Consider two Girsanov measures $\mu_1$ and $\mu_2$ corresponding to drifts $F_1(t)$ and $F_2(t)$ respectively. By this, I mean that we have that $B(t)\sim F_1(t)+\tilde B(t)$ where $\tilde B(t)$ is a Brownian motion under $\mu_1$. Similarly for $\mu_2$.

For $\lambda \in [0,1]$ we can consider the probability measure $\mu=\lambda \mu_1+(1-\lambda) \mu_2$. $\mu$ is also a Girsanov measure so it corresponds to a drift $F(t)$. What is $F$ in terms of $F_1,F_2$?

I know if $F_1, F_2, F$ are all deterministic then $$F(t)=E_\mu[B(t)]=\lambda F_1(t)+(1-\lambda)F_2(t)$$.

What about in general?

Consider two Girsanov measures $\mu_1$ and $\mu_2$ corresponding to drifts $F_1(t)$ and $F_2(t)$ respectively. By this, I mean that we have that $B(t)\sim F_1(t)+\tilde B(t)$ where $\tilde B(t)$ is a Brownian motion under $\mu_1$. Similarly for $\mu_2$.

For $\lambda \in [0,1]$ we can consider the probability measure $\mu=\lambda \mu_1+(1-\lambda) \mu_2$. $\mu$ is also a Girsanov measure so it corresponds to a drift $F(t)$. What is $F$ in terms of $F_1,F_2$?

I know if $F_1, F_2, F$ are all deterministic then $$F(t)=E_\mu[B(t)]=\lambda F_1(t)+(1-\lambda)F_2(t)$$.

What about in general?

Even in the case where $F_1,F_2$ are deterministic can we say that $F$ is? This itself is pretty tricky.

Consider two Girsanov measures $\mu_1$ and $\mu_2$ corresponding to drifts $F_1(t)$ and $F_2(t)$ respectively. For $\lambda \in [0,1]$ we can consider the probability measure $\mu=\lambda \mu_1+(1-\lambda) \mu_2$. $\mu$ is also a Girsanov measure so it corresponds to a drift $F(t)$. What is $F$ in terms of $F_1,F_2$?

I know if $F_1, F_2, F$ are all deterministic then $$F(t)=E_\mu[B(t)]=\lambda F_1(t)+(1-\lambda)F_2(t)$$.

What about in general?

Consider two Girsanov measures $\mu_1$ and $\mu_2$ corresponding to drifts $F_1(t)$ and $F_2(t)$ respectively. By this, I mean that we have that $B(t)\sim F_1(t)+\tilde B(t)$ where $\tilde B(t)$ is a Brownian motion under $\mu_1$. Similarly for $\mu_2$.

For $\lambda \in [0,1]$ we can consider the probability measure $\mu=\lambda \mu_1+(1-\lambda) \mu_2$. $\mu$ is also a Girsanov measure so it corresponds to a drift $F(t)$. What is $F$ in terms of $F_1,F_2$?

I know if $F_1, F_2, F$ are all deterministic then $$F(t)=E_\mu[B(t)]=\lambda F_1(t)+(1-\lambda)F_2(t)$$.

What about in general?