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            <title xml:lang="en">A generic construction for high order approximation schemes of semigroups using random grids</title>
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            <funder>Aurélien Alfonsi benefited from the support of the “Chaire Risques Financiers”, Fondation du Risque.</funder>
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            <idno type="halRefHtml">&lt;i&gt;Numerische Mathematik&lt;/i&gt;, 2021, &lt;a target="_blank" href="https://dx.doi.org/10.1007/s00211-021-01219-2"&gt;&amp;#x27E8;10.1007/s00211-021-01219-2&amp;#x27E9;&lt;/a&gt;</idno>
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                <term xml:lang="en">Random grids</term>
                <term xml:lang="en">Approximation schemes</term>
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              <p>Our aim is to construct high order approximation schemes for general semigroups of linear operators $P_{t},t\geq 0$. 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 $\alpha &gt;0$. Then, we consider random time grids $\Pi (\omega )=\{t_0(\omega )=0&lt;t_{1}(\omega )&lt;...&lt;t_{m}(\omega )=T\}$ 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 $P_{T}^{\Pi (\omega )}=Q_{l_{n}}...Q_{l_{1}}.$ Our main result is the following: for any approximation order $\nu $, we can construct random grids $\Pi_{i}(\omega )$ and coefficients $c_{i}$, with $i=1,...,r$ such that \[ P_{t}f=\sum_{i=1}^{r}c_{i}\mathbb{E}(P_{t}^{\Pi _{i}(\omega )}f(x))+O(n^{-\nu}) \]% with the expectation concerning the random grids $\Pi _{i}(\omega ).$ Besides, $\text{Card}(\Pi _{i}(\omega ))=O(n)$ 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.</p>
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