Variance reduction by antithetic random numbers of Monte Carlo methods for unrestricted and reflecting diffusions

The main discretization schemes for diffusion processes, both unrestricted and reflecting in a hyper-rectangle, are considered. For every discretized path, an `antithetic' path is obtained by changing the sign of the driving random variables, which are chosen symmetric. It is shown that, under suitable monotonicity assumptions on the coefficients and boundary data, the mean of a sample of values of a monotone functional evaluated on M independent discretized paths and on the M corresponding antithetic paths has a smaller variance than the mean of a sample of values of the same functional evaluated on 2M independent paths. An example, obtained by reflecting the diffusion process of the well-known Black and Scholes model of finance, is discussed. The results of some numerical tests are also presented.