论文标题
一般扩散过程作为时间空间马尔可夫链的极限
General diffusion processes as the limit of time-space Markov chains
论文作者
论文摘要
我们证明了网格价值随机步行定律的融合，可以看作是一般扩散过程的定律。这包括具有粘性特征的过程，反映或吸收边界和偏斜行为。我们证明，对于任何$ p $ -Wasserstein距离，就电网的最大单元格大小而言，这种收敛性的任何速度都严格较低至$（1/4）\ wedge（1/p）$。我们还表明，如果网格适应扩散的速度度量，则可以实现严格低于$（1/2）\ wedge（2/p）$的任何速率，这对于$ p \ le 4 $是最佳的。这个结果使我们能够为一般扩散过程渐近地设置最佳最佳近似方案。最后，我们在数字上进行了表现出各种特征的扩散。
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 $(1/4) \wedge (1/p)$ 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 $(1/2) \wedge (2/p)$ if the grid is adapted to the speed measure of the diffusion, which is optimal for $p\le 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.
