In many sources of literature (e.g., [1]), the portfolio variance is defined as follows: $$\sigma_p^2 = w^\top \Sigma w,$$ where $\Sigma$ is the covariance matrix of portfolio returns and $w$ is a vector of weights of assets calculated by scaling dollar positions $W_i$ by the total initial investment, that is $$w_i = \frac{W_i}{\sum_i^N W_i},$$ where $N$ is the number of assets.
But dollar positions change over time, and it is not obvious what weights should be taken. For example, there were 2 stocks in 80\$ and 20\$ dollar positions on the first trading day with weights $w_1 = 0.8$ and $w_2 = 0.2$, respectively. On the second trading day, the stocks rose in price and cost 90\$ and 25\$, the weights became $w_1 \approx 0.78$, $w_2 \approx 0.22$. The choice of weights on a particular trading day may lead to inadequate variance if dollar positions change significantly during the observation. It seems the weights can be represented as a matrix ($N\times T$), where $T$ is the number of days, but I cannot find any references to such definitions.
[1] Jorion, Philippe. Value at risk: the new benchmark for managing financial risk. The McGraw-Hill Companies, Inc., 2007.