×

zbMATH — the first resource for mathematics

Absolute continuity for some one-dimensional processes. (English) Zbl 1248.60062
Summary: We introduce an elementary method for proving the absolute continuity of 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 the method 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 is in general not Malliavin differentiable.

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G30 Continuity and singularity of induced measures
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
PDF BibTeX XML Cite
Full Text: DOI Euclid arXiv
References:
[1] Aronson, D.G. (1968). Non-negative solutions of linear parabolic equations. Ann. Sc. Norm. Super Pisa Cl. Sci. (5) 22 607-694. · Zbl 0182.13802 · numdam:ASNSP_1968_3_22_4_607_0 · eudml:83474
[2] Bally, V. (2008). Malliavin calculus for locally smooths laws and applications to diffusion processes with jumps.
[3] Bally, V., Gyongy, I. and Pardoux, E. (1994). White noise driven parabolic SPDEs with measurable drift. J. Funct. Anal. 120 484-510. · Zbl 0801.60049 · doi:10.1006/jfan.1994.1040
[4] Bally, V. and Pardoux, E. (1998). Malliavin calculus for white noise driven parabolic SPDEs. Potential Anal. 9 27-64. · Zbl 0928.60040 · doi:10.1023/A:1008686922032
[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. · Zbl 0835.60053 · doi:10.1214/aop/1176988383
[6] Bichteler, K., Gravereaux, J.B. and Jacod, J. (1987). Malliavin Calculus for Processes With Jumps. Stochastics Monographs 2 . New York: Gordon & Breach. · Zbl 0706.60057
[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. · Zbl 0525.60067 · numdam:SPS_1983__17__132_0 · eudml:113430
[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. · Zbl 0642.60044 · doi:10.1007/BFb0075717 · numdam:SPS_1986__20__131_0 · eudml:113542
[9] Dellacherie, C. and Meyer, P.A. (1982). Probability and Potentials B . Amsterdam: North Holland. · Zbl 0494.60002
[10] Denis, L. (2000). A criterion of density for solutions of Poisson-driven SDEs. Probab. Theory Related Fields 118 406-426. · Zbl 0969.60064 · doi:10.1007/s004400000082
[11] Fouque, J.P., Papanicolaou, G. and Sircar, K. (2000). Derivatives in Financial Markets With Stochastic Volatility . Cambridge: Cambridge Univ. Press. · Zbl 0954.91025
[12] Gatarek, D. and Goldys, B. (1994). On weak solutions of stochastic equations in Hilbert spaces. Stochastics Stochastics Rep. 46 41-51. · Zbl 0824.60052
[13] Gyongy, I. (1986). Mimicking the one-dimensional marginal distributions of processes having an Itô differential. Probab. Theory Related Fields 71 501-516. · Zbl 0579.60043 · doi:10.1007/BF00699039
[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. · Zbl 1384.35131
[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. · Zbl 1107.60028 · doi:10.1016/j.spa.2006.04.013
[16] Jacod, J. (1979). Calcul stochastique et problèmes de martingales. Lecture Notes in Mathematics 714 . Berlin: Springer. · Zbl 0414.60053 · doi:10.1007/BFb0064907
[17] Kahane, J.P. and Salem, R. (1963). Ensembles parfaits et séries trigonométriques. Actualités Sci. Indust. 1301 . Paris: Hermann. · Zbl 0112.29304
[18] Karatzas, I. and Shreve, S.E. (1988). Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics 113 . New York: Springer. · Zbl 0638.60065
[19] Kulik, A. (2006). Malliavin calculus for Lévy processes with arbitrary Lévy measure. Theory Probab. Math. Statist. 72 75-92. · Zbl 1123.60040 · www.ams.org
[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.org
[21] Malliavin, P. (1997). Stochastic Analysis. Grundlehren der Mathematischen Wissenschaften 313 . Berlin: Springer. · Zbl 0878.60001
[22] Nourdin, I. and Simon, T. (2006). On the absolute continuity of Lévy processes with drift. Ann. Probab. 34 1035-1051. · Zbl 1099.60045 · doi:10.1214/009117905000000620
[23] Nualart, D. (1995). The Malliavin Calculus and Related Topics . New York: Springer. · Zbl 0837.60050
[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. · Zbl 0777.60046 · doi:10.1006/jfan.1993.1040
[25] Picard, J. (1996). On the existence of smooth densities for jump processes. Probab. Theory Related Fields 105 481-511. · Zbl 0853.60064 · doi:10.1007/BF01191910
[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. · Zbl 0608.60060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.