Efficiency of Multivariate Control Variates in Monte Carlo Simulation

Abstract

This paper considers some statistical aspects of applying control variates to achieve variance reduction in the estimation of a vector of response variables in Monte Carlo simulation. It gives a result that quantifies the loss in variance reduction caused by the estimation of the optimal control matrix. For the one-dimensional case, we derive analytically the optimal size of the vector of control variates under specific assumptions on the covariance matrix. For the multidimensional case, our numerical results show that good variance reduction is achieved when the number of control variates is relatively small (approximately of the same order as the number of unknown parameters). Finally, we give some recommendations for future research.

Chapter 10 A Hilbert Space Approach to Variance Reduction

“…SAN Numerical Example well-known that square integrable martingale differences have a simple interpretation in the Hilbert space framework, which suggests that it might be possible to obtain additional insights when dealing with such techniques. Another area of interest is the Hilbert space formulation of CVs in the multi-response setting, where Y is a random vector; see Rubinstein and Marcus (1985) for relevant results. The combination of importance sampling (cf.…”

“…As to control variates, several publications furnish a broader picture; see, for example, Lavenberg and Welch (1981), Lavenberg et al (1982), Wilson (1984), Rubinstein and Marcus (1985), Venkatraman and Wilson (1986), Law and Kelton (2000), Nelson (1990), Loh (1995), and Glasserman (2004). For additional material on other variance reduction techniques examined here, refer to the items in the References section and to references therein.…”

Vacation Model

“…The method can be easily extended to multiple control variables (Rubinstein and Marcus 1985). A well-known application of control variates is pricing of Asian options.…”

Control variate method for stationary processes

“…Then Assumptions 2.1, 2.2 and 2.3 imply 8) where the summation ν is defined as in (2.4) (see, Brillinger (2001), p48). From Assumptions 2.1 and 2.2, it is seen that the process {Z(t)} becomes a stationary process with nonsingular spectral density matrix (e.g., Brillinger (2001)).…”