A generic construction for high order approximation schemes of semigroups using random grids
Abstract
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 $h_{l}=\frac{T}{n^{l}},l\in N$ 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 $\Pi (\omega )=\{t_0(\omega )=0<t_{1}(\omega
)<...<t_{m}(\omega )=T\}$ such that for all $1\le k\le m$, $t_{k}(\omega
)-t_{k-1}(\omega )=h_{l_{k}}$ for some $l_{k}\in N$, 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 $P_{t}f=\sum_{i=1}^{r}c_{i} E(P_{t}^{\Pi _{i}(\omega )}f(x))+O(n^{-\nu})$
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.