I'm looking for an efficient algorithm for finding a root of a function which can only be evaluated using Monte Carlo.The function is monotone and has exactly one root in the interval [0, 1].I think there must be clever ways for doing this, eg. some sort of a search algorithm which increases the number of samples as it closes in on the root.Any suggestions for how to do this? Or books which might discuss this kind of problem?

You might try an MC based on a low-discrepancy series like Sobol' series. Then each time you run the sim with the same series, you will get the same result. Your MC-only function is now like a normal (nonstatistical) function, and as you adjust your parameter continuously you will get a continuously changing result. This will allow you to use a standard root-finding routine to zero in and find your root.Or instead of using the more rapidly converging low discrepancy series, you could simply re-use the same (large number of) randoms. This would lead you to a root.Next generate another very large set of randoms (which you will re-use) and do the same thing. This will give you another (hopefully very similar) root.Finally, average these roots.

Thanks erstwhile, I'll try that.