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Constructive Quantization and Multilevel Algorithms for Quadrature of SDEs III: Multilevel Monte Carlo Algorithms for L\'evy driven SDEs
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We consider the computational problem of evaluating an expectation $\mathbb{E}[f(Y)]$, where $Y=(Y_t)_{t\in[0,1]}$ is the solution of a stochastic differential equation driven by a square integrable L\'evy process and $f:D[0,1] \to \mathbb{R}$. Here, $D[0,1]$ denotes the Skorohod space of c\`adl\`ag functions. Computational problems of this kind arise, e.g., in computational finance for valuations of path-dependent options.

We introduce multilevel Monte Carlo algorithms for the Euler scheme where the small jumps of the L\'evy process are neglected. Upper bounds for the worst case error of these multilevel algorithms are provided in terms of their computational cost. Here, the worst case is taken over all functionals $f$, that are Lipschitz continuous with Lipschitz constant one with respect to supremum norm. Specifically, if the driving L\'evy process has Blumenthal-Getoor index $\beta$, errors of order $n^{-(\frac 1{\beta\vee 1}-\frac12)}$ are achieved with cost $n$. In the case $\beta>1$, this rate can even be improved by using a Gaussian correction term for the neglected small jumps.

In this talk, we emphasize on the implementation of the multilevel algorithm and we present numerical experiments complementing the theoretical results.