Pathwise Convergent Approximation for the Fractional SDEs
Fractional stochastic differential equation (FSDE)-based random processes are used in a wide spectrum of scientific disciplines. However, in the majority of cases, explicit solutions for these FSDEs do not exist and approximation schemes have to be applied. In this paper, we study one-dimensional st...
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MDPI AG
2022-02-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/4/669 |
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| author | Kęstutis Kubilius Aidas Medžiūnas |
| author_facet | Kęstutis Kubilius Aidas Medžiūnas |
| author_sort | Kęstutis Kubilius |
| collection | DOAJ |
| description | Fractional stochastic differential equation (FSDE)-based random processes are used in a wide spectrum of scientific disciplines. However, in the majority of cases, explicit solutions for these FSDEs do not exist and approximation schemes have to be applied. In this paper, we study one-dimensional stochastic differential equations (SDEs) driven by stochastic process with Hölder continuous paths of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><mn>2</mn><mo><</mo><mi>γ</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>. Using the Lamperti transformation, we construct a backward approximation scheme for the transformed SDE. The inverse transformation provides an approximation scheme for the original SDE which converges at the rate <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>h</mi><mrow><mn>2</mn><mi>γ</mi></mrow></msup></semantics></math></inline-formula>, where <i>h</i> is a time step size of a uniform partition of the time interval under consideration. This approximation scheme covers wider class of FSDEs and demonstrates higher convergence rate than previous schemes by other authors in the field. |
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| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T21:30:34Z |
| publishDate | 2022-02-01 |
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| spelling | doaj.art-7c893bff62e04467b0e4921f6366b06c2023-11-23T20:58:20ZengMDPI AGMathematics2227-73902022-02-0110466910.3390/math10040669Pathwise Convergent Approximation for the Fractional SDEsKęstutis Kubilius0Aidas Medžiūnas1Faculty of Mathematics and Informatics, Vilnius University, Akademijos g. 4, LT-08412 Vilnius, LithuaniaFaculty of Mathematics and Informatics, Vilnius University, Akademijos g. 4, LT-08412 Vilnius, LithuaniaFractional stochastic differential equation (FSDE)-based random processes are used in a wide spectrum of scientific disciplines. However, in the majority of cases, explicit solutions for these FSDEs do not exist and approximation schemes have to be applied. In this paper, we study one-dimensional stochastic differential equations (SDEs) driven by stochastic process with Hölder continuous paths of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><mn>2</mn><mo><</mo><mi>γ</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>. Using the Lamperti transformation, we construct a backward approximation scheme for the transformed SDE. The inverse transformation provides an approximation scheme for the original SDE which converges at the rate <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>h</mi><mrow><mn>2</mn><mi>γ</mi></mrow></msup></semantics></math></inline-formula>, where <i>h</i> is a time step size of a uniform partition of the time interval under consideration. This approximation scheme covers wider class of FSDEs and demonstrates higher convergence rate than previous schemes by other authors in the field.https://www.mdpi.com/2227-7390/10/4/669stochastic differential equationsfractional Brownian motionbackward approximationLamperti transformation |
| spellingShingle | Kęstutis Kubilius Aidas Medžiūnas Pathwise Convergent Approximation for the Fractional SDEs Mathematics stochastic differential equations fractional Brownian motion backward approximation Lamperti transformation |
| title | Pathwise Convergent Approximation for the Fractional SDEs |
| title_full | Pathwise Convergent Approximation for the Fractional SDEs |
| title_fullStr | Pathwise Convergent Approximation for the Fractional SDEs |
| title_full_unstemmed | Pathwise Convergent Approximation for the Fractional SDEs |
| title_short | Pathwise Convergent Approximation for the Fractional SDEs |
| title_sort | pathwise convergent approximation for the fractional sdes |
| topic | stochastic differential equations fractional Brownian motion backward approximation Lamperti transformation |
| url | https://www.mdpi.com/2227-7390/10/4/669 |
| work_keys_str_mv | AT kestutiskubilius pathwiseconvergentapproximationforthefractionalsdes AT aidasmedziunas pathwiseconvergentapproximationforthefractionalsdes |