We denote by [$]F(t_i, t_j)[$] the forward price for delivery day [$]t_j[$] observed on date [$]t_i[$].
    $$F(t_i, t_j)=F(0, t_j)\exp\left(-\frac{1}{2}\sigma(t_i,t_j)^2t_i+\sigma(t_i,t_j)W(t_i,m)\right)$$
    where
    - [$]m[$] is the month containing period [$]t_j[$]
    -[$]\sigma(t_i, t_j)[$] is the volatility for period [$]t_j[$] and expiry date [$]t_i[$]. It depends on the at-the-money monthly volatility [$]\sigma(t_i, m)[$] for expiry on [$]t_i[$] and on the forward volatility [$]\sigma_{fwd}(m)[$]
    $$\sigma(t_i, t_j) =\sqrt[]{\frac{\sigma(t_i(m),m)^2 t_i(m)+\sigma_{fwd}(m)^2 \Delta t_i}{t_i(m)+\Delta t_i}}$$
    where [$]t_i(m)[$] is the last day of the month preceding [$]m[$] and [$]\Delta t_i[$] is a time constant representing 15 days. By convention, [$]\sigma(t_i,m)[$] is the standard expiry volatility when [$]t_i[$] is past the standard expiry date.
 - [$]W(.,m)[$] is a standard Brownian motion. We denote by [$]\rho(m_1;m_2)[$] the time spread correlation between the standard Brownian motions [$]W(.,m_1)[$] and [$]W(.,m_2)[$] associated to months [$]m_1[$] and [$]m_2[$].

Given the previous process for my forward price, when using MCS for my prices, how is the correlation between the Brownian motions taking into account ?
I do know that [$]W^1_t=\rho W^2_t +\sqrt{1-\rho^2}Z_t[$] if [$]W^1_t[$] and [$]W^1_t[$] have correlation [$]\rho[$] and are independent of [$]Z_t[$], but then how is it accounted for in the MCS?