Structure-preserving stochastic numerical methods

In this talk we aim to analyze conservation properties of numerical methods for stochastic differential equations (SDEs) that exhibit a-priori known qualitative behaviors. Such properties are sometimes hidden behind conditional stability issues of the numerical methods and do not have to be artificially conveyed in the numerical scheme. We first consider stochastic Hamiltonian problems of Ito type , for which a linear drift of the expected energy is visible along the exact dynamics (differently from the deterministic case, for which the energy is preserved). We aim to study the behaviour of stochastic Runge-Kutta methods in order to understand their natural ability in retaining the same energy behaviour also along the numerical solutions. The analysis is driven through ε-expansion of the solutions, where ε is the amplitude of the stochastic fluctuation. Then, we analyze the numerical approximation of nonlinear SDEs of Ito type with exponential mean-square contractive solutions. We aim to understand if such an inequality still holds true when the solutions are computed by stochastic linear multistep methods, in order to provide stepsize restrictions ensuring analogous exponential mean-square properties also numerically, without adding further constraints on the numerical method itself. The talk deals with a selection of results related to joint research with Evelyn Buckwar (“Johannes Kepler” University of Linz), and Beatrice Paternoster (University of Salerno). [1] Burrage, P.M.; Burrage, K. Structure-preserving Runge-Kutta methods for stochastic Hamiltonian equations with additive noise. Numer. Algor. 65(3), 519–532 (2014). [2] Burrage, P.M.; Burrage, K. Low rank RungeKutta methods, symplecticity and stochastic Hamilto- nian problems with additive noise, J. Comput. Appl. Math. 236(16), 3920–3930 (2012). [3] Higham D.J.; Kloeden, P.E. Numerical methods for nonlinear stochastic differential equations with jumps, Numer. Math. 101(1), 101–119 (2005).