1
$\begingroup$

Do there exist stochastic walk process that are bounded? For example the value of the cumulative sum of the random values never exceeds some bound x, or goes lower than some y.

  • 0
    Do you mean bounded by definition (as in e.g. reflected Brownian motion) or bounded (maybe with some probability) by its properties (e.g. geometric Brownian motion)? – 2017-01-26
  • 0
    The answer is yes! A random walk is by definition a stochastic process with integer domain. \\ Look at a Markov chain with transistion matrix $A\in\mathbb{R}^{d\times d}$ (that is a matrix giving the probability of going from one state to another) this will give you a bounded random walk if the states are bounded (in this case you have $d$-states). – 2017-01-26
  • 0
    @user3658307 I think I mean the first, bounded by definition: 0 <= f(x) <= 1, where f(x) is something akin to `f(x) = f(x-1) + noise` is the random process.. I have not heard of reflected Brownian motion, but that looks like what I am hoping for! – 2017-01-26
  • 0
    @Robert glad to know; btw the two I mentioned are continuous space-time stochastic processes, whereas Vincent gives a good example of how to straightforwardly define a discrete one. – 2017-01-27

1 Answers 1

2

The only bounded random walk is the boring one where $S_n=0$ for all $n$. This follows from Theorem 4.1.2 (p. 155) of Probability: Theory and Examples (4th edition) by Richard Durrett. The book is freely available at the author's website.

Theorem 4.1.2. For a random walk on $\mathbb{R}$, there are only four possibilities, one of which has probability one.

  1. $S_n=0$ for all $n$.

  2. $S_n\to-\infty$.

  3. $S_n\to\infty$.

  4. $-\infty=\liminf S_n<\limsup S_n=\infty$.

There are lots of stochastic processes that are bounded, but they are not random walks.

  • 0
    This explains why "bounded random walk" doesn't turn up anything. Can you suggest a few stochastic processes that have this sort of bound? – 2017-01-30
  • 0
    @Robert Random walks on $\mathbb{Z}$ with two boundaries are good examples. – 2017-01-30