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.


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