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The integral of the squared Gaussian process

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  • Reus, Lorenzo

Abstract

This work studies the random variable defined by X≔∫tTZs′AZsds, with A a real matrix of size N×N, and Zs∈RN Gaussian processes. The results show that X is a constant variable when Zs is time-independent. When Zs∈R follows a Brownian motion, a closed-form moment generating function (MGF) of X is derived, which does not match the MGFs of known distributions. Finally, a portfolio problem is presented to show how the MGF of X is needed for finding the optimal solution in closed form.

Suggested Citation

  • Reus, Lorenzo, 2024. "The integral of the squared Gaussian process," Chaos, Solitons & Fractals, Elsevier, vol. 179(C).
    • Handle: RePEc:eee:chsofr:v:179:y:2024:i:c:s096007792301319x
    DOI: 10.1016/j.chaos.2023.114417
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    References listed on IDEAS

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