Numerical conservation issues for stochastic differential equations (Raffaele D'Ambrosio, University of L'Aquila)

05.10.2021 14:00

This seminar is devoted to sharing recent advances in the numerical preservation of invariant laws characterizing the underlying dynamics of stochastic differential problems. We first focus on stochastic Hamiltonian problems, with the aim to provide estimates on the long-term energy conservation via suitable energy preserving schemes [1,5]. We finally consider the nonlinear stability analysis of stochastic theta-methods with respect to mean-square dissipative nonlinear test problems, generating a mean-square contractive behaviour [6]. The pursued aim is that of making the same property visible also alog the numerical discretization via stochastic theta-methods: this issue is translated into sharp stepsize restrictions depending on some parameters of the problem, accurately estimated [2,3,4]. A selection of numerical tests confirming the effectiveness of the analysis and its sharpness is also provided. The talk is based on a series of joint papers with Stefano Di Giovacchino (University of L'Aquila).

References
[1] C. Chen, D. Cohen, R. D’Ambrosio, A. Lang, Drift-preserving numerical inte-grators for stochastic Hamiltonian systems, Adv. Comput. Math. 46, article number 27 (2020).
[2] R. D’Ambrosio, Numerical approximation of differential problems, Springer (to appear).
[3] R. D’Ambrosio, S. Di Giovacchino, Mean-square contractivity of stochastic theta-methods, Comm. Nonlin. Sci. Numer. Simul. 96, article number 105671 (2021).
[4] R. D’Ambrosio, S. Di Giovacchino, Nonlinear stability issues for stochastic Runge-Kutta methods, Comm. Nonlin. Sci. Numer. Simul. 94, article number 105549 (2021).
[5] R. D'Ambrosio, S. Di Giovacchino, Long-term analysis of stochastic Hamiltonian systems under time discretizations.
[6] D.J. Higham, P.E. Kloeden, Numerical methods for nonlinear stochastic differential equations with jumps. Numer. Math. 101(1), 101–119 (2005).

Lieu

Bâtiment: Conseil Général 7-9

Room 1-05, Séminaire d'analyse numérique

entrée libre