The law of the Euler scheme for stochastic differential equations. II: Convergence rate of the density.

*(English)*Zbl 0866.60049Summary: In the first part of this work [Probab. Theory Relat. Fields 104, No. 1, 43-60 (1996; Zbl 0838.60051)] we have studied the approximation problem of \(\mathbb{E} f(X_T)\) by \(\mathbb{E} f(X^n_T)\), where \((X_t)\) is the solution of a stochastic differential equation, \((X_t^n)\) is defined by the Euler discretization scheme with step \(T/n\), and \(f(\cdot)\) is a given function, only supposed measurable and bounded; we have proven that the error can be expanded in terms of powers of \(1/n\), under a nondegeneracy condition of Hörmander type for the infinitesimal generator of \((X_t)\).

In this second part, we consider the density of the law of a small perturbation of \(X^n_T\) and we compare it to the density of the law of \(X_T\): we prove that the difference between the densities can also be expanded in terms of \(1/n\). The results of this paper had been announced in special issues of journals devoted to the Proceedings of Conferences: see authors and P. Protter [same title, Z. Angew. Math. Mech. (1995)] and the authors [Math. Comput. Simul. 38, No. 1-3, 35-41 (1995; Zbl 0824.60056)].

In this second part, we consider the density of the law of a small perturbation of \(X^n_T\) and we compare it to the density of the law of \(X_T\): we prove that the difference between the densities can also be expanded in terms of \(1/n\). The results of this paper had been announced in special issues of journals devoted to the Proceedings of Conferences: see authors and P. Protter [same title, Z. Angew. Math. Mech. (1995)] and the authors [Math. Comput. Simul. 38, No. 1-3, 35-41 (1995; Zbl 0824.60056)].

##### MSC:

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

65C99 | Probabilistic methods, stochastic differential equations |

65C05 | Monte Carlo methods |

60J60 | Diffusion processes |