Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Pathwise solvability of stochastic integral equations with generalized drift and non-smooth dispersion functions

Ioannis Karatzas and Johannes Ruf

Full-text: Open access

Abstract

We study one-dimensional stochastic integral equations with non-smooth dispersion coëfficients, and with drift components that are not restricted to be absolutely continuous with respect to Lebesgue measure. In the spirit of Lamperti, Doss and Sussmann, we relate solutions of such equations to solutions of certain ordinary integral equations, indexed by a generic element of the underlying probability space. This relation allows us to solve the stochastic integral equations in a pathwise sense.

Résumé

Nous étudions des équations intégrales stochastiques unidimensionnelles avec coefficient de diffusion non-régulier, et avec termes de dérive non nécessairement absolument continus par rapport à la mesure de Lebesgue. En s’inspirant de Lamperti, Doss et Sussmann, la résolution de ces équations se ramène à la résolution de certaines équations intégrales ordinaires, paramétrées par un élément $\omega$ variant dans l’espace de probabilité de base. Ce lien nous permet de résoudre les équations intégrales stochastiques d’une façon trajectorielle.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 2 (2016), 915-938.

Dates
Received: 13 December 2013
Revised: 16 July 2014
Accepted: 28 October 2014
First available in Project Euclid: 4 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1462367900

Digital Object Identifier
doi:10.1214/14-AIHP660

Mathematical Reviews number (MathSciNet)
MR3498016

Zentralblatt MATH identifier
1346.45010

Subjects
Primary: 34A99: None of the above, but in this section 45J05: Integro-ordinary differential equations [See also 34K05, 34K30, 47G20] 60G48: Generalizations of martingales 60H10: Stochastic ordinary differential equations [See also 34F05]

Keywords
Stochastic integral equation Ordinary integral equation Pathwise solvability Existence Uniqueness Generalized drift Wong–Zakai approximation Support theorem Comparison theorem Stratonovich integral

Citation

Karatzas, Ioannis; Ruf, Johannes. Pathwise solvability of stochastic integral equations with generalized drift and non-smooth dispersion functions. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 2, 915--938. doi:10.1214/14-AIHP660. https://projecteuclid.org/euclid.aihp/1462367900


Export citation

References

  • [1] S. Aida and K. Sasaki. Wong–Zakai approximation of solutions to reflecting stochastic differential equations on domains in Euclidean spaces. Stochastic Process. Appl. 123 (2013) 3800–3827.
  • [2] X. Bardina and C. Rovira. On Itô’s formula for elliptic diffusion processes. Bernoulli 13 (3) (2007) 820–830.
  • [3] M. Barlow and E. Perkins. One-dimensional stochastic differential equations involving a singular increasing process. Stochastics 12 (3–4) (1984) 229–249.
  • [4] R. F. Bass and Z.-Q. Chen. One-dimensional stochastic differential equations with singular and degenerate coefficients. Sankhyā 67 (1) (2005) 19–45.
  • [5] R. F. Bass, B. M. Hambly and T. J. Lyons. Extending the Wong–Zakai theorem to reversible Markov processes. J. Eur. Math. Soc. (JEMS) 4 (3) (2002) 237–269.
  • [6] V. E. Beneš. Nonexistence of strong nonanticipating solutions to stochastic DEs: Implications for functional DEs, filtering, and control. Stochastic Process. Appl. 5 (3) (1977) 243–263.
  • [7] V. E. Beneš. Realizing a weak solution on a probability space. Stochastic Process. Appl. 7 (2) (1978) 205–225.
  • [8] J. Bertoin. Les processus de Dirichlet et tant qu’espace de Banach. Stochastics 18 (2) (1986) 155–168.
  • [9] J. Bertoin. Temps locaux et intégration stochastique pour les processus de Dirichlet. In Séminaire de Probabilités XXI 191–205. Lecture Notes in Math. 1247. Springer, Berlin, 1987.
  • [10] K. Bichteler. Stochastic integration and $L^{p}$-theory of semimartingales. Ann. Probab. 9 (1) (1981) 49–89.
  • [11] S. Blei and H.-J. Engelbert. One-dimensional stochastic differential equations with generalized and singular drift. Stochastic Process. Appl. 123 (12) (2013) 4337–4372.
  • [12] S. Blei and H.-J. Engelbert. A note on one-dimensional stochastic differential equations with generalized drift. Theory Probab. Appl. 58 (3) (2014) 345–357. DOI: 10.1137/S0040585X97986655.
  • [13] N. Bouleau and M. Yor. Sur la variation quadratique des temps locaux de certaines semimartingales. C. R. Acad. Sci. Sér. I Math. 292 (1981) 491–494.
  • [14] R. Catellier and M. Gubinelli. Averaging along irregular curves and regularisation of ODEs. Preprint, 2014. Available at arXiv:1205.1735.
  • [15] E. Çinlar, J. Jacod, P. Protter and M. Sharpe. Semimartingales and Markov processes. Z. Wahrsch. Verw. Gebiete 54 (2) (1980) 161–219.
  • [16] E. A. Coddington and N. Levinson. Theory of Ordinary Differential Equations. McGraw-Hill, New York, 1955.
  • [17] R. Coviello and F. Russo. Nonsemimartingales: Stochastic differential equations and weak Dirichlet processes. Ann. Probab. 35 (1) (2007) 255–308.
  • [18] P. Da Pelo, A. Lanconelli and A. I. Stan. An Itô formula for a family of stochastic integrals and related Wong–Zakai theorems. Stochastic Process. Appl. 123 (2013) 3183–2300.
  • [19] A. Davie. Uniqueness of solutions of stochastic differential equations. Int. Math. Res. Notes IMRN 2007 (2007) 1–26.
  • [20] H. Doss. Liens entre équations différentielles stochastiques et ordinaires. Ann. Inst. Henri Poincaré Probab. Stat. 13 (2) (1977) 99–125.
  • [21] K. Dupoiron, P. Mathieu and J. San Martin. Formule d’Itô pour des diffusions uniformément elliptiques, et processus de Dirichlet. Potential Anal. 21 (2004) 7–33.
  • [22] H.-J. Engelbert. On the theorem of T. Yamada and S. Watanabe. Stochastics 36 (3–4) (1991) 205–216.
  • [23] H.-J. Engelbert and W. Schmidt. One-dimensional stochastic differential equations with generalized drift. In Stochastic Differential Systems. Filtering and Control 143–155. Lecture Notes in Control and Information Sciences 69. Springer, Berlin, 1985.
  • [24] H.-J. Engelbert and W. Schmidt. Strong Markov continuous local martingales and solutions of one-dimensional stochastic differential equations. III. Math. Nachr. 151 (1) (1991) 149–197.
  • [25] H.-J. Engelbert and J. Wolf. Strong Markov local Dirichlet processes and stochastic differential equations. Theory Probab. Appl. 43 (2) (1999) 189–202.
  • [26] M. Errami and F. Russo. $n$-covariation, generalized Dirichlet processes and calculus with respect to finite cubic variation processes. Stochastic Process. Appl. 104 (2003) 259–299.
  • [27] M. Errami, F. Russo and P. Vallois. Itô’s formula for ${C}^{1,\lambda}$-functions of a càdlàg process and related calculus. Probab. Theory Related Fields 122 (2002) 191–221.
  • [28] E. Fedrizzi and F. Flandoli. Pathwise uniqueness and continuous dependence for SDEs with non-regular drift. Stochastics 83 (3) (2011) 241–257.
  • [29] F. Flandoli. Random Perturbation of PDEs and Fluid dynamic Models. École d’Été de Probabilités de Saint-Flour XL, 2010. Lecture Notes in Mathematics 2015. Springer, Heidelberg, 2011.
  • [30] F. Flandoli. Regularizing properties of Brownian paths and a result of Davie. Stoch. Dyn. 11 (2–3) (2011) 323–331.
  • [31] F. Flandoli. Topics on regularization by noise. Lecture notes, Univ. Pisa, 2013.
  • [32] F. Flandoli, F. Russo and J. Wolf. Some SDEs with distributional drift. I. General calculus. Osaka J. Math. 40 (2) (2003) 493–542.
  • [33] F. Flandoli, F. Russo and J. Wolf. Some SDEs with distributional drift. II. Lyons–Zheng structure, Itô’s formula and semimartingale characterization. Random Oper. Stoch. Equ. 12 (2) (2004) 145–184.
  • [34] H. Föllmer. Calcul d’Itô sans probabilités. In Séminaire de Probabilités XV 143–150. Lecture Notes in Mathematics 850. Springer, Berlin, 1981.
  • [35] H. Föllmer. Dirichlet processes. In Stochastic Integrals 476–778. D. Williams (Ed.). Lecture Notes in Mathematics 851. Springer, Berlin, 1981.
  • [36] M. Fukushima, Y. Oshima and M. Takeda. Dirichlet Forms and Symmetric Markov Processes. de Gruyter, Berlin, 1994.
  • [37] R. Ghomrasni and G. Peskir. Local time–space calculus and extensions of Itô’s formula. In High Dimensional Probablity III 177–192. J. Hoffmann-Joergensen, M. B. Marcus and J. A. Wellner (Eds). Birkhäuser, Basel, 2003.
  • [38] J. Harrison and L. Shepp. On skew Brownian motion. Ann. Probab. 9 (2) (1981) 309–313.
  • [39] P. Hartman. Ordinary Differential Equations, 2nd edition. Birkhäuser, Boston, 1982.
  • [40] N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes, 2nd edition. North-Holland Publishing, Amsterdam, 1989.
  • [41] J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes, 2nd edition. Springer, Berlin, 2003.
  • [42] R. L. Karandikar. On pathwise stochastic integration. Stochastic Process. Appl. 57 (1) (1995) 11–18.
  • [43] I. Karatzas and J. Ruf. Distribution of the time-to-explosion for one-dimensional diffusions. Probab. Theory Related Fields 164 (2016) 1027–1069.
  • [44] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus, 2nd edition. Springer, New York, 1991.
  • [45] N. V. Krylov and M. Röckner. Strong solutions of stochastic equations with singular time dependent drift. Probab. Theory Related Fields 131 (2005) 154–196.
  • [46] T. G. Kurtz, É. Pardoux and P. Protter. Stratonovich stochastic differential equations driven by general semimartingales. Ann. Inst. Henri Poincaré Probab. Stat. 31 (1995) 351–377.
  • [47] J. Lamperti. A simple construction of certain diffusion processes. Kyoto J. Math. 4 (1) (1964) 161–170.
  • [48] J.-F. Le Gall. One-dimensional stochastic differential equations involving the local times of the unknown process. In Stochastic Analysis and Applications 51–82. A. Truman and D. Williams (Eds). Lecture Notes in Mathematics 1095. Springer, Berlin, 1984.
  • [49] A. Lejay and M. Martinez. A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients. Ann. Appl. Probab. 16 (1) (2006) 107–139.
  • [50] G. Lowther. Nondifferentiable functions of one-dimensional semimartingales. Ann. Probab. 38 (1) (2010) 76–101.
  • [51] T. J. Lyons and Z. Qian. System Control and Rough Paths. Oxford Univ. Press, Oxford, 2002.
  • [52] S. I. Marcus. Modeling and approximation of stochastic differential equations driven by semimartingales. Stochastics 4 (3) (1981) 223–245.
  • [53] E. McShane. Stochastic differential equations. J. Multivariate Anal. 5 (2) (1975) 121–177.
  • [54] A. Mijatović and M. Urusov. Convergence of integral functionals of one-dimensional diffusions. Electron. Commun. Probab. 17 (61) (2012) 1–13.
  • [55] M. Nutz. Pathwise construction of stochastic integrals. Electron. Commun. Probab. 17 (24) (2012) 1–7.
  • [56] Y. Ouknine. “Skew-Brownian motion” and derived processes. Theory Probab. Appl. 35 (1) (1991) 163–169.
  • [57] N. Perkowski and D. J. Prömel. Pathwise stochastic integrals for model free finance. Preprint, 2013. Available at arXiv:1311.6187.
  • [58] P. E. Protter. On the existence, uniqueness, convergence and explosions of solutions of systems of stochastic integral equations. Ann. Probab. 5 (2) (1977) 243–261.
  • [59] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Springer, Berlin, 1999.
  • [60] F. Russo and G. Trutnau. Some parabolic PDEs whose drift is an irregular random noise in space. Ann. Probab. 35 (6) (2007) 2213–2262.
  • [61] H. M. Soner, N. Touzi and J. Zhang. Quasi-sure stochastic analysis through aggregation. Electron. J. Probab. 16 (2) (2011) 1844–1879.
  • [62] D. W. Stroock and S. R. Varadhan. On the support of diffusion processes with applications to the strong maximum principle. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971) 3 333–359. Univ. California Press, Berkeley, CA, 1972.
  • [63] D. W. Stroock and S. R. S. Varadhan. Multidimensional Diffusion Processes. Springer, Berlin, 2006. Reprint of the 1997 edition.
  • [64] D. W. Stroock and M. Yor. Some remarkable martingales. In Séminaire de Probabilités XV 590–603. Springer, Berlin, 1981.
  • [65] H. J. Sussmann. On the gap between deterministic and stochastic ordinary differential equations. Ann. Probab. 6 (1) (1978) 19–41.
  • [66] B. Tsirel’son. An example of a stochastic differential equation having no strong solution. Theory Probab. Appl. 20 (1975) 416–418.
  • [67] A. J. Veretennikov. On strong solutions and explicit formulas for solutions of stochastic integral equations. Math. USSR Sb. 39 (3) (1981) 387–403.
  • [68] J. B. Walsh. A diffusion with a discontinuous local time. Astérisque 52 (53) (1978) 37–45.
  • [69] J. Wolf. An Itô formula for local Dirichlet processes. Stochastics 62 (1–2) (1997) 103–115.
  • [70] J. Wolf. Transformations of semi-martingales and local Dirichlet processes. Stochastics 62 (1–2) (1997) 65–101.
  • [71] E. Wong and M. Zakai. On the convergence of ordinary integrals to stochastic integrals. Ann. Math. Stat. 36 (5) (1965) 1560–1564.
  • [72] E. Wong and M. Zakai. On the relation between ordinary and stochastic differential equations. Internat. J. Engrg. Sci. 3 (2) (1965) 213–229.
  • [73] T. Zhang. Strong convergence of Wong–Zakai approximations of reflected SDEs in a multidimensional general domain. Potential Anal. 41 (3) (2014) 783–815.
  • [74] A. K. Zvonkin. A transformation of the phase space of a diffusion process that removes the drift. Math. USSR Sb. 22 (1) (1974) 129–149.