Adaptive time-stepping strategies for nonlinear stochastic systems

Abstract

We introduce a class of adaptive timestepping strategies for stochastic differential equations with non-Lipschitz drift coefficients. These strategies work by controlling potential unbounded growth in solutions of a numerical scheme due to the drift. We prove that the Euler-Maruyama scheme with an adaptive timestepping strategy in this class is strongly convergent. Specific strategies falling into this class are presented and demonstrated on a selection of numerical test problems. We observe that this approach is broadly applicable, can provide more dynamically accurate solutions than a drift-tamed scheme with fixed stepsize, and can improve MLMC simulations.

On Cubic Difference Equations with Variable Coefficients and Fading Stochastic Perturbations

“…In this section we illustrate the asymptotic behaviour of solutions of the unperturbed equation (9) with summable and non-summable timestep sequences, as described in Lemmas 8 & 9, and the stochastically perturbed equation (49) with unbounded Gaussian noise as described in Theorem 2. Figure 1, parts (a) and (b) provide three solutions of the unperturbed deterministic equation (9) corresponding to the initial values x 0 = 1.1, 0.5, −1.1, with timestep sequence h n = 1/n 10 , so that ∑ ∞ i=1 h i < ∞. We observe that all three solutions appear to converge to different finite limits, as predicted by Lemma 8.…”

“…Several recent publications are devoted to the use of adaptive timestepping in a explicit Euler-Maruyama discretization of nonlinear equations: for example [3,9,12,11]. In [9] (see also [7]) it was shown that suitably designed adaptive timestepping strategies could be used to ensure strong convergence of order 1/2 for a class of equations with non-globally Lipschitz drift, and globally Lipschitz diffusion. These strategies work by controlling the extent of the nonlinear drift response in discrete time and required that the timesteps depend on solution values.…”

Adaptive timestepping for pathwise stability and positivity of strongly discretised nonlinear stochastic differential equations

“…Nor do we impose maximum and minimum stepsizes h max and h min for the theoretical analysis here. Such bounds are essential to the convergence analysis in [9], but are not required for an analysis of discrete dynamics.…”

“…An analysis of the ability of explicit numerical methods with adaptive timesteps to reproduce the dynamics of solutions of (1) is important because explicit Euler methods of the form (2) with constant stepsize h n ≡ h are known (see [11]) to fail to converge strongly to solutions of (1) if either f or g grows superlinearly, as is the case for (6). Fixed-step taming methods were introduced first in [12] to provide an alternative class of strongly convergent explicit methods for such equations, but may not provide an optimal reproduction of qualitative behaviour: see [21,9]. It was recently shown (see [7,9]) that, for equations with one-sided Lipschitz drift and globally Lipschitz diffusion coefficients, adaptive timestepping strategies can be used to ensure strong convergence of solutions of the explicit Euler method with variable stepsizes, and therefore their effect on the dynamics of solutions is of interest: see [8].…”

On arbitrarily slow convergence rates for strong numerical approximations of Cox–Ingersoll–Ross processes and squared Bessel processes

“…Theorem 1 is an immediate consequence of Theorem 34 in Section 5 below. Upper error bounds for strong approximation of CIR processes and squared Bessel processes, i.e., the opposite question of Theorem 1, have been intensively studied in the literature; see, e.g., Delstra & Delbaan [10], Alfonsi [1], Higham & Mao [22], Berkaoui, Bossy, & Diop [3], Gyöngy & Rásonyi [15], Dereich, Neuenkirch, & Szpruch [11], Alfonsi [2], Hutzenthaler, Jentzen, & Noll [25], Neuenkirch & Szpruch [35], Bossy & Olivero Quinteros [5], Hutzenthaler & Jentzen [26], Chassagneux, Jacquier, & Mihaylov [6], Hefter & Herzwurm [17], and Hefter & Herzwurm [18] (for further approximation results, see, e.g., Milstein & Schoenmakers [33], Cozma & Reisinger [9], and Kelly & Lord [31]). In the following we relate our result to these results.…”