBM) with view towards possible applications to the construction of the stochastic integral with respect to IBM. We prove that the 4-th variation of IBM is a deterministic linear function. This clearly means that the quadratic variation is infinite (although we do not prove this). We show that, in a weak sense, the "signed quadratic variation" of IBM is distributed like Brownian motion.

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What this paper is about

In this paper, we study higher order variations of iterated Brownian motion (IBM) with view towards possible applications to the construction of the stochastic integral with respect to IBM. We prove that the 4-th variation of IBM is a deterministic linear function. This clearly means that the quadratic variation is infinite (although we do not prove this). We show that, in a weak sense, the "signed quadratic variation" of IBM is distributed like Brownian motion.

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OpenAlex reports 64 citations for this work. Citation counts describe recorded attention and do not establish research quality.

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Available abstract

In this paper, we study higher order variations of iterated Brownian motion (IBM) with view towards possible applications to the construction of the stochastic integral with respect to IBM. We prove that the 4-th variation of IBM is a deterministic linear function. This clearly means that the quadratic variation is infinite (although we do not prove this). We show that, in a weak sense, the "signed quadratic variation" of IBM is distributed like Brownian motion.

Key concepts: Quadratic variation, Mathematics, Brownian motion, Stochastic process, Applied mathematics, IBM, Materials science, Statistics

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