Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations (Q373839)

The authors are concerned with the multi-level Monte-Carlo method based on the Euler-Maruyama scheme for stochastic differential equations NEWLINE\[NEWLINE dX_t = \mu(X_t)dt+\sigma(X_t)dW_t NEWLINE\]NEWLINE with a smooth, at most polynomially growing, globally one-sided Lipschitz continuous drift coefficient \(\mu\) and a globally Lipschitz continuous diffusion coefficient \(\sigma\). They show that the method diverges when used for the approximation of the expectation \(E f(X_T)\), \(T>0\), for a smooth function \(f\) with at most polynomially growing derivatives if the drift coefficient grows superlinearly. The divergence is shown by a simple counter example (\( dX_t = -X_t^5dt \) with random initial condition). The divergence can only be proven for this example and the more general divergence assumption is stated as a conjecture only. This divergence result recalls the previously shown strong and numerically weak divergence of the Euler-Maruyama method under these assumptions case. Used in a classical Monte-Carlo method, however, the Euler-Maruyama method is known to converge in this case. In order to overcome the multi-level Monte-Carlo method's possible divergence in applications, the authors propose to employ a strongly convergent method, e.g., a tamed Euler scheme, in multi-level Monte-Carlo methods. This particular method, then, is shown to converge almost surely and strongly. Moreover, it preserves the improved order of convergence w.r.t.~computational effort of multi-level over the classical Monte-Carlo method. That is, one may conclude that the divergence is not a defect of the multi-level Monte-Carlo approach but of the discretization scheme employed, i.e., the very simple explicit Euler-Maruyama scheme.NEWLINENEWLINEThroughout the paper simulation results are used to illustrate the results.