  For a Markov process the detailed balance condition is equivalent to the
time-reversibility of the process. For stochastic differential equations
(SDE's) time discretization numerical schemes usually destroy the property of
time-reversibility. 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 the discrete-time approximation process. 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 {\em 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. Additionally, we analyze the entropy production for the
BBK integrator of the Langevin processes. We show that entropy production is an
observable that distinguishes between different numerical schemes in terms of
their discretization-induced irreversibility. Furthermore, our results show
that the type of the noise critically affects the behavior of the entropy
production rate.
