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<p>In this paper we present a general mathematical construction that allows us to define a parametric class of non-Gaussian processes, namely generalized grey Brownian motion (ggBm) and multi-mixed generalized grey Brownian motions (mmggBm). For this purpose, we write down explicitly all the finite dimensional probability density functions and we provide different new properties associated with the ggBm characterizations. Next, the so-called multi-mixed generalized grey Brownian motions (mmggBm) are constructed by mixing by superimposing or mixing (infinitely many) independent generalized grey Brownian motions. Their existence as <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1840_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> processes is proved, and their path properties, viz. long-range dependence and Hölder continuity are studied. As applications, we derive an explicit solutions of the Generalized Grey Ornstein-Uhlenbeck and of a linear Wick-type stochastic differential equation driven respectively by generalized grey Brownian motion and Multi-mixed generalized grey Brownian motions. The solutions are characterized to be in a suitable distribution space.</p>

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On the Generalized and Multi-mixed Generalized Grey Brownian Motions

  • 期刊论文

  • 发表日期：

    2025年10月24日

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  • Mohamed Abdelkader,
  • Mohamed Rhaima

摘要

In this paper we present a general mathematical construction that allows us to define a parametric class of non-Gaussian processes, namely generalized grey Brownian motion (ggBm) and multi-mixed generalized grey Brownian motions (mmggBm). For this purpose, we write down explicitly all the finite dimensional probability density functions and we provide different new properties associated with the ggBm characterizations. Next, the so-called multi-mixed generalized grey Brownian motions (mmggBm) are constructed by mixing by superimposing or mixing (infinitely many) independent generalized grey Brownian motions. Their existence as \(L^{2}\) L 2 processes is proved, and their path properties, viz. long-range dependence and Hölder continuity are studied. As applications, we derive an explicit solutions of the Generalized Grey Ornstein-Uhlenbeck and of a linear Wick-type stochastic differential equation driven respectively by generalized grey Brownian motion and Multi-mixed generalized grey Brownian motions. The solutions are characterized to be in a suitable distribution space.