• Bernoulli
  • Volume 16, Number 2 (2010), 343-360.

Absolute continuity for some one-dimensional processes

Nicolas Fournier and Jacques Printems

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We introduce an elementary method for proving the absolute continuity of the time marginals of one-dimensional processes. It is based on a comparison between the Fourier transform of such time marginals with those of the one-step Euler approximation of the underlying process. We obtain some absolute continuity results for stochastic differential equations with Hölder continuous coefficients. Furthermore, we allow such coefficients to be random and to depend on the whole path of the solution. We also show how it can be extended to some stochastic partial differential equations and to some Lévy-driven stochastic differential equations. In the cases under study, the Malliavin calculus cannot be used, because the solution in generally not Malliavin differentiable.

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Bernoulli, Volume 16, Number 2 (2010), 343-360.

First available in Project Euclid: 25 May 2010

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absolute continuity Hölder coefficients Lévy processes random coefficients stochastic differential equations stochastic partial differential equations


Fournier, Nicolas; Printems, Jacques. Absolute continuity for some one-dimensional processes. Bernoulli 16 (2010), no. 2, 343--360. doi:10.3150/09-BEJ215.

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  • [1] Aronson, D.G. (1968). Non-negative solutions of linear parabolic equations. Ann. Sc. Norm. Super Pisa Cl. Sci. (5) 22 607–694.
  • [2] Bally, V. (2008). Malliavin calculus for locally smooths laws and applications to diffusion processes with jumps. Preprint.
  • [3] Bally, V., Gyongy, I. and Pardoux, E. (1994). White noise driven parabolic SPDEs with measurable drift. J. Funct. Anal. 120 484–510.
  • [4] Bally, V. and Pardoux, E. (1998). Malliavin calculus for white noise driven parabolic SPDEs. Potential Anal. 9 27–64.
  • [5] Bally, V., Millet, A. and Sanz-Solé, M. (1995). Approximation and support in Hölder norm for parabolic stochastic partial differential equations. Ann. Probab. 23 178–222.
  • [6] Bichteler, K., Gravereaux, J.B. and Jacod, J. (1987). Malliavin Calculus for Processes With Jumps. Stochastics Monographs 2. New York: Gordon & Breach.
  • [7] Bichteler, K. and Jacod, J. (1983). Calcul de Malliavin pour les diffusions avec sauts: Existence d’une densité dans le cas unidimensionnel. In Seminar on Probability, XVII. Lecture Notes in Math. 986 132–157. Berlin: Springer.
  • [8] Bouleau, N. and Hirsch, F. (1986). Propriétés d’absolue continuité dans les espaces de Dirichlet et application aux E.D.S. In Séminaire de Probabilités, XX, 1984/85. Lecture Notes in Math. 1204 131–161. Berlin: Springer.
  • [9] Dellacherie, C. and Meyer, P.A. (1982). Probability and Potentials B. Amsterdam: North Holland.
  • [10] Denis, L. (2000). A criterion of density for solutions of Poisson-driven SDEs. Probab. Theory Related Fields 118 406–426.
  • [11] Fouque, J.P., Papanicolaou, G. and Sircar, K. (2000). Derivatives in Financial Markets With Stochastic Volatility. Cambridge: Cambridge Univ. Press.
  • [12] Gatarek, D. and Goldys, B. (1994). On weak solutions of stochastic equations in Hilbert spaces. Stochastics Stochastics Rep. 46 41–51.
  • [13] Gyongy, I. (1986). Mimicking the one-dimensional marginal distributions of processes having an Itô differential. Probab. Theory Related Fields 71 501–516.
  • [14] Heston, S. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6 327–343.
  • [15] 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.
  • [16] Jacod, J. (1979). Calcul stochastique et problèmes de martingales. Lecture Notes in Mathematics 714. Berlin: Springer.
  • [17] Kahane, J.P. and Salem, R. (1963). Ensembles parfaits et séries trigonométriques. Actualités Sci. Indust. 1301. Paris: Hermann.
  • [18] Karatzas, I. and Shreve, S.E. (1988). Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics 113. New York: Springer.
  • [19] Kulik, A. (2006). Malliavin calculus for Lévy processes with arbitrary Lévy measure. Theory Probab. Math. Statist. 72 75–92.
  • [20] Kulik, A. (2007). Stochastic calculus of variations for general Lévy processes and its applications to jump-type SDEs with non-degenerated drift. Available at arXiv:math/0606427v2.
  • [21] Malliavin, P. (1997). Stochastic Analysis. Grundlehren der Mathematischen Wissenschaften 313. Berlin: Springer.
  • [22] Nourdin, I. and Simon, T. (2006). On the absolute continuity of Lévy processes with drift. Ann. Probab. 34 1035–1051.
  • [23] Nualart, D. (1995). The Malliavin Calculus and Related Topics. New York: Springer.
  • [24] Pardoux, E. and Zhang, T.S. (1993). Absolute continuity of the law of the solution of a parabolic SPDE. J. Funct. Anal. 112 447–458.
  • [25] Picard, J. (1996). On the existence of smooth densities for jump processes. Probab. Theory Related Fields 105 481–511.
  • [26] Walsh, J.B. (1986). An introduction to stochastic partial differential equations. In Ecole d’été de probabilités de Saint-Flour, XIV, 1984. Lecture Notes in Math. 1180 265–439. Berlin: Springer.