A generalized mean-reverting equation and applications
ESAIM: Probability and Statistics, Tome 18 (2014), pp. 799-828.

Consider a mean-reverting equation, generalized in the sense it is driven by a 1-dimensional centered Gaussian process with Hölder continuous paths on [0,T] (T> 0). Taking that equation in rough paths sense only gives local existence of the solution because the non-explosion condition is not satisfied in general. Under natural assumptions, by using specific methods, we show the global existence and uniqueness of the solution, its integrability, the continuity and differentiability of the associated Itô map, and we provide an Lp-converging approximation with a rate of convergence (p ≫ 1). The regularity of the Itô map ensures a large deviation principle, and the existence of a density with respect to Lebesgue's measure, for the solution of that generalized mean-reverting equation. Finally, we study a generalized mean-reverting pharmacokinetic model.

DOI : https://doi.org/10.1051/ps/2014002
Classification : 60H10
Mots clés : stochastic differential equations, rough paths, large deviation principle, mean-reversion, gaussian processes
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author = {Marie, Nicolas},
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Marie, Nicolas. A generalized mean-reverting equation and applications. ESAIM: Probability and Statistics, Tome 18 (2014), pp. 799-828. doi : 10.1051/ps/2014002. http://www.numdam.org/articles/10.1051/ps/2014002/

[1] R.J. Adler, An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes. Inst. Math. Stat., Lect. Notes Monogr. Vol. 38, Ser. 12 (1990). | MR 1088478 | Zbl 0747.60039

[2] L. Coutin, Rough Paths via Sewing Lemma. ESAIM: PS 16 (2012) 479-526. | Numdam | Zbl 1277.47081

[3] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications. Stoch. Model. Appl. Probab. Springer-Verlag, New-York (1998). | MR 1619036 | Zbl 0896.60013

[4] M. Delattre and M. Lavielle, Pharmacokinetics and Stochastic Differential Equations: Model and Methodology. Annual Meeting of the Population Approach Group in Europe (2011).

[5] H. Doss, Liens entre équations différentielles stochastiques et ordinaires. C.R. Acad. Sci. Paris Ser. A-B 283 (1976) A939-A942. | MR 426152 | Zbl 0352.60044

[6] J. Feng, J.-P. Fouque and R. Kumar, Small-Time Asymptotics for Fast Mean-Reverting Stochastic Volatility Models (2010). Preprint arXiv:1009.2782. | MR 2985169 | Zbl 1266.60049

[7] E. Fournié, J.-M. Lasry, J. Lebuchoux, P.-L. Lions and N. Touzi, Applications of Malliavin Calculus to Monte-Carlo Methods in Finance. Finance Stoch. 3 (1999) 391-412. | MR 1842285 | Zbl 0947.60066

[8] P. Friz and N. Victoir, Multidimensional Stochastic Processes as Rough Paths: Theory and Applications. Vols 120 of Camb. Stud. Appl. Math. Cambridge University Press, Cambridge (2010). | MR 2604669 | Zbl 1193.60053

[9] Y. Jacomet, Pharmacocinétique. Cours et Exercices. Université de Nice, U.E.R. de Médecine, Service de pharmacologie expérimentale et clinique, Ellipses (1989).

[10] S. Karlin and H.M. Taylor, A Second Course in Stochastic Processes. Academic Press Inc., Harcourt Brace Jovanovich Publishers (1981). | MR 611513 | Zbl 0469.60001

[11] K. Kalogeropoulos, N. Demiris and O. Papaspiliopoulos, Diffusion-driven Models for Physiological Processes. Int. Workshop on Appl. Probab. IWAP (2008).

[12] M. Ledoux, Isoperimetry and Gaussian Analysis. Ecole d'été de probabilité de Stain-Flour (1994). | Zbl 0874.60005

[13] A. Lejay, Controlled Differential Equations as Young Integrals: A Simple Approach. J. Differ. Eqs. 248 (2010) 1777-1798. | MR 2679003 | Zbl 1216.34058

[14] T. Lyons and Z. Qian, System Control and Rough Paths. Oxford University Press (2002). | MR 2036784 | Zbl 1029.93001

[15] F. Malrieu, Convergence to Equilibrium for Granular Media Equations and their Euler Schemes. Ann. Appl. Probab. 13 (2003) 540-560. | MR 1970276 | Zbl 1031.60085

[16] F. Malrieu and D. Talay, Concentration Inequalities for Euler Schemes. Springer-Verlag (2006) 355-371. | MR 2208718 | Zbl 1097.65012

[17] X. Mao, A. Truman and C. Yuan, Euler-Maruyama Approximations in Mean-Reverting Stochastic Volatility Model under Regime-Switching. J. Appl. Math. Stoch. Anal. (2006). | MR 2237177 | Zbl 1147.60320

[18] N. Marie, Sensitivities via Rough Paths (2011). Preprint arXiv:1108.0852.

[19] J. Neveu, Processus aléatoires gaussiens. Presses de l'Université de Montréal (1968). | MR 272042 | Zbl 0192.54701

[20] D. Nualart, The Malliavin Calculus and Related Topics. Second Edition. Probab. Appl. Springer (2006). | MR 2200233 | Zbl 0837.60050

[21] N. Tien Dung, Fractional Geometric Mean Reversion Processes. J. Math. Anal. Appl. 330 (2011) 396-402. | MR 2786210 | Zbl 1215.60030

[22] M. Sanz-Solé and I. Torrecilla-Tarantino, A Large Deviation Principle in Hölder Norm for Multiple Fractional Integrals. (2007). Preprint arXiv:0702049. | MR 2339298

[23] N. Simon, Pharmacocinétique de population. Collection Pharmacologie médicale, Solal (2006).

[24] H.J. Sussman, On the Gap between Deterministic and Stochastic Ordinary Differential Equations. Ann. Probab. 6 (1978) 19-41. | MR 461664 | Zbl 0391.60056

[25] F. Wu, X. Mao and K. Chen, A Highly Sensitive Mean-Reverting Process in Finance and the Euler-Maruyama Approximations. J. Math. Anal. Appl. 348 (2008) 540-554. | MR 2449373 | Zbl 1158.60033

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