General diffusion processes as limit of time-space Markov chains. (English) Zbl 1530.60067
Summary: We prove the convergence of the law of grid-valued random walks, which can be seen as time-space Markov chains, to the law of a general diffusion process. This includes processes with sticky features, reflecting or absorbing boundaries and skew behavior. We prove that the convergence occurs at any rate strictly inferior to \((1 / 4) \wedge(1 / \mathit{p})\) in terms of the maximum cell size of the grid, for any \(p\)-Wasserstein distance. We also show that it is possible to achieve any rate strictly inferior to \((1 / 2) \wedge(2 / \mathit{p})\) if the grid is adapted to the speed measure of the diffusion, which is optimal for \(\mathit{p} \leq 4\). This result allows us to set up asymptotically optimal approximation schemes for general diffusion processes. Last, we experiment numerically on diffusions that exhibit various features.
MSC:
| 60J60 | Diffusion processes |
| 60J10 | Markov chains (discrete-time Markov processes on discrete state spaces) |
| 60G50 | Sums of independent random variables; random walks |
Keywords:
Markov chain approximation; random walk; singular diffusion; sticky; skew; slow reflection; Donsker’s invariance principle; Wasserstein distanceReferences:
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