Firstly, apologies if some of the notation or terminology is odd, or if I am defining functions that have standard notation associated with them already - I am not familiar with the concepts in this question. If there are any mistakes, please edit my post! I am only too happy to clarify anything that isn't clear.

I suspect naively that this problem could draw on methods found in stochastic, optimizational and computational areas of mathematics - though I could be wildly out. I am not expecting a definitive answer and accordingly have not given exact functions (I don't know exactly what they are in some cases!) but any help is much appreciated e.g. ideas, pointers etc.

Given $0\leq p\leq 1$, let $\delta(p)$ denote the random variable such that $P(\delta(p) = 1)=p$ and $P(\delta(p)=0)=1-p$. Then I have the following recursive process which can be described by the equations: for $1\leq i\leq n$:

$y_1=0$

$x_i = \omega(y_i)$

$y_i = \delta((1-p)^{\kappa\cdot x_{i-1}})\cdot(\alpha(y_{i-1})\cdot x_{i-1} + y_{i-1})$

where $\omega$ is a known power-type function, $p$ is small ($p\approx 0.00001$) and $\kappa$ is a constant ($\kappa = 1/2$ say). Given $n$, I am looking to define the function $\alpha$ such that:

$\Sigma_{i=1}^n(1-\alpha(y_i))\cdot x_i$ is maximal. I have experience with **R** if you think this may be useful.

Thanks,

Mullefa