The Euler-Maruyama (EM) scheme looks at least superficially like the most natural discretization of an SDE, just like the Euler-method for ODEs. However, it's strong order of converge is only 1/2. As I recall from (Milstein & Tretyakov, stochastic numerics for mathematical physics), this implies that during a single timestep h, also a deviation of order h occurs. The EM scheme is thus not 'tangential' to the exact solution. Given this situation, is an EM actually unique to a certain SDE, or can different SDEs share the same EM scheme? This confuses me a bit.
As a more practical follow-up, consider a set of two coupled stochastic variables x(t), y(t) and compare the following situations
- y(0)=0, and after every timestep, $y(t+\Delta t)$ is projected back to 0.
- the equation for y(t) is replaced by y(t)=0 at all times
Would these correspond to the same process in the limit $\Delta t\rightarrow0$ as they share the same EM scheme?