No abstract

In this paper, we implement a weak Milstein Scheme to simulate low-dimensional stochastic differential equations (SDEs). We prove that combining the antithetic multilevel Monte-Carlo (MLMC) estimator introduced by Giles and Szpruch with the MLMC approach for weak SDE approximation methods by Belomestny and Nagapetyan, we can achieve a quadratic computational complexity in the inverse of the Root Mean Square Error (RMSE) when estimating expected values of smooth functionals of SDE solutions, without simulating Lévy areas and without requiring any strong convergence of the underlying SDE approximation method. By using appropriate discrete variables this approach allows us to calculate the expectation on the coarsest level of resolution by enumeration, which, for low-dimensional problems, results in a reduced computational effort compared to standard MLMC sampling. These theoretical results are also confirmed by a numerical experiment.

The lac promoter is one of the most commonly used promoters for expression control of recombinant genes in E. coli. In the absence of galactosides, the lac promoter is repressed by its repressor protein LacI. Since the lac promoter is regulated by a repressor, overexpression of LacI is necessary for regulation when the promoter is introduced on a high-copy plasmid. For that purpose, a modified variant of LacI, a LVA-tagged LacI, was submitted to the Registry of Standard Biological Parts and has been used for more than 500 constructs since then. We have found, however, that natural LacI is superior to the LVA-tagged LacI as controller of expression.

No abstract

In this paper, we consider a class of stochastic midpoint and trapezoidal Lawson schemes for the numerical discretization of highly oscillatory stochastic differential equations. These Lawson schemes incorporate both the linear drift and diffusion terms in the exponential operator. We prove that the midpoint Lawson schemes preserve quadratic invariants and discuss this property as well for the trapezoidal Lawson scheme. Numerical experiments demonstrate that the integration error for highly oscillatory problems is smaller than that of some standard methods.