Suppose I am given a probability distribution only via its characteristic or moment generating function and I want to sample from that distribution to generate paths in a Monte Carlo simulation. Is there a way to sample from the distribution other than numerically inverting the transform and then using conventional methods like the inversion method or acceptance rejection? In my specific case I want to sample paths of the integrated CIR process for which the Laplace transform of the transition probabilities is known. Since the transition probabilities of the CIR processs are knwon explicitely I could approximate the transition probabilities of the integrated CIR process by using the Trapez rule for example. But what do I do if a want an unbiased sample?
Some time ago I worked on something similar. If you are still interested in an implementation of the Devoye (1981) method you can give a look here https://www.kent.ac.uk/smsas/personal/msr/webfiles/rlaptrans/rdevroye.r This is the code by Martin Ridout.
Prof. Ridout also wrote a really interesting paper on the topic (I advice you to give a look at it, https://www.kent.ac.uk/smsas/personal/msr/webfiles/rlaptrans/SimRandom4.pdf).
Finally, there is a more recent paper by SG Walker (https://link.springer.com/article/10.1007/s11222-016-9631-8)
Hope my answer helps.