[sixhigh]

If I understand you correctly you mean the following: If the coin comes heads the new value (x_(n+1)) will be x_n*1.10, if the coin comes tails the new value will be x_n*0.90.

In this case the probability of reaching any point above the initial value will be less than 100%. That is because the whole process is a supermartingale - i.e. the expected value of x_(n+1) is lower than the ev of x_n

E[x_(n+1) | x_n] = 1/2 * x_n * 1.1 + 1/2 * x_y * 0.9 = 0.99 * x_n < x_n

thus E[x_n] --> 0 (for n --> \infty)

Now there are ways to calculate the exact probability of x_n reaching a certain value at least once, but iirc this is quite difficult.

I don't know what you mean by using a certain distribution to determine x_(n+1). Should the probability of moving up/down be based on the value of x?

And the answer to the first questions is yes - you will reach every value infinitely often in a one-dimensional random walk.