Using antithetic variates to improve the Monte Carlo approximation of $\mathbb E^F[h(X)]$ mean generating correlated realisations from $F$, $X_1,\ldots,X_n$ such that$$\text{var}(h(X_1)+\cdots+h(X_n))<\text{var}(h(X_1))+\cdots+\text{var}(h(X_n))\tag{1}$$ While the idea is appealing, it is difficult to implement in realistically complex settings since establishing the reduction of variance for a given $h$ [or a collection of $h$'s] is challenging.

Assuming such an antithetic scheme (1) has been constructed, if a control variate is available for the model, i.e. a function $h_0(\cdot)$ such that $\mathbb E^F[h_0(X)]=0$ and $\text{corr}(h(X),h_0(X))\ne 0$, the (overall) negative correlation between the $h(X_i)$'s does not automatically transfer to an (overall) negative correlation between the $h(X_i)+\alpha h_0(X_i)$'s. Hence, even if $\alpha$ is chosen such that $$\text{var}(h(X_i)+\alpha h_0(X_i))<\text{var}(h(X_i))\tag{2}$$ it does not necessarily imply that $$\text{var}(\sum_{i=1}^n h(X_i)+\alpha h_0(X_i))<\sum_{i=1}^n \text{var}(h(X_i))$$ because the $h(X_i)+\alpha h_0(X_i)$'s may turn out to be positively correlated.

Using antithetic variates to improve the Monte Carlo approximation of $\mathbb E^F[h(X)]$ mean generating correlated realisations from $F$, $X_1,\ldots,X_n$ such that$$\text{var}(h(X_1)+\cdots+h(X_n))<\text{var}(h(X_1))+\cdots+\text{var}(h(X_n))\tag{1}$$ While the idea is appealing, it is difficult to implement in realistically complex settings since establishing the reduction of variance for a given $h$ [or a collection of $h$'s] is challenging.

Assuming such an antithetic scheme (1) has been constructed, if a control variate is available for the model, i.e. a function $h_0(\cdot)$ such that $\mathbb E^F[h_0(X)]=0$ and $\text{corr}(h(X),h_0(X))\ne 0$, the (overall) negative correlation between the $h(X_i)$'s does not automatically transfer to an (overall) negative correlation between the $h(X_i)+\alpha h_0(X_i)$'s. Hence, even if $\alpha$ is chosen such that $$\text{var}(h(X_i)+\alpha h_0(X_i))<\text{var}(h(X_i))\tag{2}$$ it does not necessarily imply that $$\text{var}\left\{\sum_{i=1}^n h(X_i)+\alpha h_0(X_i)\right\}<\sum_{i=1}^n \text{var}(h(X_i))$$ because the $h(X_i)+\alpha h_0(X_i)$'s may turn out to be positively correlated.

Using antithetic variates to improve the Monte Carlo approximation of $\mathbb E^F[h(X)]$ mean generating correlated realisations from $F$, $X_1,\ldots,X_n$ such that$$\text{var}(h(X_1)+\cdots+h(X_n))<\text{var}(h(X_1))+\cdots+\text{var}(h(X_n))\tag{1}$$ While the idea is appealing, it is difficult to implement in realistic complex settings since establishing the reduction of variance for a given $h$ [or a collection of $h$'s] is challenging.

Assuming such an antithetic scheme (1) has been constructed, if a control variate is available for the model, i.e. a function $h_0(\cdot)$ such that $\mathbb E^F[h_0(X)]=0$ and $\text{corr}(h(X),h_0(X))\ne 0$, the (overall) negative correlation between the $h(X_i)$'s does not automatically transfer to an (overall) negative correlation between the $h(X_i)+\alpha h_0(X_i)$'s. Hence, even if $\alpha$ is chosen such that $$\text{var}(h(X_i)+\alpha h_0(X_i))<\text{var}(h(X_i))\tag{2}$$ it does not necessarily imply that $$\text{var}(\sum_{i=1}^n h(X_i)+\alpha h_0(X_i))<\sum_{i=1}^n \text{var}(h(X_i))$$ because the $h(X_i)+\alpha h_0(X_i)$'s may turn out to be positively correlated.

Using antithetic variates to improve the Monte Carlo approximation of $\mathbb E^F[h(X)]$ mean generating correlated realisations from $F$, $X_1,\ldots,X_n$ such that$$\text{var}(h(X_1)+\cdots+h(X_n))<\text{var}(h(X_1))+\cdots+\text{var}(h(X_n))\tag{1}$$ While the idea is appealing, it is difficult to implement in realistically complex settings since establishing the reduction of variance for a given $h$ [or a collection of $h$'s] is challenging.

Assuming such an antithetic scheme (1) has been constructed, if a control variate is available for the model, i.e. a function $h_0(\cdot)$ such that $\mathbb E^F[h_0(X)]=0$ and $\text{corr}(h(X),h_0(X))\ne 0$, the (overall) negative correlation between the $h(X_i)$'s does not automatically transfer to an (overall) negative correlation between the $h(X_i)+\alpha h_0(X_i)$'s. Hence, even if $\alpha$ is chosen such that $$\text{var}(h(X_i)+\alpha h_0(X_i))<\text{var}(h(X_i))\tag{2}$$ it does not necessarily imply that $$\text{var}(\sum_{i=1}^n h(X_i)+\alpha h_0(X_i))<\sum_{i=1}^n \text{var}(h(X_i))$$ because the $h(X_i)+\alpha h_0(X_i)$'s may turn out to be positively correlated.