The Annals of Probability

Densities for SDEs driven by degenerate $\alpha$-stable processes

Xicheng Zhang

Full-text: Open access

Abstract

In this work, by using the Malliavin calculus, under Hörmander’s condition, we prove the existence of distributional densities for the solutions of stochastic differential equations driven by degenerate subordinated Brownian motions. Moreover, in a special degenerate case, we also obtain the smoothness of the density. In particular, we obtain the existence of smooth heat kernels for the following fractional kinetic Fokker–Planck (nonlocal) operator: \[\mathscr{L}^{(\alpha)}_{b}:=\Delta^{\alpha/2}_{\mathrm{v}}+\mathrm{v} \cdot \nabla_{x}+b(x,\mathrm{v})\cdot\nabla_{\mathrm{v}},\qquad x,\mathrm{v}\in\mathbb{R}^{d},\] where $\alpha\in(0,2)$ and $b:\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ is smooth and has bounded derivatives of all orders.

Article information

Source
Ann. Probab., Volume 42, Number 5 (2014), 1885-1910.

Dates
First available in Project Euclid: 29 August 2014

Permanent link to this document
https://projecteuclid.org/euclid.aop/1409319469

Digital Object Identifier
doi:10.1214/13-AOP900

Mathematical Reviews number (MathSciNet)
MR3262494

Zentralblatt MATH identifier
1307.60090

Subjects
Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 35Q84: Fokker-Planck equations

Keywords
Malliavin calculus Hörmander’s condition $\alpha$-stable process distributional density SDE

Citation

Zhang, Xicheng. Densities for SDEs driven by degenerate $\alpha$-stable processes. Ann. Probab. 42 (2014), no. 5, 1885--1910. doi:10.1214/13-AOP900. https://projecteuclid.org/euclid.aop/1409319469


Export citation

References

  • [1] Applebaum, D. (2009). Lévy Processes and Stochastic Calculus, 2nd ed. Cambridge Studies in Advanced Mathematics 116. Cambridge Univ. Press, Cambridge.
  • [2] Bally, V. and Clément, E. (2011). Integration by parts formula and applications to equations with jumps. Probab. Theory Related Fields 151 613–657.
  • [3] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge.
  • [4] Bichteler, K., Gravereaux, J.-B. and Jacod, J. (1987). Malliavin Calculus for Processes with Jumps. Stochastics Monographs 2. Gordon and Breach Science Publishers, New York.
  • [5] Bismut, J.-M. (1983). Calcul des variations stochastique et processus de sauts. Z. Wahrsch. Verw. Gebiete 63 147–235.
  • [6] Bondarchuk, S. V. and Kulik, O. M. (2008). Conditions for the existence and smoothness of a density of distribution for Ornstein–Uhlenbeck processes with Lévy noise. Theory Probab. Math. Statist. 79 20–33.
  • [7] Cass, T. (2009). Smooth densities for solutions to stochastic differential equations with jumps. Stochastic Process. Appl. 119 1416–1435.
  • [8] Ishikawa, Y. and Kunita, H. (2006). Malliavin calculus on the Wiener–Poisson space and its application to canonical SDE with jumps. Stochastic Process. Appl. 116 1743–1769.
  • [9] Komatsu, T. and Takeuchi, A. (2001). On the smoothness of PDF of solutions to SDE of jump type. Int. J. Differ. Equ. Appl. 2 141–197.
  • [10] Kunita, H. (2011). Analysis of nondegenerate Wiener–Poisson functionals and its applications to Itô’s SDE with jumps. Sankhya A 73 1–45.
  • [11] Kunita, H. (2013). Nondegenerate SDE’s with jumps and their hypoelliptic properties. J. Math. Soc. Japan 65 993–1035.
  • [12] Kusuoka, S. (2010). Malliavin calculus for stochastic differential equations driven by subordinated Brownian motions. Kyoto J. Math. 50 491–520.
  • [13] Léandre, R. (1988). Régularité de processus de sauts dégénérés. II. Ann. Inst. Henri Poincaré Probab. Stat. 24 209–236.
  • [14] Malliavin, P. (1978). Stochastic calculus of variation and hypoelliptic operators. In Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976) 195–263. Wiley, New York.
  • [15] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Springer, Berlin.
  • [16] Picard, J. (1996). On the existence of smooth densities for jump processes. Probab. Theory Related Fields 105 481–511.
  • [17] Priola, E. and Zabczyk, J. (2009). Densities for Ornstein–Uhlenbeck processes with jumps. Bull. Lond. Math. Soc. 41 41–50.
  • [18] Protter, P. E. (2004). Stochastic Integration and Differential Equations, 2nd ed. Applications of Mathematics (New York) 21. Springer, Berlin.
  • [19] Sato, K.-i. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge.
  • [20] Takeuchi, A. (2002). The Malliavin calculus for SDE with jumps and the partially hypoelliptic problem. Osaka J. Math. 39 523–559.
  • [21] Zhang, X. (2013). Derivative formulas and gradient estimates for SDEs driven by $\alpha$-stable processes. Stochastic Process. Appl. 123 1213–1228.