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A generalized mean-reverting equation andapplications

Published online by Cambridge University Press:  22 October 2014

Nicolas Marie
Nicolas Marie*
Affiliation:
Laboratoire Modal’X, Université Paris-Ouest, 200 Avenue de la République, 92000 Nanterre, France. [email protected]
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Abstract

Consider a mean-reverting equation, generalized in the sense it is driven by a1-dimensional centeredGaussian process with Hölder continuous paths on [0,T] (T> 0). Taking thatequation in rough paths sense only gives local existence of the solution because thenon-explosion condition is not satisfied in general. Under natural assumptions, by usingspecific methods, we show the global existence and uniqueness of the solution, itsintegrability, the continuity and differentiability of the associated Itô map, and weprovide an Lp-convergingapproximation with a rate of convergence (p ≫ 1). The regularity of the Itô map ensures a largedeviation 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 ageneralized mean-reverting pharmacokinetic model.

Keywords

Stochastic differential equationsrough pathslarge deviation principlemean-reversionGaussian processes
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014 , pp. 799 - 828
Copyright
© EDP Sciences, SMAI 2014

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References

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).
Coutin, L., Rough Paths via Sewing Lemma. ESAIM: PS 16 (2012) 479–526. Google Scholar
A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications. Stoch. Model. Appl. Probab. Springer-Verlag, New-York (1998).
M. Delattre and M. Lavielle, Pharmacokinetics and Stochastic Differential Equations: Model and Methodology. Annual Meeting of the Population Approach Group in Europe (2011).
Doss, H., Liens entre équations différentielles stochastiques et ordinaires. C.R. Acad. Sci. Paris Ser. A-B 283 (1976) A939–A942. Google Scholar
J. Feng, J.-P. Fouque and R. Kumar, Small-Time Asymptotics for Fast Mean-Reverting Stochastic Volatility Models (2010). Preprint arXiv:1009.2782.
Fournié, E., Lasry, J.-M., Lebuchoux, J., Lions, P.-L. and Touzi, N., Applications of Malliavin Calculus to Monte-Carlo Methods in Finance. Finance Stoch. 3 (1999) 391–412. Google Scholar
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).
Y. Jacomet, Pharmacocinétique. Cours et Exercices. Université de Nice, U.E.R. de Médecine, Service de pharmacologie expérimentale et clinique, Ellipses (1989).
S. Karlin and H.M. Taylor, A Second Course in Stochastic Processes. Academic Press Inc., Harcourt Brace Jovanovich Publishers (1981).
K. Kalogeropoulos, N. Demiris and O. Papaspiliopoulos, Diffusion-driven Models for Physiological Processes. Int. Workshop on Appl. Probab. IWAP (2008).
M. Ledoux, Isoperimetry and Gaussian Analysis. Ecole d’été de probabilité de Stain-Flour (1994).
Lejay, A., Controlled Differential Equations as Young Integrals: A Simple Approach. J. Differ. Eqs. 248 (2010) 1777–1798. Google Scholar
T. Lyons and Z. Qian, System Control and Rough Paths. Oxford University Press (2002).
Malrieu, F., Convergence to Equilibrium for Granular Media Equations and their Euler Schemes. Ann. Appl. Probab. 13 (2003) 540–560. Google Scholar
F. Malrieu and D. Talay, Concentration Inequalities for Euler Schemes. Springer-Verlag (2006) 355–371.
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).
N. Marie, Sensitivities via Rough Paths (2011). Preprint arXiv:1108.0852.
J. Neveu, Processus aléatoires gaussiens. Presses de l’Université de Montréal (1968).
D. Nualart, The Malliavin Calculus and Related Topics. Second Edition. Probab. Appl. Springer (2006).
Tien Dung, N., Fractional Geometric Mean Reversion Processes. J. Math. Anal. Appl. 330 (2011) 396–402. Google Scholar
M. Sanz-Solé and I. Torrecilla-Tarantino, A Large Deviation Principle in Hölder Norm for Multiple Fractional Integrals. (2007). Preprint arXiv:0702049.
N. Simon, Pharmacocinétique de population. Collection Pharmacologie médicale, Solal (2006).
Sussman, H.J., On the Gap between Deterministic and Stochastic Ordinary Differential Equations. Ann. Probab. 6 (1978) 19–41. Google Scholar
Wu, F., Mao, X. and Chen, K., A Highly Sensitive Mean-Reverting Process in Finance and the Euler-Maruyama Approximations. J. Math. Anal. Appl. 348 (2008) 540–554. Google Scholar