Bernstein Fractional Derivatives: Censoring and Stochastic Processes

Abstract We define censored fractional Bernstein derivatives on the positive half-line based on the Bernstein–Riemann–Liouville fractional derivative. The censored fractional derivative turns out to be the generator of the censored decreasing subordinator $$S^c = (S_t^c)_{t\geqslant 0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>S</mml:mi> <mml:mi>c</mml:mi> </mml:msup> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>t</mml:mi> <mml:mi>c</mml:mi> </mml:msubsup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> . We provide two constructions of the censored subordinator: (i) pathwise by removing those jumps from the decreasing subordinator $$(x-S_t)_{t\geqslant 0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>-</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math> , $$x&gt;0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , that drive the path into negative territory. (ii) via a semigroup approach and the Hille–Yosida theorem. Then we show that the censored decreasing subordinator has only finite life-time, and we identify various probability distributions related to $$S^c.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>S</mml:mi> <mml:mi>c</mml:mi> </mml:msup> <mml:mo>.</mml:mo> </mml:mrow> </mml:math>