Stability of Densities for Perturbed Diffusions and Markov Chains
Konakov, Valentin1 ; Kozhina, Anna2 ; Menozzi, Stéphane3
1 Higher School of Economics, Shabolovka 31, Moscow, Russian Federation.
2 Higher School of Economics, Shabolovka 31, Moscow, Russian Federation and RTG 1953, Institute of Applied Mathematics, Heidelberg University, Germany.
3 Higher School of Economics, Shabolovka 31, Moscow, Russian Federation and LaMME, UMR CNRS 8070, Université d’Evry Val d’Essonne, 23 Boulevard de France, 91037 Evry, France.
ESAIM: Probability and Statistics, Tome 21 (2017), pp. 88-112.

We study the sensitivity of the densities of non degenerate diffusion processes and related Markov Chains with respect to a perturbation of the coefficients. Natural applications of these results appear in models with misspecified coefficients or for the investigation of the weak error of the Euler scheme with irregular coefficients.

Reçu le :
Accepté le :
DOI : 10.1051/ps/2016028
Classification : 60H10, 65C30
Mots clés : Diffusion processes, Markov chains, parametrix, Hölder coefficients, bounded drifts
Konakov, Valentin 1 ; Kozhina, Anna 2 ; Menozzi, Stéphane 3

1 Higher School of Economics, Shabolovka 31, Moscow, Russian Federation.
2 Higher School of Economics, Shabolovka 31, Moscow, Russian Federation and RTG 1953, Institute of Applied Mathematics, Heidelberg University, Germany.
3 Higher School of Economics, Shabolovka 31, Moscow, Russian Federation and LaMME, UMR CNRS 8070, Université d’Evry Val d’Essonne, 23 Boulevard de France, 91037 Evry, France. 
@article{PS_2017__21__88_0,
     author = {Konakov, Valentin and Kozhina, Anna and Menozzi, St\'ephane},
     title = {Stability of {Densities} for {Perturbed} {Diffusions} and {Markov} {Chains}},
     journal = {ESAIM: Probability and Statistics},
     pages = {88--112},
     publisher = {EDP-Sciences},
     volume = {21},
     year = {2017},
     doi = {10.1051/ps/2016028},
     mrnumber = {3630604},
     zbl = {1371.60136},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps/2016028/}
}
TY  - JOUR
AU  - Konakov, Valentin
AU  - Kozhina, Anna
AU  - Menozzi, Stéphane
TI  - Stability of Densities for Perturbed Diffusions and Markov Chains
JO  - ESAIM: Probability and Statistics
PY  - 2017
SP  - 88
EP  - 112
VL  - 21
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ps/2016028/
DO  - 10.1051/ps/2016028
LA  - en
ID  - PS_2017__21__88_0
ER  - 
%0 Journal Article
%A Konakov, Valentin
%A Kozhina, Anna
%A Menozzi, Stéphane
%T Stability of Densities for Perturbed Diffusions and Markov Chains
%J ESAIM: Probability and Statistics
%D 2017
%P 88-112
%V 21
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ps/2016028/
%R 10.1051/ps/2016028
%G en
%F PS_2017__21__88_0
Konakov, Valentin; Kozhina, Anna; Menozzi, Stéphane. Stability of Densities for Perturbed Diffusions and Markov Chains. ESAIM: Probability and Statistics, Tome 21 (2017), pp. 88-112. doi : 10.1051/ps/2016028. http://archive.numdam.org/articles/10.1051/ps/2016028/

D.G. Aronson, The fundamental solution of a linear parabolic equation containing a small parameter. Ill. J. Math. 3 (1959) 580–619. | MR | Zbl

E. Benhamou, E. Gobet and M. Miri, Expansion formulas for European options in a local volatility model. Inter. J. Theor. Appl. Fin. 13 (2010) 602–634. | DOI | MR | Zbl

R.F. Bass and E.A. Perkins, A new technique for proving uniqueness for martingale problems. From Probability to Geometry (I): Volume in Honor of the 60th Birthday of Jean–Michel Bismut (2009) 47–53. | Numdam | MR | Zbl

R. Bhattacharya and R. Rao, Normal approximations and asymptotic expansions. Wiley and sons (1976). | MR | Zbl

V. Bally and D. Talay, The law of the Euler scheme for stochastic differential equations, II. Convergence rate of the density. Monte-Carlo Methods and Appl. 2 (1996) 93–128. | DOI | MR | Zbl

F. Corielli, P. Foschi and Pascucci, Parametrix approximation of diffusion transition densities. SIAM J. Fin. Math. 1 (2010) 833–867. | DOI | MR

H. Cramér and M.R. Leadbetter, Stationary and related stochastic processes: Sample function properties and their applications. Reprint of the 1967 original. Dover Publications, Inc., Mineola, NY (2004). | MR

F. Delarue and S. Menozzi, Density estimates for a random noise propagating through a chain of differential equations. J. Funct. Anal. 259 (2010) 1577–1630. | DOI | MR | Zbl

E.B. Dynkin, Markov Processes. Springer Verlag (1965). | MR | Zbl

A. Friedman, Partial Differential Equations of Parabolic Type. Prentice-Hall (1964). | MR | Zbl

A. Friedman, Stochastic differential equations. Chapmann-Hall (1975).

V. Genon−Catalot and J. Jacod, On the estimation of the diffusion coefficient for multi-dimensional diffusion processes. Annal. l’I.H.P. Probab. Stat. 29 (1993) 119–151. | Numdam | MR | Zbl

A.M. Il’In, A.S. Kalashnikov and O.A. Oleinik, Second-order linear equations of parabolic type. Uspehi Mat. Nauk 17 (1962) 3–146. | MR

V. Konakov and E. Mammen, Local limit theorems for transition densities of Markov chains converging to diffusions. Prob. Theory Relat. Fields 117 (2000) 551–587. | DOI | MR | Zbl

V. Konakov and E. Mammen, Edgeworth type expansions for Euler schemes for stochastic differential equations. Monte Carlo Methods Appl. 8 (2002) 271–285. | DOI | MR | Zbl

V. Konakov and S. Menozzi, Weak error for the Euler scheme of a diffusion with non-regular coefficients. Preprint (2016). | arXiv | MR

V. Konakov, S. Menozzi and S. Molchanov, Explicit parametrix and local limit theorems for some degenerate diffusion processes. Ann. Inst. Henri Poincaré, Série B 46 (2010) 908–923. | Numdam | MR | Zbl

V. Kolokoltsov, Symmetric stable laws and stable-like jump diffusions. Proc. London Math. Soc. 80 (2000) 725–768. | DOI | MR | Zbl

N.V. Krylov, Lectures on elliptic and parabolic equations in Hölder spaces. Graduate Studies in Mathematics. Vol. 12. AMS (1996). | MR | Zbl

S. Menozzi, Parametrix techniques and martingale problems for some degenerate Kolmogorov equations. Electr. Commun. Prob. 17 (2011) 234–250. | MR | Zbl

R. Mikulevičius and E. Platen, Rate of convergence of the Euler approximation for diffusion processes. Math. Nachr. 151 (1991) 233–239. | DOI | MR | Zbl

H.P. Mckean and I.M. Singer, Curvature and the eigenvalues of the Laplacian. J. Differ. Geom. 1 (1967) 43–69. | DOI | MR | Zbl

S.J. Sheu, Some estimates of the transition density of a nondegenerate diffusion Markov process. Ann. Probab. 19 (1991) 538–561. | MR | Zbl

A.N. Shiryaev, Probability, 2nd Edition. Graduate Texts in Mathematics. Vol. 95. Springer-Verlag, New York (1996). | MR | Zbl

D.W. Stroock and S.R.S. Varadhan, Multidimensional diffusion processes. Springer-Verlag Berlin Heidelberg New York (1979). | MR | Zbl

Cité par Sources :