I have a set of random variables denoted by $a_t, b_t, c_t, d_t$ with standard deviation as ${\sigma}_{t, a_t}, {\sigma}_{t, b_t}, {\sigma}_{t, c_t}, {\sigma}_{t, d_t}$, and mean as ${\mu}_{t, a_t}, {\mu}_{t, b_t}, {\mu}_{t, c_t}, {\mu}_{t, d_t}$. All these means and standard deviations are time dependent, however among them only ${\sigma}_{t, d_t}$ is increasing over time, and rests are stationary.
These random variable are connected to each other with below equations
$a_t = \frac{\omega_1b_t+\omega_2c_t}{\omega_1+\omega_2}$
$b_t = c_t+d_t$
The random variables $c_t, d_t$ are uncorrelated.
The goal is to chose $\omega_1, \omega_2$ in such a way that at the beginning of time $t$, $a_t$ will stay at around middle point between $\left(b_t, c_t\right)$, but as time progresses, it will move closer and closer to $c_t$.
One trivial value for them may be $\omega_2 = {\sigma}_{t, b_t}, \omega_1 = {\sigma}_{t, c_t}$, just because ${\sigma}_{t, b_t}$ would be increasing over time. However this choice does not seem to follow the condition that at the beginning of time $t$, $a_t$ will stay at around middle point between $\left(b_t, c_t\right)$. Additionally it also appears that this r.v. is converging to $c_t$ to soon whereas I wanted the convergence to be a bit slow and parabolic in nature.
Is there any other choices for $\omega_1, \omega_2$ satisfying above conditions?
Any pointer will be very helpful
