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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)
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