Adaptive Euler methods for stochastic systems with non-globally Lipschitz coefficients

Abstract

We present strongly convergent explicit and semi-implicit adaptive numerical schemes for systems of semi-linear stochastic differential equations (SDEs) where both the drift and diffusion are not globally Lipschitz continuous. Numerical instability may arise either from the stiffness of the linear operator or from the perturbation of the nonlinear drift under discretization, or both. Typical applications arise from the space discretization of an SPDE, stochastic volatility models in finance, or certain ecological models. Under conditions that include montonicity, we prove that a timestepping strategy which adapts the stepsize based on the drift alone is sufficient to control growth and to obtain strong convergence with polynomial order. The order of strong convergence of our scheme is (1 − ε)/2, for ε ∈ (0,1), where ε becomes arbitrarily small as the number of finite moments available for solutions of the SDE increases. Numerically, we compare the adaptive semi-implicit method to a fully drift-implicit method and to three other explicit methods. Our numerical results show that overall the adaptive semi-implicit method is robust, efficient, and well suited as a general purpose solver.

A splitting method for SDEs with locally Lipschitz drift: Illustration on the FitzHugh-Nagumo model

“…The DTEM method yields the largest RMSEs, and for that method we only observe a convergence of order 1/2. The latter finding is in agreement with the observations in [30]. All RMSEs are also reported in Table 1.…”

“…Taming/truncating perturbations do not improve this behaviour. Even worse, taming perturbations may also lead to a nonpreservation of frequencies of oscillations [29,30]. This lack of amplitude and frequency preservation is confirmed by our numerical experiments on the FHN model.…”

“…Moreover, for some values of X 0 , we observe that the DTrEM method (24) may produce spurious oscillations (figures not shown). See [30,62], where such a behaviour has also been observed.…”

“…In contrast, both tamed Euler-Maruyama methods underestimate the frequency and overestimate the amplitude of the neuronal oscillations as ∆ increases, for both values of γ under consideration. For similar observations regarding tamed methods, we refer to [29,30]. Note also that their paths already start deviating from the reference paths for ∆ = 2 • 10 −3 , performing thus worse than the truncated Euler-Maruyama methods.…”

“…guaranteeing Condition (5). We refer to [5,7,30,48] and to [17,36] for the consideration of the elliptic and hypoelliptic FHN model, respectively, and to [13] for an investigation of both cases. Moreover, it has been proved that the FHN model is ergodic, see, e.g., [7,36].…”