The question is NOT about real trading, but about simplified mathematical models for trading.
One of the main problems in trading is that asset prices are not correctly described by the some random processes.
Let us consider idealized situation - assume that asset's price $p(t)$ is given by some random process, which is known to trader. It roughly speaking means that probabilities of all events $p(t_1) = p_1, p(t_2) = p_2, ..., p(t_k) = p_k$ are known to trader. Both for time $t$ in future or in the past.
What are some theoretical results on optimal or (sub)-optimal trading strategies in this idealized setup? What are some theoretical results on profit estimates ? (I.e. some bounds - we cannot earn more than ...)
If nothing like that is known - what is the reason - is it difficult or "no one needs"?
By "optimal" I mean the following. Of course, intuitively it means that trader will gain the most profit, but here is subtlety - our price $p(t)$ is a random variable, so profit is also a random variable, so it should be specified what means "the most".
It can be $E(p(T))$ (mean value for some $t=T$), or can be $\frac{E(p(t))}{std(p(t))}$ - or what whatever, any mathematically correct result is very welcome.
Let me emphasize that from my point of view this is question mathematically well-defined, and I would expect that many mathematically rigorous theorems (and/or conjectures) should be known on this for experts. If someone doubts that question is mathematically rigorous, please let us discuss in comments.
It might be that such kind of results known only for some special kind of random processes - e.g. Brownian motion or whatever - any information is welcome, I am novice in the field.
For example assume that our random process is actually deterministic process - i.e. only one trajectory $p(t) = p_0(t)$ has probability $1$, all other trajectories has probability $0$. Then the optimal trading strategy is to buy at local minimums and sell at local maximums; profit is variation of $p_0(t)$.
Well, this is of course over-simplified situation, but I just put it to demonstrate that there exist rigorous mathematical results.
Also this example gives some bound on maximal possible mean profit for general random process - we should sum over all possible trajectories their variations with the weight - probability of the trajectory (a'la Feynmann's path-integral).
This bound should not be sharp - it seems it is impossible to achieve it in general - is it correct? What are the sharper bounds?