## Recursive computation of the invariant distribution of a diffusion.(English)Zbl 1006.60074

Consider a stochastic differential equation on $$\mathbb{R}^d$$: $dY_t= \sigma(Y_t) dW_t+ b(Y_t) dt,\tag{$$*$$}$ with more or less classical growth and stability conditions on $$(\sigma,b)$$, and admitting a unique invariant measure $$\nu$$. Choose a sequence $$(\gamma_n)$$ such that $$\gamma_n\geq 0$$, $$\lim_n\gamma_n= 0$$, $$\sum^\infty_1 \gamma_n= \infty$$. Discretize the equation $$(*)$$ by defining (for an i.i.d. sequence $$(U_n)$$): $X_{n+1}:= X_n+ \gamma_{n+1} b(X_n)+ \sqrt{\gamma_{n+1}} \sigma(X_n) U_{n+1}$ for any $$n\in\mathbb{N}$$. Approximate $$\nu$$ by $$\nu_n:= \sum^n_{j=1} \gamma_j\delta_{X_{j-1}}/\sum^n_{j=1} \gamma_j$$, so that recursively $\nu_{n+1}(b)= \nu_n(b)+ [f\circ X_n- \nu_n(f)]\times \gamma_{n+1}\Biggl/\sum^{n+1}_{j= 1}\gamma_j.$ Then the main result asserts that for continuous, not too fast growing function $$f$$, $$\nu_n(f)$$ converges to $$\nu(f)$$. This result implies the almost sure central limit theorem. The rate of convergence is also studied.

### MSC:

 60J60 Diffusion processes 60F15 Strong limit theorems 60F05 Central limit and other weak theorems 60H35 Computational methods for stochastic equations (aspects of stochastic analysis)