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I have this equation:

$$f(k)=argmax_{0\le T\le T_{max}} \left(X+\left\lceil\frac{k-X}{T}\right\rceil T\right)$$

$X$ is uniformly distributed between $(0,T)$. I guess this is called a stochastic optimisation problem, but I am unable to figure out which method to use to solve this problem.

Is there a way to characterise the random variable $f(k)$? I am not sure if I could substitute $\frac{T}{2}$ in place of $X$ and solve the problem.

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Can you solve the problem acting as if $X$ is an ordinary unknown? Then once you have done that, reinterpret your solution with $X$ as a stochastic variable. Additional note: your $f$ actually also still depends on $X$. So you should write $f(k,X)$. – Raskolnikov Jun 1 '12 at 9:14
@Raskolnikov: A question: isn't $X$ absorbed into $T$ during the maximisation? – Bravo Jun 1 '12 at 9:19
@Raskolnikov: Actually I tried what you said and the answer when $X$ is not random is $T_{max}$ - the first part is irrelevant to the max, the ceil in the second part also becomes immaterial. – Bravo Jun 1 '12 at 9:21
By the way, are you sure you want the $\max$ and not the $\sup$ ? – Raskolnikov Jun 1 '12 at 9:32
If the ceiling function is immaterial, then the answer should always be $k$, independently of $X$. And it therefore should not matter wether $X$ is stochastic or not. – Raskolnikov Jun 1 '12 at 9:35

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