Numerical preservation of stochastic dissipativity

Helena Biščević (Gran Sasso Science Institute), Raffaele D’Ambrosio, Stefano Di Giovacchino

Standard numerical analysis for stochastic differential equations has a clear understanding of stability in the linear case or when the drift coefficient satisfies a one-sided Lipschitz condition and the diffusion term is globally Lipschitz. By looking at many applications, it is obvious that we need a deeper mathematical and numerical insight into stability of problems with non-global Lipschitz coefficients.

This talk is aimed to analyze nonlinear stability properties of \(\theta\)-methods for stochastic differential equations under non-global Lipschitz conditions on the coefficients. In particular, the concept of exponential mean-square contractivity is introduced for the exact dynamics; additionally, stepsize restrictions in order to inherit the contractive behaviour over the discretized dynamics are also given. A selection of numerical tests confirming the theoretical expectations is also presented.

Moreover, we will briefly tackle current work concerning numerical dissipativity for stochastic partial differential equations.

References

  1. H. Biscevic, R. D’Ambrosio, S. Di Giovacchino, Contractivity of stochastic \(\theta\)-methods under non-global Lipschitz conditions, submitted for publication.

  2. R. D’Ambrosio, Numerical approximation of ordinary differential problems - From deterministic to stochastic numerical methods, Springer (2023).

  3. R. D’Ambrosio, S. Di Giovacchino, Mean-square contractivity of stochastic theta-methods, Comm. Nonlin. Sci. Numer. Simul. 96, 105671 (2021).

  4. W. E, D. Liu, Gibbsian dynamics and invariant measures for stochastic dissipative PDEs, J. Stat. Phys. 108(5–6), 1125–1156 (2002).

  5. D.J. Higham, X. Mao, A.M. Stuart, Exponential mean-square stability of numerical solutions to stochastic differential equations, LMS J. Comput. Math. 6, 297–313 (2003).

  6. M. Hutzenthaler, A. Jentzen, Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients, Mem. Amer. Math. Soc. 236, 1112 (2015).

  7. X. Mao, Exponential stability of stochastic differential equations, CRC Press (1994).

  8. M.V. Tretyakov, Z. Zhang, A Fundamental Mean-Square Convergence Theorem for SDEs with Locally Lipschitz Coefficients and Its Applications, SIAM J. Numer. Anal. 51(6), 3135–3162 (2013).

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