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The law of the Euler scheme for stochastic differential equations. II: Convergence rate of the density. (English) Zbl 0866.60049
Summary: 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)].

##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 65C99 Probabilistic methods, stochastic differential equations 65C05 Monte Carlo methods 60J60 Diffusion processes
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