The law of total variance states that for a random variable $X$ defined in $(\Omega , \mathbb{P}, \mathcal{F})$ with $\mathcal{F}=(\mathcal{F}_t)_{t \in [0,T]}$ $$Var(X) = E[Var(X|\mathcal{F}_t)] + Var(E[X|\mathcal{F}_t])$$ Suppose we have two equivalent measures $\mathbb{P}_1 \sim \mathbb{P}_2$. I was wondering what can we say about the following: $$Var^{\mathbb{P}_1}(X) = E^{\mathbb{P}_1}[Var^{\mathbb{P}_2}(X|\mathcal{F}_t)] + Var^{\mathbb{P}_1}(E^{\mathbb{P}_2}[X|\mathcal{F}_t]) + k$$ where $k$ is a correction term. In particular:
- Can we say whether $k \gtreqqless 0$ ?
- Can we bound $k$?
EDIT: suppose we also know that $E^{\mathbb{P}_1}[X]>E^{\mathbb{P}_2}[X]$ and $E^{\mathbb{P}_1}[X|\mathcal{F}_t]>E^{\mathbb{P}_2}[X|\mathcal{F}_t]$