Multidimensional Milstein scheme for solving a stochastic model for prebiotic evolution

A multi-dimensional system of stochastic differential equations is presented for modeling prebiotic evolution. The system consists of replication of several reacting species using activated monomers and inactivated residues. The stochastic effects are included in the system as excitations which can be caused by the prebiotic medium where the reactions are taking place. To numerically solve the considered system we consider a generalized Milstein method for multi-dimensional stochastic differential equations. A first-order accuracy is ensured in the approximation of double Itô integrals by using a truncation in the Fourier series expansion. To verify the accuracy of the proposed method we consider a system of two stochastic differential equations with known analytical solution. Numerical results are presented for a reduced prototype system of the prebiotic model.