Optimal approximation of stochastic differential equations by adaptive step-size control

Abstract

Abstract. We study the pathwise (strong) approximation of scalar stochastic differential equations with respect to the global error in the L 2 -norm. For equations with additive noise we establish a sharp lower error bound in the class of arbitrary methods that use a fixed number of observations of the driving Brownian motion. As a consequence, higher order methods do not exist if the global error is analyzed. We introduce an adaptive step-size control for the Euler scheme which performs asymptotically optimally. In particular, the new method is more efficient than an equidistant discretization. This superiority is confirmed in simulation experiments for equations with additive noise, as well as for general scalar equations.

Convergence Rates for Adaptive Weak Approximation of Stochastic Differential Equations

“…Becker and Rannacher [5,6], Eriksson et al [15], Johnson and Szepessy [22], and Moon et al [30], but the theoretical understanding of convergence rates of adaptive algorithms is not as well developed; there are, however, recent important contributions. The work of Hofmann et al [19,20] and Müller-Gronbach [34] prove optimal convergence rates for strong approximation of stochastic differential equations. DeVore studies in [12] the efficiency of adaptive approximation of functions, including wavelet expansions, based on smoothness conditions in Besov spaces.…”

Construction of a Mean Square Error Adaptive Euler–Maruyama Method With Applications in Multilevel Monte Carlo

“…For example in [17], where the step-size depends on the size of the diffusion coefficient for a MSE Euler-Maruyama adaptive algorithm; in [23], the step-size is controlled by the variation in the size of the drift coefficient in the constructed EulerMaruyama adaptive algorithm, which preserves the long-term ergodic behavior of the true solution for many SDE problems; and in [19], a local error based adaptive Milstein algorithm is developed for solving multi-dimensional chemical Langevin equations.…”

A Variable Step Size Riemannian Sum for an Itô Integral

“…This includes, but is not restricted to, the condition that, for any n and i, t (n) i+1 must be F (t (n) i )-adapted (which requires that the length of each interval would need to be fully determined by the starting time of that step), described in [11, p. 321]. Indeed, there are variable step size numerical schemes for SDEs which implement nonrandom step size selection (see, for instance, [18]) or nonanticipating random-step selection (see, for instance, [8] or [9]). More generally, we may use spatial discretisation schemes such as given in [1] or [16], since first hitting times for a continuous diffusion process are stopping times.…”