First passage times over stochastic boundaries for subdiffusive processes

Constantinescu, C ORCID: 0000-0002-5219-3022, Loeffen, R and Patie, P (2022) First passage times over stochastic boundaries for subdiffusive processes Transactions of the American Mathematical Society, 375 (3). pp. 1629-1652. ISSN 0002-9947, 1088-6850

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Abstract

<p>Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper X equals left-parenthesis double-struck upper X Subscript t Baseline right-parenthesis Subscript t greater-than-or-equal-to 0"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">X</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">X</mml:mi> </mml:mrow> <mml:mi>t</mml:mi> </mml:msub> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>t</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {X}=(\mathbb {X}_t)_{t\geq 0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the subdiffusive process defined, for any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t greater-than-or-equal-to 0"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">t\geq 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper X Subscript t Baseline equals upper X Subscript script l Sub Subscript t"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">X</mml:mi> </mml:mrow> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {X}_t = X_{\ell _t}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X equals left-parenthesis upper X Subscript t Baseline right-parenthesis Subscript t greater-than-or-equal-to 0"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>t</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">X=(X_t)_{t\geq 0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a Lévy process and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l Subscript t Baseline equals inf left-brace s greater-than 0 semicolon script upper K Subscript s Baseline greater-than t right-brace"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mo movablelimits="true" form="prefix">inf</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>s</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> <mml:mo>;</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">K</mml:mi> </mml:mrow> <mml:mi>s</mml:mi> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:mi>t</mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\ell _t=\inf \{s&gt;0; \mathcal {K}_s&gt;t \}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper K equals left-parenthesis script upper K Subscript t Baseline right-parenthesis Subscript t greater-than-or-equal-to 0"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">K</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">K</mml:mi> </mml:mrow> <mml:mi>t</mml:mi> </mml:msub> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>t</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {K}=(\mathcal {K}_t)_{t\geq 0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a subordinator independent of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We start by developing a composite Wiener-Hopf factorization to characterize the law of the pair <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis double-struck upper T Subscript a Superscript left-parenthesis b right-parenthesis Baseline comma left-parenthesis double-struck upper X minus b right-parenthesis Subscript double-struck upper T Sub Subscript a Sub Superscript left-parenthesis b right-parenthesis Subscript Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> <mml:mi>a</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>b</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo>,</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">X</mml:mi> </mml:mrow> <mml:mo>−</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>b</mml:mi> </mml:mrow> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> <mml:mi>a</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>b</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\mathbb {T}_a^{({b})}, (\mathbb {X} - {b})_{\mathbb {T}_a^{({b})}})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper T Subscript a Superscript left-parenthesis b right-parenthesis Baseline equals inf left-brace t greater-than 0 semicolon double-struck upper X Subscript t Baseline greater-than a plus b Subscript t Baseline right-brace"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> <mml:mi>a</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>b</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo>=</mml:mo> <mml:mo movablelimits="true" form="prefix">inf</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>t</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> <mml:mo>;</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">X</mml:mi> </mml:mrow> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:mi>a</mml:mi> <mml:mo>+</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>b</mml:mi> </mml:mrow> <mml:mi>t</mml:mi> </mml:msub> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\begin{equation*}\mathbb {T}_a^{({b})} = \inf \{t&gt;0; \mathbb {X}_t &gt; a+ {b}_t \} \end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a element-of double-struck upper R"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">a \in \mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b equals left-parenthesis b Subscript t Baseline right-parenthesis Subscript t greater-than-or-equal-to 0"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>b</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>b</mml:mi> </mml:mrow> <mml:mi>t</mml:mi> </mml:msub> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>t</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{b}=({b}_t)_{t\geq 0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a (possibly degenerate) subordinator independent of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper K"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">K</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {K}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We proceed by providing a detailed analysis of the cases where either <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper X"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a self-similar or is spectrally negative. For the later, we show the fact that the process <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis double-struck upper T Subscript a Superscript left-parenthesis b right-parenthesis Baseline right-parenthesis Subscript a greater-than-or-equal-to 0"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> <mml:mi>a</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>b</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msubsup> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>a</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">(\mathbb {T}_a^{({b})})_{a\geq 0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a subordinator. Our proofs hinge on a variety of techniques including excursion theory, change of measure, asymptotic analysis and on establishing a link between subdiffusive processes and a subclass of semi-regenerative processes. In particular, we show that the variable <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper T Subscript a Superscript left-parenthesis b right-parenthesis"> <mml:semantics> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> <mml:mi>a</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>b</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msubsup> <mml:annotation encoding="application/x-tex">\mathbb {T}_a^{({b})}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has the same law as the first passage time of a <italic>semi-regenerative process of Lévy type</italic>, a terminology that we introduce to mean that this process satisfies the Markov property of Lévy processes for stopping times whose graph is included in the associated regeneration set.</p>

Item Type:	Article
Divisions:	Faculty of Science & Engineering > School of Physical Sciences
Depositing User:	Symplectic Admin
Date Deposited:	15 Nov 2021 08:18
Last Modified:	06 Dec 2024 20:41
DOI:	10.1090/tran/8534
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URI:	https://livrepository.liverpool.ac.uk/id/eprint/3143132
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• FIRST PASSAGE TIMES OVER STOCHASTIC BOUNDARIES FOR SUBDIFFUSIVE PROCESSES (deposited 18 Oct 2021 07:47)
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