I want to regulate the speed of a clock but with each regulation there is a random change whose size increases with the size of the intended change. Formally speaking, let $a_n$ be the error in the speed of the clock after $n$ corrections. Knowing the value of $a_n$ I correct the speed by $b_n$, then the error becomes $a_{n+1}=a_n-b_n+e_{n+1}$ where $e_{n+1}$ is normally distributed with zero mean and variance $\alpha b_n^2$. After all changes I leave the clock with an error of $a$, this costs me $a^2$.
Assume I make exactly $x$ regulations with an initial error of $a$. How can I find the optimal policy for the regulation? I have no idea.
I think the dynamic programming equation can be written as $F(x,n)=\inf\{a^2+\mathbb E^e[F(x-u+e,n+1)]\}$
Any help or suggestions would be really great!
