Abstract
We prove a uniform bound for the density, p t (x), of the solution at time t∈(0, 1] of a 1-dimensional stochastic differential equation, under hypoellipticity conditions. A similar bound is obtained for an expression involving the distributional derivative (with respect to x) of p t (x). These results are applied to extend the Itô formula to the composition of a function (satisfying slight regularity conditions) with a hypoelliptic diffusion process in the spirit of the work of Föllmer et al. (5)
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REFERENCES
Bardina, X., and Jolis, M. (1997). An extension of Itô's formula for elliptic diffusion processes. Stochastic Processes Appl. 69, 83-109.
Caballero, M. E., Fernández, B., and Nualart, D. (1998). Estimations of densities and applications. J. Theoret. Probab. 11(3), 831-851.
Flandoli, F., Russo, F., and Wolf, J. (2000). Some stochastic differential equations with distributional drift. Preprint.
Föllmer, H., and Protter, P. (2000). On Itô's formula for multidimensional Brownian motion. Probab. Theory Relat. Fields 116(1), 1-20.
Föllmer, H., Protter, P., and Shiryayev, A. (1995). Quadratic covariatiom and an extension of Itô's formula. Bernoulli 1, 149-169.
Haussmann, U. G., and Pardoux, E. (1986). Time reversal of diffusions. Ann. Probab. 14(4), 1188-1205.
Millet, A., Nualart, D., and Sanz, M. (1989). Integration by parts and time reversal for diffusion processes. Ann. Probab. 17(1)(1), 206-238.
Moret, S., and Nualart, D. (2000). Quadratic covariation and Itô's formula for smooth non-degenerate martingales. J. Theoret. Probab. 13(1), 193-224.
Moret, S., and Nualart, D. (2001). Generalization of Itô's formula for smooth nondegenerate martingales. Stochastic Process. Appl. 91(1), 115-149.
Nualart, D. (1995). The Malliavin Calculus and Related Topics, Springer Verlag.
Nualart, D. (1998). Analysis on Wiener Space ancd Anticipating Stochastic Calculus, Lecture Notes in Math., Vol. 1690, Springer Verlag.
Pardoux, E. (1986). Grossissement d'une filtration et retournement du temps d'une diffusion, Séminaire de Proba. XX, Lecture Notes in Math., Vol. 1204, Springer Verlag, pp. 48-55.
Rozkosz, A. (1996). Stochastic representation of diffusions corresponding to divergence form operators. Stochastic Processes Appl. 63, 11-33.
Russo, F., and Vallois, P. (1996). Itô formula for \(C^1 \)-functions of semimartingales. Probab. Theory Relat. Fields 104(1), 27-41.
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Bardina, X., Jolis, M. Estimation of the Density of Hypoelliptic Diffusion Processes with Application to an Extended Itô's Formula. Journal of Theoretical Probability 15, 223–247 (2002). https://doi.org/10.1023/A:1013899603656
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DOI: https://doi.org/10.1023/A:1013899603656