TY - JOUR
T1 - A splitting method for SDEs with locally Lipschitz drift
T2 - Illustration on the FitzHugh-Nagumo model
AU - Buckwar, Evelyn
AU - Samson, Adeline
AU - Tamborrino, Massimiliano
AU - Tubikanec, Irene
PY - 2022
Y1 - 2022
N2 - In this article, we construct and analyse an explicit numerical splitting method for a class of semi-linear stochastic differential equations (SDEs) with additive noise, where the drift is allowed to grow polynomially and satisfies a global one-sided Lipschitz condition. The method is proved to be mean-square convergent of order 1 and to preserve important structural properties of the SDE. First, it is hypoelliptic in every iteration step. Second, it is geometrically ergodic and has an asymptotically bounded second moment. Third, it preserves oscillatory dynamics, such as amplitudes, frequencies and phases of oscillations, even for large time steps. Our results are illustrated on the stochastic FitzHugh-Nagumo model and compared with known mean-square convergent tamed/truncated variants of the Euler-Maruyama method. The capability of the proposed splitting method to preserve the aforementioned properties may make it applicable within different statistical inference procedures. In contrast, known Euler-Maruyama type methods commonly fail in preserving such properties, yielding ill-conditioned likelihood-based estimation tools or computationally infeasible simulation-based inference algorithms.
AB - In this article, we construct and analyse an explicit numerical splitting method for a class of semi-linear stochastic differential equations (SDEs) with additive noise, where the drift is allowed to grow polynomially and satisfies a global one-sided Lipschitz condition. The method is proved to be mean-square convergent of order 1 and to preserve important structural properties of the SDE. First, it is hypoelliptic in every iteration step. Second, it is geometrically ergodic and has an asymptotically bounded second moment. Third, it preserves oscillatory dynamics, such as amplitudes, frequencies and phases of oscillations, even for large time steps. Our results are illustrated on the stochastic FitzHugh-Nagumo model and compared with known mean-square convergent tamed/truncated variants of the Euler-Maruyama method. The capability of the proposed splitting method to preserve the aforementioned properties may make it applicable within different statistical inference procedures. In contrast, known Euler-Maruyama type methods commonly fail in preserving such properties, yielding ill-conditioned likelihood-based estimation tools or computationally infeasible simulation-based inference algorithms.
KW - Ergodicity
KW - FitzHugh-Nagumo model
KW - Hypoellipticity
KW - Locally Lipschitz drift
KW - Mean-square convergence
KW - Splitting methods
KW - Stochastic differential equations
U2 - 10.1016/j.apnum.2022.04.018
DO - 10.1016/j.apnum.2022.04.018
M3 - Article
AN - SCOPUS:85129915562
SN - 0168-9274
VL - 179
SP - 191
EP - 220
JO - Applied Numerical Mathematics
JF - Applied Numerical Mathematics
ER -