For a stationary Markov process the detailed balance condition is equivalent to the
time-reversibility of the process. For stochastic differential equations (SDE’s), the time
discretization of numerical schemes usually destroys the time-reversibility property.
Despite an extensive literature on the numerical analysis for SDE’s, their stability
properties, strong and/or weak error estimates, large deviations and infinite-time
estimates, no quantitative results are known on the lack of reversibility of discrete-time
approximation processes. In this paper we provide such quantitative estimates by using the
concept of entropy production rate, inspired by ideas from non-equilibrium statistical
mechanics. The entropy production rate for a stochastic process is defined as the relative
entropy (per unit time) of the path measure of the process with respect to the path
measure of the time-reversed process. By construction the entropy production rate is
nonnegative and it vanishes if and only if the process is reversible. Crucially, from a
numerical point of view, the entropy production rate is an a posteriori
quantity, hence it can be computed in the course of a simulation as the ergodic
average of a certain functional of the process (the so-called Gallavotti−Cohen (GC) action functional). We compute
the entropy production for various numerical schemes such as explicit Euler−Maruyama and explicit Milstein’s for
reversible SDEs with additive or multiplicative noise. In addition we analyze the entropy
production for the BBK integrator for the Langevin equation. The order (in the
time-discretization step Δt) of the entropy production rate provides a tool to
classify numerical schemes in terms of their (discretization-induced) irreversibility. Our
results show that the type of the noise critically affects the behavior of the entropy
production rate. As a striking example of our results we show that the Euler scheme for
multiplicative noise is not an adequate scheme from a reversibility
point of view since its entropy production rate does not decrease with
Δt.