Consider a discrete one-dimensional random walk with $(-1, +1)$ increments and $n$ steps. We'll assume that $p_{+1}$ describes the fixed probability that in each of the $n$ steps the increment $+1$ will be randomly drawn. Furthermore, I have some values $high$, $low$, $end$ that give the highest, lowest and final excursion of the walk from the starting point $0$, as well as the corresponding last hitting times for these values, $t_{high}$, $t_{low}$ and $t_{end}$.
How can one estimate the most likely probability $p_{+1}$ that created the walk with the given 6 numbers $high$, $low$, $end$, $t_{high}$, $t_{low}$ and $t_{end}$? Is it even possible to estimate a most likely parameter $p_{+1}$ that produced these readings?
And can this estimation procedure be generalized for random walks with other symmetrical distributions for the increments (e.g. continuous normal, uniform, lognormal distributions)?