In this manuscript, we consider stochastic expressions of the parametrix method for solutions of -dimensional stochastic differential equations (SDEs) with drift coefficients which belong to . We prove the existence and Hölder continuity of probability density functions for distributions of solutions at fixed points and obtain an explicit expansion via (stochastic) parametrix methods. We also obtain Gaussian type upper and lower bounds for these probability density functions.
Mots-clés : Stochastic differential equation, singular drift, density function, Gaussian two-sided bounds, parametrix method
@article{PS_2018__22__178_0,
author = {Kohatsu-Higa, Arturo and Y\^uki, G\^o},
title = {Stochastic formulations of the parametrix method},
journal = {ESAIM: Probability and Statistics},
pages = {178--209},
publisher = {EDP-Sciences},
volume = {22},
year = {2018},
doi = {10.1051/ps/2018013},
mrnumber = {3891754},
zbl = {1403.60047},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2018013/}
}
TY - JOUR AU - Kohatsu-Higa, Arturo AU - Yûki, Gô TI - Stochastic formulations of the parametrix method JO - ESAIM: Probability and Statistics PY - 2018 SP - 178 EP - 209 VL - 22 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2018013/ DO - 10.1051/ps/2018013 LA - en ID - PS_2018__22__178_0 ER -
Kohatsu-Higa, Arturo; Yûki, Gô. Stochastic formulations of the parametrix method. ESAIM: Probability and Statistics, Tome 22 (2018), pp. 178-209. doi : 10.1051/ps/2018013. http://www.numdam.org/articles/10.1051/ps/2018013/
, and , Gaussian estimates for elliptic operators with unbounded drift. J. Math. Anal. Appl. 338 (2008) 505–517. | DOI | MR | Zbl
and , A probabilistic interpretation of the parametrix method. Ann. Appl. Probab. 25 (2015) 3095–3138. | DOI | MR | Zbl
and , Optimal density bounds for marginals of Itô processes. Commun. Stoch. Anal. 10 (2016) 1. | MR
and , Hölder continuous densities of solutions of SDEs with measurable and path dependent drift coefficients. Stoch. Proc. Appl. 127 (2017) 1785–1799. | DOI | MR | Zbl
and , Singular Stochastic Differential Equations, Vol. 1858. Springer Science & Business Media, Berlin (2005). | DOI | MR | Zbl
, and Parametrix approximation of diffusion transition densities. SIAM J. Finan. Math. 1 (2010) 833–867. | DOI | MR
, , and , Strong uniqueness for stochastic evolution equations in hilbert spaces perturbed by a bounded measurable drift. Ann. Probab. 41 (2013) 3306–3344. | DOI | MR | Zbl
, , and , Strong uniqueness for stochastic evolution equations with unbounded measurable drift term. J. Theor. Probab. 28 (2015) 1571–1600. | DOI | MR | Zbl
, Smoothness and asymptotic estimates of densities for sdes with locally smooth coefficients and applications to square root-type diffusions. Ann. Appl. Probab. 21 (2011) 1282–1321. | DOI | MR | Zbl
and , On a class of degenerate parabolic equations of kolmogorov type. AMRX Appl. Math. Res. Express 3 (2005) 77–116. | DOI | MR
, Partial Differential Equations of Parabolic Type. Prentice-Hall, Inc., Englewood Cliffs, NJ (1964). | MR | Zbl
, and , Local Hölder continuity property of the densities of solutions of SDEs with singular coefficients. J. Theoret. Probab. 26 (2013) 1117–1134. | DOI | MR | Zbl
, and , Hölder continuity property of the densities of SDEs with singular drift coefficients. Electron. J. Probab. 19 (2014) 1–22. | DOI | MR | Zbl
and , Brownian Motion and Stochastic Calculus, 2nd Edn. Vol. 113 of Graduate Texts in Mathematics. Springer-Verlag, New York (1991). | MR | Zbl
and , Two-sided estimates on the density of brownian motion with singular drift. Illinois J. Math. 50 (2006) 635–688. | MR | Zbl
and , Estimates for the density of functionals of SDEs with irregular drift. Stoch. Process. Appl. 123 (2013) 1716–1728. | DOI | MR | Zbl
and , A Malliavin calculus method to study densities of additive functionals of SDE’s with irregular drifts. Ann.Inst. Henri Poincaré Probab. Stat. 48 (2012) 871–883. | DOI | Numdam | MR | Zbl
and , Strong solutions of stochastic equations with singular time dependent drift. Probab. Theory Relat. Fields 131 (2005) 154–196. | DOI | MR | Zbl
, Continuity and gaussian two-sided bounds of the density functions of the solutions to path-dependent stochastic differential equations via perturbation. Stoch. Process. Appl. (2016) 359–384. | MR | Zbl
and Applications of the Malliavin calculus. II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 (1985) 1–76. | MR | Zbl
, Existence and Gaussian bounds for the density of Brownian motion with random drift. Commun. Stoch. Anal. 10 (2016) 151–162. | MR
, The Malliavin Calculus and Related Topics, 2nd Edn. Probability and its Applications (New York). Springer-Verlag, Berlin (2006). | MR | Zbl
and , The Parametrix Method for Parabolic Spdes. Preprint (2018). | arXiv | MR
, Generalized Diffusion Processes. Translations of Mathematical Monographs. American Mathematical Society, Providence,RI (1990). | DOI | MR | Zbl
and , A representation formula for transition probability densities of diffusions and applications. Stoch. Process. Appl. 111 (2004) 57–76. | DOI | MR | Zbl
, and , Comparison theorem and estimates for transition probability densities of diffusion processes. Probab. Theory Relat. Fields 127 (2003) 388–406. | DOI | MR | Zbl
, A Simple Method for the Existence of a Density for Stochastic Evolutions with Rough Coefficients. Preprint (2017). | arXiv | MR
and , On singular stochastic differential equations and dirichlet forms. Rev. Roumaine Math. Pures Appl. 62 (2017a) 217–258. | MR | Zbl
and , On the stochastic regularity of distorted brownian motions. Trans. Am. Math. Soc. 369 (2017b) 7883–7915. | DOI | MR | Zbl
and , Explicit formulae for the solutions of stochastic equations. Mat. Sb. (N.S.) 100 (1976) 266–284. | MR | Zbl
Cité par Sources :
