{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,18]],"date-time":"2026-03-18T00:52:18Z","timestamp":1773795138586,"version":"3.50.1"},"reference-count":41,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2018,1,16]],"date-time":"2018-01-16T00:00:00Z","timestamp":1516060800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we investigate analytical solutions of multi-time scale fractional stochastic differential equations driven by fractional Brownian motions. We firstly decompose homogeneous multi-time scale fractional stochastic differential equations driven by fractional Brownian motions into independent differential subequations, and give their analytical solutions. Then, we use the variation of constant parameters to obtain the solutions of nonhomogeneous multi-time scale fractional stochastic differential equations driven by fractional Brownian motions. Finally, we give three examples to demonstrate the applicability of our obtained results.<\/jats:p>","DOI":"10.3390\/e20010063","type":"journal-article","created":{"date-parts":[[2018,1,17]],"date-time":"2018-01-17T04:23:44Z","timestamp":1516163024000},"page":"63","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":23,"title":["Analytical Solutions for Multi-Time Scale Fractional Stochastic Differential Equations Driven by Fractional Brownian Motion and Their Applications"],"prefix":"10.3390","volume":"20","author":[{"given":"Xiao-Li","family":"Ding","sequence":"first","affiliation":[{"name":"Department of Mathematics, Xi\u2019an Polytechnic University, Xi\u2019an 710048, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8202-6578","authenticated-orcid":false,"given":"Juan","family":"Nieto","sequence":"additional","affiliation":[{"name":"Departamento de Estat\u00edstica, An\u00e1lisis Matem\u00e1tico y Optimizaci\u00f3n, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,1,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1403","DOI":"10.1029\/2000WR900031","article-title":"Application of a fractional advection-dispersion equation","volume":"36","author":"Benson","year":"2000","journal-title":"Water Resour. Res."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Hilfer, R. (2000). Applications of Fractional Calculus in Physics, World Scientific.","DOI":"10.1142\/9789812817747"},{"key":"ref_3","unstructured":"Morel, J.-M., Cachan, ENS, Teissier, B., and Paris 7, U. (2017). The mathematical theories of diffusion: Nonlinear and fractional diffusion. Lecture Notes in Mathematics, Springer."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1002\/mrm.21453","article-title":"From diffusion-weighted MRI to anomalous diffusion imaging","volume":"59","author":"Hall","year":"2008","journal-title":"Magn. Reson. Med."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2344","DOI":"10.1016\/j.aml.2012.06.029","article-title":"Anomalous transport of particles in plasma physics","volume":"25","author":"Cesbron","year":"2012","journal-title":"Appl. Math. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0370-1573(00)00070-3","article-title":"The random walk\u2019s guide to anomalous diffusion: A fractional dynamics approach","volume":"339","author":"Metzler","year":"2000","journal-title":"Phys. Rep."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"5095","DOI":"10.1016\/j.physa.2012.06.002","article-title":"Model of lateral diffusion in ultrathin layered films","volume":"391","author":"Postnikov","year":"2012","journal-title":"Phys. A"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Wang, Y., Zheng, S., Zhang, W., and Wang, J. (2017). Complex and Entropy of Fluctuations of Agent-Based Interacting Financial Dynamics with Random Jump. Entropy, 19.","DOI":"10.3390\/e19100512"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"San-Millan, A., Feliu-Talegon, D., Feliu-Batlle, V., and Rivas-Perez, R. (2017). On the Modelling and Control of a Laboratory Prototype of a Hydraulic Canal Based on a TITO Fractional-Order Model. Entropy, 19.","DOI":"10.3390\/e19080401"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1177\/1687814015618127","article-title":"Fractional electrical circuits","volume":"7","author":"Alsaedi","year":"2015","journal-title":"Adv. Mech. Eng."},{"key":"ref_11","first-page":"269","article-title":"Development of nonlinear stochastic models by using stock price data and basic statistics","volume":"18","author":"Ladde","year":"2010","journal-title":"Neutral Parallel Sci. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"334","DOI":"10.1016\/j.jmaa.2012.07.062","article-title":"Fractional stochastic differential equations with applications to finance","volume":"397","author":"Tien","year":"2013","journal-title":"J. Math. Anal. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2066","DOI":"10.1016\/j.camwa.2017.02.031","article-title":"Derivation of a new Merton\u2019s optimal problem presented by fractional stochastic stock price and its applications","volume":"73","author":"Farhadi","year":"2017","journal-title":"Comput. Math. Appl."},{"key":"ref_14","unstructured":"Arnold, L. (1974). Stochastic Differential Equations: Theory and Applications, Wiley."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1137\/1010093","article-title":"Fractional Brownian motions, fractional noises and applications","volume":"10","author":"Mandelbrot","year":"1968","journal-title":"SIAM Rev."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1007\/s10959-014-0578-4","article-title":"Solving a nonlinear fractional stochastic partial differential equation with fractional noise","volume":"29","author":"Liu","year":"2016","journal-title":"J. Theor. Probab."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1186\/s13662-017-1151-0","article-title":"Some properties of the solution to fractional heat equation with a fractional Brownian noise","volume":"2017","author":"Xia","year":"2017","journal-title":"Adv. Differ. Equ."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"336","DOI":"10.1016\/j.cam.2017.02.027","article-title":"Numerical solution of stochastic fractional integro-differential equation by the spectral collocation method","volume":"321","author":"Taheri","year":"2017","journal-title":"J. Comput. Appl. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1016\/j.amc.2017.02.013","article-title":"Moment stability via resolvent operators of fractional stochastic differential inclusions driven by fractional Brownian motion","volume":"305","author":"Tamilalagan","year":"2017","journal-title":"Appl. Math. Comput."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1354","DOI":"10.1016\/j.spa.2016.08.002","article-title":"Intermittency fronts for space-time fractional stochastic partial differential equations in (d + 1) dimensions","volume":"127","author":"Asogwa","year":"2017","journal-title":"Stoch. Process. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"346","DOI":"10.4208\/jcm.1607-m2015-0329","article-title":"Error Estimates of Finite Element Methods for Stochastic Fractional Differential Equations","volume":"35","author":"Li","year":"2017","journal-title":"J. Comput. Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"273","DOI":"10.15388\/NA.2017.2.9","article-title":"The rate of convergence of the Hurst index estimate for a stochastic differential equation","volume":"22","author":"Kubilius","year":"2017","journal-title":"Nonlinear Anal. Model. Control"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1393","DOI":"10.1515\/fca-2016-0072","article-title":"On the solution of two-sided fractional ordinary differential equations of Caputo type","volume":"19","author":"Kolokoltsov","year":"2016","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1016\/j.jmaa.2016.03.033","article-title":"Stochastic solution of fractional Fokker\u2013Planck equations with space\u2013time-dependent coefficients","volume":"442","author":"Nane","year":"2016","journal-title":"J. Math. Anal. Appl."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Kloeden, P.E., and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations, Springer.","DOI":"10.1007\/978-3-662-12616-5"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1934","DOI":"10.1016\/j.jde.2011.07.032","article-title":"Stochastic generalized Burgers equations driven by fractional noises","volume":"252","author":"Jiang","year":"2011","journal-title":"J. Differ. Equ."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1007\/s00245-001-0001-2","article-title":"Heat equations with fractional white noise potentials","volume":"43","author":"Hu","year":"2001","journal-title":"Appl. Math. Optim."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1080\/1045112031000078917","article-title":"Stochastic integration with respect to the fractional Brownian motion","volume":"75","author":"Nualart","year":"2003","journal-title":"Stoch. Stoch. Rep."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1016\/j.chaos.2011.12.009","article-title":"Stochastic fractional differential equations: Modeling, method and analysis","volume":"45","author":"Pedjeu","year":"2012","journal-title":"Chaos Solitons Fractals"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Singh, J., Kumar, D., Qurashi, M.A., and Baleanu, D. (2017). A Novel Numerical Approach for a Nonlinear Fractional Dynamical Model of Interpersonal and Romantic Relationships. Entropy, 19.","DOI":"10.3390\/e19070375"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.cnsns.2016.07.023","article-title":"Implicit Euler approximation of stochastic evolution equations with fractional Brownian motion","volume":"44","author":"Kamrani","year":"2017","journal-title":"Commun. Nonlinear Sci. Numer. Simulat."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"557","DOI":"10.1007\/s10543-016-0639-4","article-title":"Sharp mean-square regularity results for SPDEs with fractional noise and optimal convergence rates for the numerical approximations","volume":"57","author":"Wang","year":"2016","journal-title":"BIT Numer. Math."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1117","DOI":"10.1016\/j.jmaa.2011.12.055","article-title":"Analytical solutions for the multi-term time\u2013space Caputo\u2013Riesz fractional advection\u2013diffusion equations on a finite domain","volume":"389","author":"Jiang","year":"2012","journal-title":"J. Math. Anal. Appl."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1410","DOI":"10.1016\/j.aim.2010.07.016","article-title":"Fractional Laplacian in conformal geometry","volume":"226","author":"Chang","year":"2011","journal-title":"Adv. Math."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1026","DOI":"10.1016\/j.nonrwa.2012.08.014","article-title":"Analytical solutions for the multi-term time\u2013space fractional advection\u2013diffusion equations with mixed boundary conditions","volume":"14","author":"Ding","year":"2013","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/j.cnsns.2017.04.020","article-title":"Analytical solutions for coupling fractional partial differential equations with Dirichlet boundary conditions","volume":"52","author":"Ding","year":"2017","journal-title":"Commun. Nonlinear Sci. Numer. Simulat."},{"key":"ref_37","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier Science B.V.. North-Holland Mathematics Studies."},{"key":"ref_38","unstructured":"It\u00f4, K. (1978). Stochastic Differential Equations, Wiley Interscience."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/S0304-4149(02)00212-0","article-title":"An It\u00f4 formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter","volume":"104","author":"Bender","year":"2003","journal-title":"Stoch. Process. Appl."},{"key":"ref_40","unstructured":"Nualart, D. (2006). The Malliavin Calculus and Related Topics, Springer-Verlag."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1016\/j.chaos.2017.05.013","article-title":"Pricing of basket options in subdiffusive fractional Black-Scholes model","volume":"102","author":"Karipova","year":"2017","journal-title":"Chaos Solitons Fractals"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/1\/63\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T14:51:30Z","timestamp":1760194290000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/1\/63"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1,16]]},"references-count":41,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2018,1]]}},"alternative-id":["e20010063"],"URL":"https:\/\/doi.org\/10.3390\/e20010063","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1,16]]}}}