Rough differential equations with path-dependent coefficients
[Équations différentielles rugueuses avec coefficients dépendant du chemin]
Ananova, Anna1
1 Mathematical Institute/ University of Oxford/ Radcliffe Observatory, Andrew Wiles Building, Woodstock Rd, Oxford OX2 6GG (United Kingdom)
Annales Henri Lebesgue, Tome 6 (2023), pp. 1-29

We establish the existence of solutions to path-dependent rough differential equations with non-anticipative coefficients. Regularity assumptions on the coefficients are formulated in terms of horizontal and vertical Dupire derivatives.

Nous montrons l’existence de solutions aux équations différentielles rugueuses dépendant du chemin avec des coefficients non anticipatifs. Les hypothèses de régularité sur les coefficients sont formulées en termes de dérivées de Dupire horizontales et verticales.

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DOI : 10.5802/ahl.157
Classification : 60H05, 60H99, 60H10, 60G17
Keywords: rough differential equation, path-dependent coefficients, functional Ito calculus, rough paths

Ananova, Anna 1

1 Mathematical Institute/ University of Oxford/ Radcliffe Observatory, Andrew Wiles Building, Woodstock Rd, Oxford OX2 6GG (United Kingdom)
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Ananova, Anna. Rough differential equations with path-dependent coefficients. Annales Henri Lebesgue, Tome 6 (2023), pp. 1-29. doi: 10.5802/ahl.157

[Ana18] Ananova, Anna Pathwise Integration and Functional Calculus for paths with finite quadratic variation, Ph. D. Thesis, Imperial College London, London, United Kingdom (2018)

[Bai14] Bailleul, Ismaël Flows Driven by Banach Space-Valued Rough Paths, Séminaire de Probabilités XLVI (Donati-Martin, Catherine; Lejay, Antoine; Rouault, Alain, eds.), Springer, 2014, pp. 195-205 | DOI | Zbl | MR

[Bai15] Bailleul, Ismaël Flows driven by rough paths, Rev. Mat. Iberoam., Volume 31 (2015) no. 3, pp. 901-934 | DOI | Zbl | MR

[CF10a] Cont, Rama; Fournié, David-Antoine Change of variable formulas for non-anticipative functionals on path space, J. Funct. Anal., Volume 259 (2010) no. 4, pp. 1043-1072 | DOI | MR | Zbl

[CF10b] Cont, Rama; Fournié, David-Antoine A functional extension of the Ito formula, C. R. Math. Acad. Sci. Paris, Volume 348 (2010) no. 1-2, pp. 57-61 | MR | DOI | Zbl | Numdam

[CF13] Cont, Rama; Fournié, David-Antoine Functional Ito Calculus and Stochastic Integral representation of Martingales, Ann. Probab., Volume 41 (2013) no. 1, pp. 109-133 | Zbl | MR

[CF16] Cont, Rama; Fournié, David-Antoine Functional Ito Calculus and functional Kolmogorov equations, Stochastic Integration by Parts and Functional Ito Calculus (Lecture Notes of the Barcelona Summer School in Stochastic Analysis, July 2012) (Advanced Courses in Mathematics – CRM Barcelona), Birkhäuser/Springer, 2016, pp. 115-208 | Zbl

[CK20] Cont, Rama; Kalinin, Alexander On the support of solutions to stochastic differential equations with path-dependent coefficients, Stochastic Processes Appl., Volume 130 (2020) no. 5, pp. 2639-2674 | DOI | Zbl | MR

[Dav08] Davie, Alexander M. Differential Equations Driven by Rough Paths: An Approach via Discrete Approximation, AMRX, Appl. Math. Res. Express, Volume 2008 (2008), abm009 | DOI | Zbl

[DGHT19] Deya, Aurelien; Gubinelli, Massimiliano; Hofmanova, Martina; Tindel, Samy One-dimensional reflected rough differential equations, Stochastic Processes Appl., Volume 129 (2019) no. 9, pp. 3261 -3281 | DOI | Zbl | MR

[Dup19] Dupire, Bruno Functional Ito calculus, Quant. Finance, Volume 19 (2019) no. 5, pp. 721-729 | DOI | Zbl | MR

[FH14] Friz, Peter K.; Hairer, Martin A course on rough paths. With an introduction to regularity structures, Springer, 2014 | DOI | MR | Zbl

[FS17] Friz, Peter K.; Shekhar, Atul General rough integration, Lévy rough paths and a Lévy-Kintchine-type formula, Ann. Probab., Volume 45 (2017) no. 4, pp. 2707-2765 | DOI | MR | Zbl

[FV10] Friz, Peter K.; Victoir, Nicolas B. Multidimensional stochastic processes as rough paths. Theory and applications, Cambridge Studies in Advanced Mathematics, 120, Cambridge University Press, 2010 | DOI | MR | Zbl

[FZ18] Friz, Peter K.; Zhang, Huilin Differential equations driven by rough paths with jumps, J. Differ. Equations, Volume 264 (2018) no. 10, pp. 6226-6301 | DOI | Zbl | MR

[GT01] Gilbarg, David; Trudinger, Neil S. Elliptic partial differential equations of second order, Classics in Mathematics, Springer, 2001 (Reprint of the 1998 edition) | MR | Zbl | DOI

[Gub04] Gubinelli, Massimiliano Controlling rough paths, J. Funct. Anal., Volume 216 (2004) no. 1, pp. 86-140 | DOI | MR | Zbl

[Gub10] Gubinelli, Massimiliano Ramification of rough paths, J. Differ. Equations, Volume 248 (2010) no. 4, pp. 693-721 | DOI | Zbl | MR

[Hai14] Hairer, Martin A theory of regularity structures, Invent. Math., Volume 198 (2014) no. 2, pp. 269-504 | Zbl | DOI | MR

[IN64] Itô, Kiyoshi; Nisio, Makiko On stationary solutions of a stochastic differential equation, J. Math. Kyoto Univ., Volume 4 (1964), pp. 1-75 | DOI | MR | Zbl

[Kal21] Kalinin, Alexander Support characterization for regular path-dependent stochastic Volterra integral equations, Electron. J. Probab., Volume 26 (2021), pp. 1-29 | DOI | MR | Zbl

[Kus68] Kushner, Harold J. On the stability of processes defined by stochastic difference-differential equations, J. Differ. Equations, Volume 4 (1968), pp. 424-443 | MR | Zbl | DOI

[KZ16] Keller, Christian; Zhang, Jianfeng Pathwise Itô calculus for rough paths and rough PDEs with path dependent coefficients, Stochastic Processes Appl., Volume 126 (2016) no. 3, pp. 735-766 | DOI | Zbl

[LCL07] Lyons, Terry J.; Caruana, Michael; Lévy, Thierry Differential equations driven by rough paths, Lecture Notes in Mathematics, 1908, Springer, 2007 (Ecole d’Été de Probabilités de Saint-Flour XXXIV – 2004. Lectures given at the 34th probability summer school, July 6–24, 2004) | MR | Zbl | DOI

[LQ02] Lyons, Terry J.; Qian, Zhongmin System control and rough paths, Oxford Mathematical Monographs, Oxford University Press, 2002 | DOI | MR | Zbl

[LV07] Lyons, Terry J.; Victoir, Nicolas B. An extension theorem to rough paths, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 24 (2007) no. 5, pp. 835-847 | DOI | MR | Zbl | Numdam

[Lyo98] Lyons, Terry J. Differential equations driven by rough signals, Rev. Mat. Iberoam., Volume 14 (1998) no. 2, pp. 215-310 | MR | Zbl | DOI

[Moh98] Mohammed, Salah-Eldin A. Stochastic differential systems with memory: theory, examples and applications, Stochastic analysis and related topics, VI (Geilo, 1996) (Progress in Probability), Volume 42, Birkhäuser, 1998, pp. 1-77 | MR | Zbl

[NNT08] Neuenkirch, Andreas; Nourdin, Ivan; Tindel, Samy Delay equations driven by rough paths, Electron. J. Probab., Volume 13 (2008), pp. 2031-2068 | DOI | Zbl | MR

[Pea90] Peano, Giuseppe Démonstration de l’intégrabilité des équations différentielles ordinaires, Math. Ann., Volume 37 (1890) no. 2, pp. 182-228 | Zbl | DOI

[PP16] Perkowski, Nicolas; Prömel, David J. Pathwise stochastic integrals for model free finance, Bernoulli, Volume 22 (2016) no. 4, pp. 2486-2520 | DOI | Zbl | MR

[Shi19] Shigeki, Aida Rough differential equations containing path-dependent bounded variation terms (2019) (https://arxiv.org/abs/1608.03083, Working Paper)

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