On the simulation of expectations of random variables depending on a stopping time

Abstract

Birkhoff's pointwise ergodic theorem with the shift operator over [0,1]¹¹yields a new practical method to compute expectations of functionals of stochastic process. Indeed converges to as Nconverges toward infinity, almost surely. By numerical simulations we will explain the efficiency of this method especially when compared to the classic Monte-Carlo one. It will furthermore be proven that under suitable assumptions a. central limit theorem holds. These assumptions are satisfied in most encountered practical problems. It will precisely be fulfilled when with a stopping time T having a moment of order p, p > 2.Moreover, under this assumptions a “weak” law of iterated logarithm applies. Such that:

Numerical simulations were processed.