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<p>The process <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((\int _0^t e^{2b_s-b_t}\, ds\;\ t\ge 0)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>t</mi> </msubsup> <msup> <mi>e</mi> <mrow> <mn>2</mn> <msub> <mi>b</mi> <mi>s</mi> </msub> <mo>-</mo> <msub> <mi>b</mi> <mi>t</mi> </msub> </mrow> </msup> <mspace width="0.166667em" /> <mi>d</mi> <mi>s</mi> <mspace width="0.277778em" /> <mspace width="4pt" /> <mi>t</mi> <mo>≥</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <i>b</i> is a real Brownian motion, is known as the geometric 2M-X Matsumoto–Yor process. Remarkably, it enjoys the Markov property. We provide a generalization of this process in the context of Jordan algebras, and we prove the Markov property for this generalization.</p><p>Our Markov process occurs as a limit of discrete-time AX+B Markov chains on the cone of squares whose invariant probability measures classically yield a Dufresne-type identity for a perpetuity. In particular, the paper provides a generalization to any symmetric cone of the matrix-valued generalization of the Matsumoto–Yor process and Dufresne identity initially developed by Rider–Valkó.</p>

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Matsumoto–Yor processes on Jordan algebras

  • 期刊论文

  • 发表日期：

    2025年11月28日

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  • Reda Chhaibi,
  • Manon Defosseux

摘要

The process \((\int _0^t e^{2b_s-b_t}\, ds\;\ t\ge 0)\) ( ∫ 0 t e 2 b s - b t d s t ≥ 0 ) , where b is a real Brownian motion, is known as the geometric 2M-X Matsumoto–Yor process. Remarkably, it enjoys the Markov property. We provide a generalization of this process in the context of Jordan algebras, and we prove the Markov property for this generalization.

Our Markov process occurs as a limit of discrete-time AX+B Markov chains on the cone of squares whose invariant probability measures classically yield a Dufresne-type identity for a perpetuity. In particular, the paper provides a generalization to any symmetric cone of the matrix-valued generalization of the Matsumoto–Yor process and Dufresne identity initially developed by Rider–Valkó.