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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 degenerate Brownian motion (dBm). This class, is made up of self-similar with stationary increments processes and depends on three real parameters <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1793_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="117" /> </InlineMediaObject> <EquationSource Format="TEX">\(\alpha \in (0, 2), \lambda &lt;0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mi>λ</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1793_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">\(\beta &gt;0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. For this purpose, we write down explicitly all the finite dimensional probability density functions and we provide different new properties associated with the dBm characterizations.</p>

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Study of the Degenerate Brownian Motion

  • 期刊论文

  • 发表日期：

    2025年8月20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 degenerate Brownian motion (dBm). This class, is made up of self-similar with stationary increments processes and depends on three real parameters \(\alpha \in (0, 2), \lambda <0\) α ∈ ( 0 , 2 ) , λ < 0 and \(\beta >0\) β > 0 . For this purpose, we write down explicitly all the finite dimensional probability density functions and we provide different new properties associated with the dBm characterizations.