Adopting Feynman–Kac Formula in Stochastic Differential Equations with (Sub-)Fractional Brownian Motion
The aim of this work is to establish and generalize a relationship between fractional partial differential equations (fPDEs) and stochastic differential equations (SDEs) to a wider class of stochastic processes, including fractional Brownian motions <inline-formula><math xmlns="http://...
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MDPI AG
2022-01-01
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| Online Access: | https://www.mdpi.com/2227-7390/10/3/340 |
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| author | Bodo Herzog |
| author_facet | Bodo Herzog |
| author_sort | Bodo Herzog |
| collection | DOAJ |
| description | The aim of this work is to establish and generalize a relationship between fractional partial differential equations (fPDEs) and stochastic differential equations (SDEs) to a wider class of stochastic processes, including fractional Brownian motions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msubsup><mi>B</mi><mi>t</mi><mi>H</mi></msubsup><mo>,</mo><mi>t</mi><mo>≥</mo><mn>0</mn><mo>}</mo></mrow></semantics></math></inline-formula> and sub-fractional Brownian motions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msubsup><mi>ξ</mi><mi>t</mi><mi>H</mi></msubsup><mo>,</mo><mi>t</mi><mo>≥</mo><mn>0</mn><mo>}</mo></mrow></semantics></math></inline-formula> with Hurst parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>∈</mo><mo>(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>. We start by establishing the connection between a fPDE and SDE via the Feynman–Kac Theorem, which provides a stochastic representation of a general Cauchy problem. In hindsight, we extend this connection by assuming SDEs with fractional- and sub-fractional Brownian motions and prove the generalized Feynman–Kac formulas under a (sub-)fractional Brownian motion. An application of the theorem demonstrates, as a by-product, the solution of a fractional integral, which has relevance in probability theory. |
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| language | English |
| last_indexed | 2024-03-09T23:32:50Z |
| publishDate | 2022-01-01 |
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| series | Mathematics |
| spelling | doaj.art-1c4d856ff2a6482fabca4f94a47d7dea2023-11-23T17:05:56ZengMDPI AGMathematics2227-73902022-01-0110334010.3390/math10030340Adopting Feynman–Kac Formula in Stochastic Differential Equations with (Sub-)Fractional Brownian MotionBodo Herzog0Economics Department, ESB Business School, Reutlingen University, 72762 Reutlingen, GermanyThe aim of this work is to establish and generalize a relationship between fractional partial differential equations (fPDEs) and stochastic differential equations (SDEs) to a wider class of stochastic processes, including fractional Brownian motions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msubsup><mi>B</mi><mi>t</mi><mi>H</mi></msubsup><mo>,</mo><mi>t</mi><mo>≥</mo><mn>0</mn><mo>}</mo></mrow></semantics></math></inline-formula> and sub-fractional Brownian motions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msubsup><mi>ξ</mi><mi>t</mi><mi>H</mi></msubsup><mo>,</mo><mi>t</mi><mo>≥</mo><mn>0</mn><mo>}</mo></mrow></semantics></math></inline-formula> with Hurst parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>∈</mo><mo>(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>. We start by establishing the connection between a fPDE and SDE via the Feynman–Kac Theorem, which provides a stochastic representation of a general Cauchy problem. In hindsight, we extend this connection by assuming SDEs with fractional- and sub-fractional Brownian motions and prove the generalized Feynman–Kac formulas under a (sub-)fractional Brownian motion. An application of the theorem demonstrates, as a by-product, the solution of a fractional integral, which has relevance in probability theory.https://www.mdpi.com/2227-7390/10/3/340Cauchy problemfractional-PDESDEfractional Brownian motionsub-fractional processesFeynman–Kac formula |
| spellingShingle | Bodo Herzog Adopting Feynman–Kac Formula in Stochastic Differential Equations with (Sub-)Fractional Brownian Motion Mathematics Cauchy problem fractional-PDE SDE fractional Brownian motion sub-fractional processes Feynman–Kac formula |
| title | Adopting Feynman–Kac Formula in Stochastic Differential Equations with (Sub-)Fractional Brownian Motion |
| title_full | Adopting Feynman–Kac Formula in Stochastic Differential Equations with (Sub-)Fractional Brownian Motion |
| title_fullStr | Adopting Feynman–Kac Formula in Stochastic Differential Equations with (Sub-)Fractional Brownian Motion |
| title_full_unstemmed | Adopting Feynman–Kac Formula in Stochastic Differential Equations with (Sub-)Fractional Brownian Motion |
| title_short | Adopting Feynman–Kac Formula in Stochastic Differential Equations with (Sub-)Fractional Brownian Motion |
| title_sort | adopting feynman kac formula in stochastic differential equations with sub fractional brownian motion |
| topic | Cauchy problem fractional-PDE SDE fractional Brownian motion sub-fractional processes Feynman–Kac formula |
| url | https://www.mdpi.com/2227-7390/10/3/340 |
| work_keys_str_mv | AT bodoherzog adoptingfeynmankacformulainstochasticdifferentialequationswithsubfractionalbrownianmotion |