Wong–Zakai approximations and support theorems for SDEs under Lyapunov conditions

Qi Li; Jianliang Zhai; Tusheng Zhang

doi:10.5802/crmath.701

Article de recherche - Probabilités

Wong–Zakai approximations and support theorems for SDEs under Lyapunov conditions
[Approximations de Wong–Zakai et théorèmes de support pour les SDEs sous conditions de Lyapunov]

Qi Li ¹ ; Jianliang Zhai ¹ ; Tusheng Zhang ²

¹ School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China
² Department of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, UK

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 617-628.

Abstract (VO)
Résumé

In this paper, we establish the Stroock–Varadhan type support theorems for stochastic differential equations (SDEs) under Lyapunov conditions, which improve the existing results in the literature where the coefficients of the SDEs are required to be globally Lipschitz. Our conditions are mild to include many important models, e.g. Threshold Ornstein–Uhlenbeck process, stochastic SIR model, stochastic Lotka–Volterra systems, stochastic Duffing–van der Pol oscillator model, which have polynomial coefficients. To obtain the support theorem, a localizing procedure plays an important role.

Dans cet article, nous établissons les théorèmes de support de type Stroock–Varadhan pour des équations différentielles stochastiques (EDS) sous conditions de Lyapunov, qui améliorent les résultats existants dans la littérature où les coefficients des EDS doivent être globalement Lipschitz. Nos conditions sont faibles et permettent d’inclure de nombreux modèles importants, par exemple le processus de Ornstein–Uhlenbeck avec seuillage, le modèle stochastique SIR, les systèmes stochastiques de Lotka–Volterra, le modèle stochastique de l’oscillateur de Duffing–van der Pol, qui ont des coefficients polynomiaux. Pour obtenir le théorème de support, une procédure de localisation joue un rôle important.

Reçu le : 2024-09-10
Accepté le : 2024-11-06
Publié le : 2025-06-23

DOI : 10.5802/crmath.701

Keywords: Wong–Zakai approximation, support theorem, local Lipschitz, Lyapunov condition
Mots-clés : Approximation de Wong–Zakai, théorème de support, fonctions localement lipschitziennes, condition de Lyapunov

Affiliations des auteurs :

Qi Li ¹ ; Jianliang Zhai ¹ ; Tusheng Zhang ²

¹ School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China
² Department of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, UK

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Qi Li and Jianliang Zhai and Tusheng Zhang},
     title = {Wong{\textendash}Zakai approximations and support theorems for {SDEs} under {Lyapunov} conditions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {617--628},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     year = {2025},
     doi = {10.5802/crmath.701},
     language = {en},
}

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Qi Li; Jianliang Zhai; Tusheng Zhang. Wong–Zakai approximations and support theorems for SDEs under Lyapunov conditions. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 617-628. doi : 10.5802/crmath.701. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.701/

Bibliographie
Cité par

[1] Shigeki Aida; Shigeo Kusuoka; Daniel W. Stroock On the support of Wiener functionals, Asymptotic problems in probability theory: Wiener functionals and asymptotics (Sanda/Kyoto, 1990) (Pitman Research Notes in Mathematics Series), Volume 284, Longman Scientific & Technical, 1993, pp. 3-34 | MR | Zbl

[2] Roy M. Anderson; Robert M. May Population biology of infectious diseases: Part I, Nature, Volume 280 (1979), pp. 361-367 | DOI

[3] Gérard Ben Arous; Mihai Grădinaru Normes hölderiennes et support des diffusions, C. R. Math., Volume 316 (1993) no. 3, pp. 283-286 | MR | Zbl

[4] Gérard Ben Arous; Mihai Grădinaru; Michel Ledoux Hölder norms and the support theorem for diffusions, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 30 (1994) no. 3, pp. 415-436 | Numdam | MR | Zbl

[5] Wanrong Cao; Zhongqiang Zhang; George Em Karniadakis Numerical methods for stochastic delay differential equations via the Wong–Zakai approximation, SIAM J. Sci. Comput., Volume 37 (2015) no. 1, p. A295-A318 | DOI | MR | Zbl

[6] Lifeng Chen; Zhao Dong; Jifa Jiang; Lei Niu; Jianliang Zhai Decomposition formula and stationary measures for stochastic Lotka–Volterra system with applications to turbulent convection, J. Math. Pures Appl. (9), Volume 125 (2019), pp. 43-93 | DOI | MR | Zbl

[7] Rama Cont; Alexander Kalinin 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 | MR | Zbl

[8] Sonja Cox; Martin Hutzenthaler; Arnulf Jentzen Local Lipschitz continuity in the initial value and strong completeness for nonlinear stochastic differential equations (2013) | arXiv

[9] István Gyöngy; T. Pröhle On the approximation of stochastic differential equation and on Stroock–Varadhan’s support theorem, Comput. Math. Appl., Volume 19 (1990) no. 1, pp. 65-70 | DOI | MR | Zbl

[10] Josef Hofbauer; Karl Sigmund Evolutionary games and population dynamics, Cambridge University Press, 1998, xxviii+323 pages | DOI | MR | Zbl

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

[12] Minoo Kamrani; Nahid Jamshidi Implicit Milstein method for stochastic differential equations via the Wong–Zakai approximation, Numer. Algorithms, Volume 79 (2018) no. 2, pp. 357-374 | DOI | MR | Zbl

[13] Franz Konecny On Wong–Zakai approximation of stochastic differential equations, J. Multivariate Anal., Volume 13 (1983) no. 4, pp. 605-611 | DOI | MR | Zbl

[14] Edward James McShane Stochastic differential equations and models of random processes, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability theory, University of California Press, 1972, pp. 263-294 | MR | Zbl

[15] Annie Millet; Marta Sanz-Solé A simple proof of the support theorem for diffusion processes, Séminaire de Probabilités, XXVIII (Lecture Notes in Mathematics), Volume 1583, Springer, 1994, pp. 36-48 | DOI | Numdam | MR | Zbl

[16] Shintaro Nakao; Yuiti Yamato Approximation theorem on stochastic differential equations, Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976) (A Wiley-Interscience Publication), John Wiley & Sons, 1978, pp. 283-296 | MR | Zbl

[17] Jiagang Ren; Jing Wu On approximate continuity and the support of reflected stochastic differential equations, Ann. Probab., Volume 44 (2016) no. 3, pp. 2064-2116 | DOI | MR | Zbl

[18] Suleyman Sengul; Zafer Bekiryazici; Mehmet Merdan Wong–Zakai method for stochastic differential equations in engineering, Therm. Sci., Volume 25 (2021), pp. 131-142 | DOI

[19] Thomas Simon Support theorem for jump processes, Stochastic Processes Appl., Volume 89 (2000) no. 1, pp. 1-30 | DOI | MR | Zbl

[20] Daniel W. Stroock; Srinivasa R. S. Varadhan On the support of diffusion processes with applications to the strong maximum principle, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability theory, University of California Press, 1972, pp. 333-359 | MR | Zbl

[21] Jian Wang; Hao Yang; Jianliang Zhai; Tusheng Zhang Large deviation principles for SDEs under locally weak monotonicity conditions, Bernoulli, Volume 30 (2024) no. 1, pp. 332-345 | DOI | MR | Zbl

[22] Eugene Wong; Moshe Zakai On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Stat., Volume 36 (1965), pp. 1560-1564 | DOI | MR | Zbl

[23] Eugene Wong; Moshe Zakai On the relation between ordinary and stochastic differential equations, Int. J. Eng. Sci., Volume 3 (1965), pp. 213-229 | DOI | MR | Zbl

[24] Jie Xu; Jiayin Gong Wong–Zakai approximations and support theorems for stochastic McKean–Vlasov equations, Forum Math., Volume 34 (2022) no. 6, pp. 1411-1432 | DOI | MR | Zbl

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