Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Invariant measures and a stability theorem for locally Lipschitz stochastic delay equations

I. Stojkovic and O. van Gaans

Full-text: Open access


We consider a stochastic delay differential equation with exponentially stable drift and diffusion driven by a general Lévy process. The diffusion coefficient is assumed to be locally Lipschitz and bounded. Under a mild condition on the large jumps of the Lévy process, we show existence of an invariant measure. Main tools in our proof are a variation-of-constants formula and a stability theorem in our context, which are of independent interest.


Nous considérons une equation différentielle stochastique retardée à dérive exponentiellement stable et conduite par un processus de Lévy général. Le coefficient de la diffusion est seulement supposé satisfaire une condition lipschitzienne locale et être borné. En supposant une condition additionnelle faible sur les grands sauts du processus de Lévy, nous démontrons l’existence d’une mesure invariante. Les principaux ingrédients de la preuve sont une formule pour les variations des constantes et un théorème de stabilité par rapport aux perturbations des conditions initiales, qui sont d’un intérêt indépendant.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 47, Number 4 (2011), 1121-1146.

First available in Project Euclid: 6 October 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34K50: Stochastic functional-differential equations [See also , 60Hxx] 60G48: Generalizations of martingales

Delay equation Invariant measure Lévy process Semimartingale Skorohod space Stability Tightness Variation-of-constants formula


Stojkovic, I.; van Gaans, O. Invariant measures and a stability theorem for locally Lipschitz stochastic delay equations. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 4, 1121--1146. doi:10.1214/10-AIHP396.

Export citation


  • [1] D. Applebaum. Lévy Processes and Stochastic Calculus. Cambridge Univ. Press, Cambridge, 2004.
  • [2] J. Appleby and X. Mao. Stochastic stabilisation of functional differential equations. Systems Control Lett. 54 (2005) 1069–1081.
  • [3] P. Billingsley. Convergence of Probability Measures, 2nd edition. Wiley, New York, 1999.
  • [4] S. Cerrai. Stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term. Probab. Theory Related Fields 125 (2003) 271–304.
  • [5] F. Confortola. Dissipative backward stochastic differential equations with locally Lipschitz nonlinearity. Stochastic Process. Appl. 117 (2007) 613–628.
  • [6] G. Da Prato and J. Zabczyk. Stochastic Equations in Infinite Dimensions. Cambridge Univ. Press, Cambridge, 1992.
  • [7] M. C. Delfour. The largest class of hereditary systems defining a C0 semigroup on the product space. Canad. J. Math. 32 (1980) 969–978.
  • [8] O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther. Delay Equations. Functional-, Complex-, and Nonlinear Analysis. Springer, New York, 1995.
  • [9] A. Es-Sarhir and W. Stannat. Invariant measures for semilinear SPDE’s with local Lipschitz drift coefficients and applications. J. Evol. Equ. 8 (2008) 129–154.
  • [10] A. Gushchin and U. Küchler. On stationary solutions of delay differential equations driven by a Lévy process. Stochastic Process. Appl. 88 (2000) 195–211.
  • [11] J. Hale. Theory of Functional Differential Equations. Springer, New York, 1977.
  • [12] E. Hausenblas. SPDEs driven by Poisson random measures with non Lipschitz coefficients: Existence results. Probab. Theory Related Fields 137 (2007) 161–200.
  • [13] J. Jacod and J. Memin. Weak and strong solutions of stochastic differential equations: Existence and stability. In Stochastic Integrals, Proceedings, LMS Durham Symposium, 1980 169–212. D. Williams (Ed.). Springer LNM 851. Springer, Berlin, 1981.
  • [14] J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes. Springer, Berlin, 2003.
  • [15] X. Mao. Stochastic Differential Equations and Their Applications. Horwood Publishing, Chichester, 1997.
  • [16] S.-E. A. Mohammed. Stochastic Functional Differential Equations. Pitman Advanced Publishing Program, Melbourne, 1984.
  • [17] S.-E. A. Mohammed and M. K. R. Scheutzow. Lyapunov exponents of linear stochastic functional differential equations. II: Examples and case studies. Ann. Probab. 25 (1997) 1210–1240.
  • [18] L. Mytnik, E. Perkins and A. Sturm. On pathwise uniqueness for stochastic heat equations with non-Lipschitz coefficients. Ann. Probab. 34 (2006) 1910–1959.
  • [19] M. Ondreját. Uniqueness for stochastic non-linear wave equations. Nonlinear Anal. 67 (2007) 3287–3310.
  • [20] P. E. Protter. Stochastic Integration and Differential Equations, 2nd edition. Springer, Berlin, 2005.
  • [21] M. Reiß, M. Riedle and O. van Gaans. Delay differential equations driven by Lévy processes: Stationarity and Feller properties. Stochastic Process. Appl. 116 (2006) 1409–1432.
  • [22] M. Reiß, M. Riedle and O. van Gaans. On Émery’s inequality and a variation-of-constants formula. Stoch. Anal. Appl. 25 (2007) 353–379.
  • [23] R. Stelzer. Multivariate COGARCH(1, 1) processes. Bernoulli 16 (2010) 80–115.
  • [24] N. N. Vakhania, V. I. Tarieladze and S. A. Chobanyan. Probability Distributions on Banach Spaces. D. Reidel Publishing Company, Dordrecht, 1987.
  • [25] S. J. Wolfe. On a continuous analogue of the stochastic difference equation Xn = ρXn−1 + Bn. Stochastic Process. Appl. 12 (1982) 301–312.