Our aim is to construct high order approximation schemes for general semigroups of linear operators \(P_{t},t\ge 0\). In order to do it, we fix a time horizon T and the discretization steps \(h_{l}=\frac{T}{n^{l}},l\in {\mathbb {N}}\) and we suppose that we have at hand some short time approximation operators \(Q_{l}\) such that \(P_{h_{l}}=Q_{l}+O(h_{l}^{1+\alpha })\) for some \(\alpha >0\). Then, we consider random time grids \(\Pi (\omega )=\{t_0(\omega )=0<t_{1}(\omega )<\cdots <t_{m}(\omega )=T\}\) such that for all \(1\le k\le m\), \(t_{k}(\omega )-t_{k-1}(\omega )=h_{l_{k}}\) for some \(l_{k}\in {\mathbb {N}}\), and we associate the approximation discrete semigroup \(P_{T}^{\Pi (\omega )}=Q_{l_{n}}\ldots Q_{l_{1}}\). Our main result is the following: for any approximation order \(\nu \), we can construct random grids \(\Pi _{i}(\omega )\) and coefficients \(c_{i}\), with \(i=1,\ldots ,r\) such that

with the expectation concerning the random grids \(\Pi _{i}(\omega ).\) Besides, \(Card (\Pi _{i}(\omega ))=O(n)\) and the complexity of the algorithm is of order n, for any order of approximation \(\nu \). The standard example concerns diffusion processes, using the Euler approximation for \(Q_l\). In this particular case and under suitable conditions, we are able to gather the terms in order to produce an estimator of \(P_tf\) with finite variance. However, an important feature of our approach is its universality in the sense that it works for every general semigroup \(P_{t}\) and approximations. Besides, approximation schemes sharing the same \(\alpha \) lead to the same random grids \(\Pi _{i}\) and coefficients \(c_{i}\). Numerical illustrations are given for ordinary differential equations, piecewise deterministic Markov processes and diffusions.

我们的目标是为线性算子\(P_{t},t\ge 0\) 的一般半群构建高阶近似方案。为了做到这一点，我们修复了时间范围T和离散化步骤\(h_{l}=\frac{T}{n^{l}},l\in {\mathbb {N}}\)并且我们假设我们手头有一些短时间近似算子\(Q_{l}\)使得\(P_{h_{l}}=Q_{l}+O(h_{l}^{1+\alpha }) \)对于某些\(\alpha >0\)。然后，我们考虑随机时间网格\(\Pi (\omega )=\{t_0(\omega )=0<t_{1}(\omega )<\cdots <t_{m}(\omega )=T\} \)使得对于所有\(1\le k\le m\)，\(t_{k}(\omega )-t_{k-1}(\omega )=h_{l_{k}}\)为一些\(l_{k}\in {\mathbb {N}}\)，我们关联近似离散半群\(P_{T}^{\Pi (\omega )}=Q_{l_{n}}\ldots Q_{l_{1}}\) . 我们的主要结果如下：对于任何近似阶\(\nu \)，我们可以构造随机网格\(\Pi _{i}(\omega )\)和系数\(c_{i}\)，其中\ (i=1,\ldots ,r\)使得

与关于随机网格的期望\(\Pi _{i}(\omega ).\)此外，\(Card (\Pi _{i}(\omega ))=O(n)\)和该算法的阶数为n，对于任何近似阶数\(\nu \)。标准示例涉及扩散过程，使用\(Q_l\)的欧拉近似 。在这种特殊情况下，在合适的条件下，我们能够收集项以产生具有有限方差的\(P_tf\)估计量。然而，我们的方法的一个重要特征是它的普遍性，因为它适用于每个一般半群\(P_{t}\)和近似值。此外，近似方案共享相同的\(\alpha \)导致相同的随机网格\(\Pi _{i}\)和系数\(c_{i}\)。给出了常微分方程、分段确定性马尔可夫过程和扩散的数值说明。