Exponential Bounds for the Density of the Law of the Solution of an SDE with Locally Lipschitz Coefficients
Under the uniform Hörmander hypothesis, we study the smoothness and exponential bounds of the density of the law of the solution of a stochastic differential equation (SDE) with locally Lipschitz drift that satisfies a monotonicity condition. We extend the approach used for SDEs with globally Lipsch...
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MDPI AG
2025-02-01
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| author | Cristina Anton |
| author_facet | Cristina Anton |
| author_sort | Cristina Anton |
| collection | DOAJ |
| description | Under the uniform Hörmander hypothesis, we study the smoothness and exponential bounds of the density of the law of the solution of a stochastic differential equation (SDE) with locally Lipschitz drift that satisfies a monotonicity condition. We extend the approach used for SDEs with globally Lipschitz coefficients and obtain estimates for the Malliavin covariance matrix and its inverse. Based on these estimates and using the Malliavin differentiability of any order of the solution of the SDE, we prove exponential bounds of the solution’s density law. These results can be used to study the convergence of implicit numerical schemes for SDEs. |
| format | Article |
| id | doaj-art-0f3f5419f7014ed2889bb8bc771de04e |
| institution | DOAJ |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-0f3f5419f7014ed2889bb8bc771de04e2025-08-20T02:59:15ZengMDPI AGMathematics2227-73902025-02-0113579810.3390/math13050798Exponential Bounds for the Density of the Law of the Solution of an SDE with Locally Lipschitz CoefficientsCristina Anton0Department of Mathematics and Statistics, Grant MacEwan University, 5-103C, 10700-104 Ave., Edmonton, AB T5J 4S2, CanadaUnder the uniform Hörmander hypothesis, we study the smoothness and exponential bounds of the density of the law of the solution of a stochastic differential equation (SDE) with locally Lipschitz drift that satisfies a monotonicity condition. We extend the approach used for SDEs with globally Lipschitz coefficients and obtain estimates for the Malliavin covariance matrix and its inverse. Based on these estimates and using the Malliavin differentiability of any order of the solution of the SDE, we prove exponential bounds of the solution’s density law. These results can be used to study the convergence of implicit numerical schemes for SDEs.https://www.mdpi.com/2227-7390/13/5/798Malliavin covariance matrixHörmander’s conditionexponential bounds for densitymonotone growthstochastic differential equation |
| spellingShingle | Cristina Anton Exponential Bounds for the Density of the Law of the Solution of an SDE with Locally Lipschitz Coefficients Mathematics Malliavin covariance matrix Hörmander’s condition exponential bounds for density monotone growth stochastic differential equation |
| title | Exponential Bounds for the Density of the Law of the Solution of an SDE with Locally Lipschitz Coefficients |
| title_full | Exponential Bounds for the Density of the Law of the Solution of an SDE with Locally Lipschitz Coefficients |
| title_fullStr | Exponential Bounds for the Density of the Law of the Solution of an SDE with Locally Lipschitz Coefficients |
| title_full_unstemmed | Exponential Bounds for the Density of the Law of the Solution of an SDE with Locally Lipschitz Coefficients |
| title_short | Exponential Bounds for the Density of the Law of the Solution of an SDE with Locally Lipschitz Coefficients |
| title_sort | exponential bounds for the density of the law of the solution of an sde with locally lipschitz coefficients |
| topic | Malliavin covariance matrix Hörmander’s condition exponential bounds for density monotone growth stochastic differential equation |
| url | https://www.mdpi.com/2227-7390/13/5/798 |
| work_keys_str_mv | AT cristinaanton exponentialboundsforthedensityofthelawofthesolutionofansdewithlocallylipschitzcoefficients |