On the Estimation of the Persistence Exponent for a Fractionally Integrated Brownian Motion by Numerical Simulations
For a fractionally integrated Brownian motion (FIBM) of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo&...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-01-01
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| Series: | Fractal and Fractional |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2504-3110/7/2/107 |
| _version_ | 1797620884881539072 |
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| author | Mario Abundo Enrica Pirozzi |
| author_facet | Mario Abundo Enrica Pirozzi |
| author_sort | Mario Abundo |
| collection | DOAJ |
| description | For a fractionally integrated Brownian motion (FIBM) of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> we investigate the decaying rate of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mo>(</mo><msubsup><mi>τ</mi><mi>S</mi><mi>α</mi></msubsup><mo>></mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo>→</mo><mo>+</mo><mo>∞</mo><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>τ</mi><mi>S</mi><mi>α</mi></msubsup><mo>=</mo><mo movablelimits="true" form="prefix">inf</mo><mrow><mo>{</mo><mi>t</mi><mo>></mo><mn>0</mn><mo>:</mo><msub><mi>X</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>≥</mo><mi>S</mi><mo>}</mo></mrow></mrow></semantics></math></inline-formula> is the first-passage time (FPT) of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> through the barrier <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>></mo><mn>0</mn><mo>.</mo></mrow></semantics></math></inline-formula> Precisely, we study the so-called persistent exponent <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>θ</mi><mo>=</mo><mi>θ</mi><mo>(</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula> of the FPT tail, such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mrow><mo>(</mo><msubsup><mi>τ</mi><mi>S</mi><mi>α</mi></msubsup><mo>></mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>t</mi><mrow><mo>−</mo><mi>θ</mi><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup><mo>,</mo></mrow></semantics></math></inline-formula> as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo>→</mo><mo>+</mo><mo>∞</mo><mo>,</mo></mrow></semantics></math></inline-formula> and by means of numerical simulation of long enough trajectories of the process <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> we are able to estimate <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>θ</mi><mo>(</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula> and to show that it is a non-increasing function of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo><mo>,</mo></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><mn>4</mn><mo>≤</mo><mi>θ</mi><mo>(</mo><mi>α</mi><mo>)</mo><mo>≤</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>.</mo></mrow></semantics></math></inline-formula> In particular, we are able to validate numerically a new conjecture about the analytical expression of the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>θ</mi><mo>=</mo><mi>θ</mi><mo>(</mo><mi>α</mi><mo>)</mo><mo>,</mo></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo><mo>.</mo></mrow></semantics></math></inline-formula> Such a numerical validation is carried out in two ways: in the first one, we estimate <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>θ</mi><mo>(</mo><mi>α</mi><mo>)</mo><mo>,</mo></mrow></semantics></math></inline-formula> by using the simulated FPT density, obtained for any <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo><mo>;</mo></mrow></semantics></math></inline-formula> in the second one, we estimate the persistent exponent by directly calculating <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mfenced separators="" open="(" close=")"><msub><mo movablelimits="true" form="prefix">max</mo><mrow><mn>0</mn><mo>≤</mo><mi>s</mi><mo>≤</mo><mi>t</mi></mrow></msub><msub><mi>X</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo><</mo><mn>1</mn></mfenced><mo>.</mo></mrow></semantics></math></inline-formula> Both ways confirm our conclusions within the limit of numerical approximation. Finally, we investigate the self-similarity property of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and we find the upper bound of its covariance function. |
| first_indexed | 2024-03-11T08:48:53Z |
| format | Article |
| id | doaj.art-6f9a16b01ced4a7e94e81ef6ed89bd9c |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-11T08:48:53Z |
| publishDate | 2023-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-6f9a16b01ced4a7e94e81ef6ed89bd9c2023-11-16T20:36:06ZengMDPI AGFractal and Fractional2504-31102023-01-017210710.3390/fractalfract7020107On the Estimation of the Persistence Exponent for a Fractionally Integrated Brownian Motion by Numerical SimulationsMario Abundo0Enrica Pirozzi1Dipartimento di Matematica, Università “Tor Vergata”, Via della Ricerca Scientifica, I-00133 Roma, ItalyDipartimento di Matematica e Applicazioni, Università di Napoli “Federico II”, Via Cintia, Complesso Monte S. Angelo, I-80126 Napoli, ItalyFor a fractionally integrated Brownian motion (FIBM) of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> we investigate the decaying rate of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mo>(</mo><msubsup><mi>τ</mi><mi>S</mi><mi>α</mi></msubsup><mo>></mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo>→</mo><mo>+</mo><mo>∞</mo><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>τ</mi><mi>S</mi><mi>α</mi></msubsup><mo>=</mo><mo movablelimits="true" form="prefix">inf</mo><mrow><mo>{</mo><mi>t</mi><mo>></mo><mn>0</mn><mo>:</mo><msub><mi>X</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>≥</mo><mi>S</mi><mo>}</mo></mrow></mrow></semantics></math></inline-formula> is the first-passage time (FPT) of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> through the barrier <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>></mo><mn>0</mn><mo>.</mo></mrow></semantics></math></inline-formula> Precisely, we study the so-called persistent exponent <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>θ</mi><mo>=</mo><mi>θ</mi><mo>(</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula> of the FPT tail, such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mrow><mo>(</mo><msubsup><mi>τ</mi><mi>S</mi><mi>α</mi></msubsup><mo>></mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>t</mi><mrow><mo>−</mo><mi>θ</mi><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup><mo>,</mo></mrow></semantics></math></inline-formula> as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo>→</mo><mo>+</mo><mo>∞</mo><mo>,</mo></mrow></semantics></math></inline-formula> and by means of numerical simulation of long enough trajectories of the process <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> we are able to estimate <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>θ</mi><mo>(</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula> and to show that it is a non-increasing function of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo><mo>,</mo></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><mn>4</mn><mo>≤</mo><mi>θ</mi><mo>(</mo><mi>α</mi><mo>)</mo><mo>≤</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>.</mo></mrow></semantics></math></inline-formula> In particular, we are able to validate numerically a new conjecture about the analytical expression of the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>θ</mi><mo>=</mo><mi>θ</mi><mo>(</mo><mi>α</mi><mo>)</mo><mo>,</mo></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo><mo>.</mo></mrow></semantics></math></inline-formula> Such a numerical validation is carried out in two ways: in the first one, we estimate <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>θ</mi><mo>(</mo><mi>α</mi><mo>)</mo><mo>,</mo></mrow></semantics></math></inline-formula> by using the simulated FPT density, obtained for any <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo><mo>;</mo></mrow></semantics></math></inline-formula> in the second one, we estimate the persistent exponent by directly calculating <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mfenced separators="" open="(" close=")"><msub><mo movablelimits="true" form="prefix">max</mo><mrow><mn>0</mn><mo>≤</mo><mi>s</mi><mo>≤</mo><mi>t</mi></mrow></msub><msub><mi>X</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo><</mo><mn>1</mn></mfenced><mo>.</mo></mrow></semantics></math></inline-formula> Both ways confirm our conclusions within the limit of numerical approximation. Finally, we investigate the self-similarity property of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and we find the upper bound of its covariance function.https://www.mdpi.com/2504-3110/7/2/107fractional integralsfirst-passage timedecaying ratetail distribution |
| spellingShingle | Mario Abundo Enrica Pirozzi On the Estimation of the Persistence Exponent for a Fractionally Integrated Brownian Motion by Numerical Simulations Fractal and Fractional fractional integrals first-passage time decaying rate tail distribution |
| title | On the Estimation of the Persistence Exponent for a Fractionally Integrated Brownian Motion by Numerical Simulations |
| title_full | On the Estimation of the Persistence Exponent for a Fractionally Integrated Brownian Motion by Numerical Simulations |
| title_fullStr | On the Estimation of the Persistence Exponent for a Fractionally Integrated Brownian Motion by Numerical Simulations |
| title_full_unstemmed | On the Estimation of the Persistence Exponent for a Fractionally Integrated Brownian Motion by Numerical Simulations |
| title_short | On the Estimation of the Persistence Exponent for a Fractionally Integrated Brownian Motion by Numerical Simulations |
| title_sort | on the estimation of the persistence exponent for a fractionally integrated brownian motion by numerical simulations |
| topic | fractional integrals first-passage time decaying rate tail distribution |
| url | https://www.mdpi.com/2504-3110/7/2/107 |
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