On fluctuation theory for spectrally negative Lévy processes with Parisian reflection below, and applications
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Abstract
As is well known, all functionals of a Markov process may be expressed in terms of the generator operator, modulo some analytic work. In the case of spectrally negative Markov processes, however, it is conjectured that everything can be expressed in a more direct way using the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W"> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding="application/x-tex">W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> scale function which intervenes in the two-sided first passage problem, modulo performing various integrals. This conjecture arises from work on Levy processes [ <bold>6, 7, 12, 16, 28–30, 50</bold> ] where the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W"> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding="application/x-tex">W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> scale function has explicit Laplace transform, and is therefore easily computable; furthermore it was found in the papers above that a second scale function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z"> <mml:semantics> <mml:mi>Z</mml:mi> <mml:annotation encoding="application/x-tex">Z</mml:annotation> </mml:semantics> </mml:math> </inline-formula> introduced in [ <bold>7</bold> ] (this is an exponential transform ( <bold>8</bold> ) of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W"> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding="application/x-tex">W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ) greatly simplifies first passage laws, especially for reflected processes. <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z"> <mml:semantics> <mml:mi>Z</mml:mi> <mml:annotation encoding="application/x-tex">Z</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a harmonic function of the Lévy process (like <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W"> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding="application/x-tex">W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ), corresponding to exterior boundary conditions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w left-parenthesis x right-parenthesis equals e Superscript theta x"> <mml:semantics> <mml:mrow> <mml:mi>w</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> θ </mml:mi> <mml:mi>x</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">w(x)=e^{\theta x}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ( <bold>9</bold> ) and is also a particular case of a “smooth Gerber–Shiu function” <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper S Subscript w"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">S</mml:mi> </mml:mrow> <mml:mi>w</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {S}_w</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The concept of the Gerber–Shiu function was introduced in [ <bold>26</bold> ]; we will use it however here in the more restricted sense of [ <bold>15</bold> ], who define this to be a “smooth” harmonic function of the process, which fits the exterior boundary condition <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>w</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">w(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and simultaneously solves the problems ( <bold>17</bold> ), ( <bold>18</bold> ). It has been conjectured that similar laws govern other classes of spectrally negativeprocesses, but it is quite difficult to find assumptions which allow proving this for general classes of Markov processes. However, we show below that in the particular case of spectrally negative Lévy processes with Parisian absorption and reflection from below [ <bold>6, 16, 21</bold> ], this conjecture holds true, once the appropriate <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W"> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding="application/x-tex">W</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="upper Z"> <mml:semantics> <mml:mi>Z</mml:mi> <mml:annotation encoding="application/x-tex">Z</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are identified (this observation seems new). This paper gathers a collection of first passage formulas for spectrally negative Parisian Lévy processes, expressed in terms of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W"> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding="application/x-tex">W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z"> <mml:semantics> <mml:mi>Z</mml:mi> <mml:annotation encoding="application/x-tex">Z</mml:annotation> </mml:semantics> </mml:math> </inli
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