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W2007431354 endingPage "2040" @default.
W2007431354 startingPage "2031" @default.
W2007431354 abstract "Let<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$B$ id=E1><mml:mi>B</mml:mi></mml:math>be a continuous additive functional for a standard process<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$(X_{t})_{tin mathbb{R}_{+}}$ id=E2><mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>ℝ</mml:mi><mml:mo>+</mml:mo></mml:msub></mml:mrow></mml:msub></mml:math>and let<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$(Y_{t})_{_{tinmathbb{R}}}$ id=E3><mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>Y</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>∈</mml:mo><mml:mi>ℝ</mml:mi></mml:mrow></mml:msub></mml:math>be a stationary Kuznetsov process with the same semigroup of transition. In this paper, we give the excursion laws of<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$(X_{t})_{tin mathbb{R}_{+}}$ id=E4><mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>ℝ</mml:mi><mml:mo>+</mml:mo></mml:msub></mml:mrow></mml:msub></mml:math>conditioned on the strict past and future without duality hypothesis. We study excursions of a general regenerative system and of a regenerative system consisting of the closure of the set of times the regular points of<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$B$ id=E5><mml:mi>B</mml:mi></mml:math>are visited. In both cases, those conditioned excursion laws depend only on two points<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$X_{g^{-}}$ id=E6><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:msup><mml:mi>g</mml:mi><mml:mo>−</mml:mo></mml:msup></mml:mrow></mml:msub></mml:math>and<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$X_{d}$ id=E7><mml:msub><mml:mi>X</mml:mi><mml:mi>d</mml:mi></mml:msub></mml:math>, where<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$]g,d[$ id=E8><mml:mrow><mml:mo>]</mml:mo><mml:mrow><mml:mi>g</mml:mi><mml:mo>,</mml:mo><mml:mi>d</mml:mi></mml:mrow><mml:mo>[</mml:mo></mml:mrow></mml:math>is an excursion interval of the regenerative set<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$M$ id=E9><mml:mi>M</mml:mi></mml:math>. We use the<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$(F_{D_{t}})$ id=E10><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>-predictable exit system to bring together the isolated points of<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$M$ id=E11><mml:mi>M</mml:mi></mml:math>and its perfect part and replace the classical optional exit system. This has been a subject in literature before (e.g., Kaspi (1988)) under the classical duality hypothesis. We define an “additive functional” for<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$(Y_{t})_{_{tin mathbb{R}}}$ id=E12><mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>Y</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>∈</mml:mo><mml:mi>ℝ</mml:mi></mml:mrow></mml:msub></mml:math>with<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$B$ id=E13><mml:mi>B</mml:mi></mml:math>, we generalize the laws cited before to<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$(Y_{t})_{_{tin mathbb{R}}}$ id=E14><mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>Y</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>∈</mml:mo><mml:mi>ℝ</mml:mi></mml:mrow></mml:msub></mml:math>, and we express laws of pairs of excursions." @default.
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W2007431354 date "2005-01-01" @default.
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W2007431354 title "Additive functionals and excursions of Kuznetsov processes" @default.
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W2007431354 doi "https://doi.org/10.1155/ijmms.2005.2031" @default.
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