A variation of constant formula for Caputo fractional stochastic differential equations
We establish and prove a variation of constant formula for Caputo fractional stochastic differential equations whose coefficients satisfy a standard Lipschitz condition. The main ingredient in the proof is to use Ito’s representation theorem and the known variation of constant formula for determinis...
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| Published in | Statistics & probability letters Vol. 145; pp. 351 - 358 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier B.V
01.02.2019
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| Abstract | We establish and prove a variation of constant formula for Caputo fractional stochastic differential equations whose coefficients satisfy a standard Lipschitz condition. The main ingredient in the proof is to use Ito’s representation theorem and the known variation of constant formula for deterministic Caputo fractional differential equations. As a consequence, for these systems we point out the coincidence between the notion of classical solutions introduced in Wang et al. (2016) and mild solutions introduced in Sakthivel et al. (2013). |
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| AbstractList | We establish and prove a variation of constant formula for Caputo fractional stochastic differential equations whose coefficients satisfy a standard Lipschitz condition. The main ingredient in the proof is to use Ito’s representation theorem and the known variation of constant formula for deterministic Caputo fractional differential equations. As a consequence, for these systems we point out the coincidence between the notion of classical solutions introduced in Wang et al. (2016) and mild solutions introduced in Sakthivel et al. (2013). |
| Author | Huong, P.T. Anh, P.T. Doan, T.S. |
| Author_xml | – sequence: 1 givenname: P.T. surname: Anh fullname: Anh, P.T. email: phamtheanhhn@gmail.com organization: Le Quy Don Technical University, 236 Hoang QuocViet, Ha Noi, Viet Nam – sequence: 2 givenname: T.S. surname: Doan fullname: Doan, T.S. email: dtson@math.ac.vn organization: Institute of Mathematics, Viet Nam Academy of Science and Technology, 18 Hoang Quoc Viet, Ha Noi, Viet Nam – sequence: 3 givenname: P.T. surname: Huong fullname: Huong, P.T. email: pthuong175@gmail.com organization: Le Quy Don Technical University, 236 Hoang QuocViet, Ha Noi, Viet Nam |
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| Cites_doi | 10.1016/j.cam.2015.12.020 10.14232/ejqtde.2016.1.39 10.1016/j.na.2012.10.009 10.1016/j.na.2016.01.020 10.1007/s11071-016-3002-z 10.1080/07362994.2018.1440243 10.3934/dcdsb.2017164 10.1016/j.aml.2011.05.035 |
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| Keywords | Fractional stochastic differential equations Mild solution Classical solution A variation of constant formula Inhomogeneous linear systems |
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| References | Cong, Doan, Tuan, Siegmund (b4) 2016; 86 Li, Peng (b8) 2011; 24 Doan, Kloeden, Huong, Tuan (b7) 2018; 36 Cong, Doan, Tuan, Siegmund (b3) 2016; 39 Wang, Xu, Kloeden (b13) 2016; 135 Podlubny (b11) 1999 Cong, Doan, Tuan, Siegmund (b5) 2017; 22 Bandyopadhyay, Kamal (b1) 2015; vol. 317 Øksendal (b10) 2000 Diethelm (b6) 2010; vol. 2004 Sakthivel, Revathi, Ren (b12) 2013; 81 Mao (b9) 2008 Benchaabane, Sakthivel (b2) 2017; 312 Diethelm (10.1016/j.spl.2018.10.010_b6) 2010; vol. 2004 Wang (10.1016/j.spl.2018.10.010_b13) 2016; 135 Bandyopadhyay (10.1016/j.spl.2018.10.010_b1) 2015; vol. 317 Doan (10.1016/j.spl.2018.10.010_b7) 2018; 36 Cong (10.1016/j.spl.2018.10.010_b4) 2016; 86 Benchaabane (10.1016/j.spl.2018.10.010_b2) 2017; 312 Øksendal (10.1016/j.spl.2018.10.010_b10) 2000 Cong (10.1016/j.spl.2018.10.010_b3) 2016; 39 Mao (10.1016/j.spl.2018.10.010_b9) 2008 Sakthivel (10.1016/j.spl.2018.10.010_b12) 2013; 81 Li (10.1016/j.spl.2018.10.010_b8) 2011; 24 Podlubny (10.1016/j.spl.2018.10.010_b11) 1999 Cong (10.1016/j.spl.2018.10.010_b5) 2017; 22 |
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| SubjectTerms | A variation of constant formula Classical solution Fractional stochastic differential equations Inhomogeneous linear systems Mild solution |
| Title | A variation of constant formula for Caputo fractional stochastic differential equations |
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