## Abstract

In this article, we study the density function of the numerical solution of the splitting averaged vector field (AVF) scheme for the stochastic Langevin equation. We first show the existence of the density function of the numerical solution by proving its exponential integrability property, Malliavin differentiability and the almost surely non-degeneracy of the associated Malliavin covariance matrix. Then the smoothness of the density function is obtained through a lower bound estimate of the smallest eigenvalue of the corresponding Malliavin covariance matrix. Meanwhile, we derive the optimal strong convergence rate in every Malliavin–Sobolev norm of the numerical solution via Malliavin calculus. Combining the strong convergence result and the smoothness of the density functions, we prove that the convergence order of the density function of the numerical scheme coincides with its strong convergence order.

Original language | English |
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Pages (from-to) | 2283-2333 |

Number of pages | 51 |

Journal | Mathematics of Computation |

Volume | 91 |

Issue number | 337 |

DOIs | |

Publication status | Published - Oct 2022 |

## Keywords

- Density function
- Malliavin calculus
- Non-globally monotone coefficient
- Splitting avf scheme
- Stochastic langevin equation
- Strong convergence

## ASJC Scopus subject areas

- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics