In a simulation one can often identify a random variable, Y, that is likely to be highly correlated with a random variable of interest, X. If μY=E(Y) is known then Y can be used as a control variate to estimate μX=E(X) more efficiently than by a direct simulation of X. We study the asymptotic properties of a method that uses Y to potentially speed up the simulation when μY is not known. The method is effective when μY can be efficiently estimated in an auxiliary simulation that does not involve X. We call Y a quasi control variate (QCV). For a simulation of length t>0 time units, we invest pt units estimating μY with the auxiliary simulation, yielding &Ymacr;pt. The remaining qt=(1–p)t units are spent on the main simulation yielding estimates (&Xtilde;qt, &Ytilde;qt) for (μX,μY). The two simulations can be interleaved so they are effectively done simultaneously. For each p ∈ (0, 1) and α ∈ ℜ we have a QCV estimator for μX, &Xcrcmflx;t(p, α)=&Xtilde;qt+α(&Ytilde;qt–&Ytilde;pt), t>0. We find p and α that minimize the asymptotic variance parameter (AVP) of &Xcirc;t(p, α) in terms of statistics that are estimated during the simulations, and then describe an easily implemented adaptive procedure that achieves the minimum AVP. The adaptive procedure evolves into the optimal QCV procedure if it is more efficient than a direct simulation, &Xmacr;t→μX; otherwise it evolves into the direct simulation. Applications in stochastic linear programming, stochastic partial differential equations (PDEs) and queuing theory are cited.
