Brazilian Journal of Probability and Statistics

Density for solutions to stochastic differential equations with unbounded drift

Christian Olivera and Ciprian Tudor

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Via a special transform and by using the techniques of the Malliavin calculus, we analyze the density of the solution to a stochastic differential equation with unbounded drift.

Article information

Braz. J. Probab. Stat., Volume 33, Number 3 (2019), 520-531.

Received: August 2017
Accepted: May 2018
First available in Project Euclid: 10 June 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Stochastic differential equations unbounded drift Malliavin calculus existence of the density


Olivera, Christian; Tudor, Ciprian. Density for solutions to stochastic differential equations with unbounded drift. Braz. J. Probab. Stat. 33 (2019), no. 3, 520--531. doi:10.1214/18-BJPS400.

Export citation


  • Bally, V. and Caramellino, L. (2017). Regularity of probability laws by using an interpolation method. Annals of Probability 45, 1110–1159.
  • Bally, V., Caramellino, L. and Cont, R. (2016). Stochastic integration by parts and functional Itô calculus. In Lecture Notes of the Barcelona Summer School on Stochastic Analysis Held in Barcelona, July 23–27, 2012. Advanced Courses in Mathematics. CRM Barcelona Cham: Birkhäuser/Springer.
  • Baños, D. and Krühner, P. (2017). Hölder continuous densities of solutions of SDEs with measurable and path dependent drift coefficients. Stochastic Processes and Their Applications 127, 1785–1799.
  • De Marco, S. (2011). Smoothness and asymptotic estimates of densities for SDEs with locally smooth coefficients and applications to square root-type diffusions. The Annals of Applied Probability 21, 1282–1321.
  • Debussche, A. and Fournier, N. (2013). Existence of densities for stable-like driven SDE’s with Hölder continuous coefficients. Journal of Functional Analysis 264, 1757–1778.
  • Flandoli, F., Gubinelli, M. and Priola, E. (2010a). Well-posedness of the transport equation by stochastic perturbation. Inventiones Mathematicae 180, 1–53.
  • Flandoli, F., Gubinelli, M. and Priola, E. (2010b). Flow of diffeomorphisms for SDEs with unbounded Hölder continuous drift. Bulletin Des Sciences Mathématiques 134, 405–422.
  • Fournier, N. and Printems, J. (2010). Absolute continuity for some one-dimension al processes. Bernoulli 16, 343–360.
  • Hayashi, M., Kohatsu-Higa, A. and Yuki, G. (2013). Local Hölder continuity property of the densities of solutions of SDEs with singular coefficients. Journal of Theoretical Probability 26, 1117–1134.
  • Kohatsu-Higa, A. (2003). Lower bounds for densities of uniformly elliptic random variables on Wiener space. Probability Theory and Related Fields 126, 421–457.
  • Kohatsu-Higa, A. and Makhlouf, A. (2013). Estimates for the density of functionals of sdes with irregular drift. Stochastic Processes and Their Applications 123, 1716–1728.
  • Kohatsu-Higa, A. and Tanaka, A. (2012). Malliavin calculus method to study densities of additive functionals of SDE’s with irregular drifts. Annales de L’IHP Probabilités et Statistiques 48, 871–883.
  • Kunita, H. (1984). Stochastic differential equations and stochastic flows of diffeomorphisms. In Ecole D’été de Probabilités de Saint-Flour, XII–1982. Lecture Notes in Math. 1097, 143–303. Berlin: Springer.
  • Nourdin, I. and Peccati, G. (2012). Normal Approximations with Malliavin Calculus from Stein’s Method to Universality. Cambridge: Cambridge University Press.
  • Nourdin, I. and Viens, F. (2009). Density formula and concentration inequalities with Malliavin calculus. Electronic Journal of Probability 14, 2287–2309.
  • Nualart, D. (2006). Malliavin Calculus and Related Topics, 2nd ed. New York: Springer.
  • Nualart, D. and Quer-Sardanyons, L. (2009). Gaussian density estimates for solutions to quasi-linear stochastic partial differential equations. Stochastic Processes and Their Applications 119, 3914–3938.
  • Romito, M. (2016). Time regularity of the densities for the Navier–Stokes equations with noise. Journal of Evolution Equations 16, 503–518.
  • Romito, M. (2017). A simple method for the existence of a density for stochastic evolutions with rough coefficients. Preprint. Available at arXiv:1707.05042.
  • Sanz-Solé, M. (1995). Malliavin Calculus. With Applications to Stochastic Partial Differential Equations. Lausanne: Fundamental Sciences, EPFL Press.
  • Shigekawa, I. (1980). Derivatives of Wiener functionals and absolute continuity of induced measures. Journal of Mathematics of Kyoto University 20, 263–289.
  • Zhang, X. (2014). Stochastic differential equations with Sobolev diffusion and singular drift. The Annals of Applied Probability 26, 2697–2732.