Testing <PERSON> process splitting in adaptive-step SDE integrators
I am investigating various methods for adaptive-step integration of stochastic differential equations and trying to implement them. All of the papers that I've seen (e.g. H. <PERSON>, J. Comp. App. Math. 161(2), 417–430 (2003)) mention in some form that one has to always preserve the sampled values of the <PERSON> process. In other words, if some steps were rejected, one still has to go through all the sampled values that are located later in time.
For example, suppose that you are at time $t_0$ and tried to step to $t_1$, sampling the <PERSON> increment $dW_1$ for that. The step turned out to have an unacceptable error and was rejected. 346
Now you try to make a step to $t_2 < t_1$. The proper procedure, it appears, is to sample an intermediate increment $dW_2$ for the step from $t_0$ to $t_2$ based on $dW_1$ (the exact formula is not important here), and take the increment for the step from $t_2$ to $t_1$ to be $dW_1 - dW_2$.
Now suppose that this step was rejected as well. Repeating the procedure above, you now have three increments, $dW_3$, $dW_2 - dW_3$ and $dW_1 - dW_2$ for the steps from $t_0$ to $t_3$, then to $t_2$ and then to $t_1$.
This part I have not seen explained, although it seems that all the authors imply that if your step to $t_3$ succeeds, your next step must not go past $t_2$ and should use the sampled increment $dW_2$ for that part. If you follow that logic, in the implementation of the integrator you will have to maintain a dynamic list of sampled increments, which increases the memory footprint (in an unpredictable way, too). This supposedly avoids bias in the sampled <PERSON> process. Let's call it approach A.
I tried to replace it with a simpler logic, where only the furthest sample in time ($dW_1$) is preserved. This way, in the situation above, after you sample $dW_3$, you forget $dW_2$, and for your next step the sampled increment only depends on $dW_1$. Let's call it approach B.
Sorry for the long explanation, now to my question. Supposedly my simplified logic should introduce bias in the solutions. The problem is that I haven't been able to detect it in any of the tests that I tried. Do you have any ideas of how to design a test that will be able to distinguish approach A from approach B? 172