Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Explicit parametrix and local limit theorems for some degenerate diffusion processes

Valentin Konakov, Stéphane Menozzi, and Stanislav Molchanov

Full-text: Open access

Abstract

For a class of degenerate diffusion processes of rank 2, i.e. when only Poisson brackets of order one are needed to span the whole space, we obtain a parametrix representation of McKean–Singer [J. Differential Geom. 1 (1967) 43–69] type for the density. We therefrom derive an explicit Gaussian upper bound and a partial lower bound that characterize the additional singularity induced by the degeneracy.

This particular representation then allows to give a local limit theorem with the usual convergence rate for an associated Markov chain approximation. The key point is that the “weak” degeneracy allows to exploit the techniques first introduced in Konakov and Molchanov [Teor. Veroyatn. Mat. Statist. 31 (1984) 51–64] and then developed in [Probab. Theory Related Fields 117 (2000) 551–587] that rely on Gaussian approximations.

Résumé

Pour une classe de processus de diffusion de rang deux, i.e. lorsque seuls des crochets de Poisson d’ordre un permettent d’engendrer l’espace, nous obtenons une représentation parametrix de type McMean–Singer [J. Differential Geom. 1 (1967) 43–69] de la densité. Nous en dérivons une borne supérieure Gaussienne explicite et une borne inférieure partielle qui caractérisent la singularité additionnelle induite par la dégénérescence.

Nous donnons ensuite un théorème limite local pour une approximation par chaîne de Markov associée. Le point crucial est que la faible dégénérescence permet d’exploiter les techniques initialement introduites par Konakov et Molchanov [Teor. Veroyatn. Mat. Statist. 31 (1984) 51–64] puis développées dans [Probab. Theory Related Fields 117 (2000) 551–587] et qui reposent sur des approximations Gaussiennes.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist. Volume 46, Number 4 (2010), 908-923.

Dates
First available: 4 November 2010

Permanent link to this document
http://projecteuclid.org/euclid.aihp/1288878329

Digital Object Identifier
doi:10.1214/09-AIHP207

Zentralblatt MATH identifier
05864057

Mathematical Reviews number (MathSciNet)
MR2744877

Subjects
Primary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60J60: Diffusion processes [See also 58J65]
Secondary: 35K65: Degenerate parabolic equations

Keywords
Degenerate diffusion processes Parametrix Markov chain approximation Local limit theorems

Citation

Konakov, Valentin; Menozzi, Stéphane; Molchanov, Stanislav. Explicit parametrix and local limit theorems for some degenerate diffusion processes. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 46 (2010), no. 4, 908--923. doi:10.1214/09-AIHP207. http://projecteuclid.org/euclid.aihp/1288878329.


Export citation

References

  • [1] D. G. Aronson. Bounds for the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc. 73 (1967) 890–896.
  • [2] P. Baldi. Premières majorations de la densité d’une diffusion sur Rm, méthode de la parametrix. Astérisques 84–85 (1978) 43–53.
  • [3] G. Ben Arous. Développement asymptotique du noyau de la chaleur hypoelliptique hors du cut-locus. Ann. Sci. École Norm. Sup. (4) 21 (1988) 307–331.
  • [4] G. Ben Arous and R. Léandre. Décroissance exponentielle du noyau de la chaleur sur la diagonale, II. Probab. Theory Related Fields 90 (1991) 377–402.
  • [5] E. Barucci, S. Polidoro and V. Vespri. Some results on partial differential equations and asian options. Math. Models Methods Appl. Sci. 3 (2001) 475–497.
  • [6] R. Bhattacharya and R. Rao. Normal Approximations and Asymptotic Expansions. Wiley, New York, 1976.
  • [7] V. Bally and D. Talay. The law of the Euler scheme for stochastic differential equations, II. Convergence rate of the density. Monte Carlo Methods Appl. 2 (1996) 93–128.
  • [8] P. Cattiaux. Calcul stochastique et opérateurs dégénérés du second ordre, I. Résolvantes, théorème de Hörmander et applications. Bull. Sci. Math. 114 (1990) 421–462.
  • [9] P. Cattiaux. Calcul stochastique et opérateurs dégénérés du second ordre, II. Problème de Dirichlet. Bull. Sci. Math. 115 (1991) 81–122.
  • [10] E. B. Dynkin. Markov Processes. Springer, Berlin, 1963.
  • [11] A. Friedman. Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, 1964.
  • [12] F. Hérau and F. Nier. Isotropic hypoellipticity and trend to equilibrium for the Fokker–Planck equation with a high-degree potential. Arch. Ration. Mech. Anal. 171 (2004) 151–218.
  • [13] L. Hörmander. Hypoelliptic second order differential operators. Acta. Math. 119 (1967) 147–171.
  • [14] V. D. Konakov and S. A Molchanov. On the convergence of Markov chains to diffusion processes. Teor. Veroyatn. Mat. Statist. (in Russian) 31 (1984) 51–64. English translation in Theory Probab. Math. Statist. 31 (1985) 59–73.
  • [15] V. Konakov and E. Mammen. Local limit theorems for transition densities of Markov chains converging to diffusions. Probab. Theory Related Fields 117 (2000) 551–587.
  • [16] V. Konakov, S. Menozzi and S. Molchanov. Explicit parametrix and local limit theorems for some degenerate diffusion processes, 2009. Available at http://hal.archives-ouvertes.fr/hal-00256588/fr/.
  • [17] A. N. Kolmogorov. Zufällige Bewegungen (zur Theorie der Brownschen Bewegung). Ann. of Math. (2) 35 (1934) 116–117.
  • [18] S. Kusuoka and D Stroock. Applications of the Malliavin calculus, III. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987) 391–442.
  • [19] H. P. McKean and I. M. Singer. Curvature and the eigenvalues of the Laplacian. J. Differential Geom. 1 (1967) 43–69.
  • [20] J. Mattingly and A. Stuart. Geometric ergodicity of some hypo-elliptic diffusions for particle motions. Inhomogeneous random systems. Markov Process. Related Fields 8 (2004) 199–214.
  • [21] S. A. Molchanov and A. N. Varchenko. Applications of the stationary phase method in limit theorems for Markov chains. Dokl. Akad. Nauk SSSR (Translated in Soviet Math. Dokl. (18) 265–269) 233 (1977) 11–14.
  • [22] J. R. Norris. Simplified Malliavin calculus. Séminaire de Probabilités, XX 101–130. Springer, Berlin, 1986.
  • [23] D. Nualart. Malliavin Calculus and Related Topics. Springer, New York, 1995.
  • [24] D. W. Stroock. Diffusion semigroups corresponding to uniformly elliptic divergence form operators. Séminaire de Probabilités, XXII 316–347. Springer, Berlin, 1988.
  • [25] D. Talay. Stochastic Hamiltonian systems: Exponential convergence to the invariant measure, and discretization by the implicit Euler scheme. Markov Process. Related Fields 8 (2002) 163–198.
  • [26] B. Lapeyre and E. Temam. Competitive Monte Carlo methods for the pricing of Asian Options. Journal of Computational Finance 5 (2001) 39–59.
  • [27] V. V. Yurinski. Estimates for the characteristic functions of certain degenerate multidimensional distributions. Teor. Verojatn. Primen. (Translated in Theory Probab. Appl. 22 101–113) 17 (1972) 99–110.