Random Walk

Table of Contents

1. Summary

A random walk is the simplest possible stochastic process: a sequence formed by summing independent, identically-distributed steps, one per discrete time period. It predates the Wiener process both historically and conceptually — the Wiener process is what a random walk becomes in the limit of infinitely many, infinitesimally small steps — and it already has the Markov property every process in the Stochastic Processes cluster relies on, in its plainest possible form. Its increments are stationary while its level is not — variance grows with the horizon and no force pulls it back — so it stands as the canonical unit-root process; and the question it is most often asked is not where it ends up, but when it first reaches a given level.

2. Layperson's mental model

Imagine flipping a coin once per second, stepping one pace right for heads and one pace left for tails, and tracking where you end up. That's a random walk: the running total of a string of independent, identical random steps. There is no rule pulling you back toward where you started — after enough flips you could, in principle, be anywhere. 

double position = 0.0;
for (int i = 0; i < n_steps; ++i) {
    position += (coin_flip() ? +1.0 : -1.0);   // one independent step
}

Shrink each step and speed up the flips — smaller paces, more of them per second — and, in the limit, this discrete jittery path becomes the smooth-looking (but still nowhere-differentiable) continuous path of a Wiener process.

3. Detail

3.1. The definition

A random walk \(S_n\) is defined by:

\begin{equation} S_n = S_0 + \sum_{i=1}^{n} X_i \end{equation}

where \(X_1, X_2, \ldots\) are independent, identically-distributed random steps. The simple random walk takes each \(X_i = \pm 1\) with equal probability — the textbook coin-flip walk — but the step distribution need not be so restricted: any independent, identically-distributed sequence of steps produces a random walk in this general sense.

3.2. Stationary steps, non-stationary level

The steps \(X_1, X_2, \ldots\) are drawn independently and from one common distribution, so the increments of a random walk are stationary: the law of the change \(X_{i+1}\) does not depend on the index \(i\). The level \(S_n\) is a different matter — it is not stationary, and its variance grows without bound:

\begin{equation} \operatorname{Var}(S_n) = n\,\sigma^2 \end{equation}

where \(\sigma^2\) is the variance of a single step. The horizon widens the distribution rather than pinning it down: uncertainty accumulates with time instead of settling to a fixed spread. That pairing — stationary increments, a non-stationary level — is the signature that makes the random walk the natural model for prices, cumulative P&L, and inventory drift rather than for a quantity that fluctuates around a fixed level. The Geometric Brownian Motion behind the synthetic market data generators is the same accumulation carried out on a logarithmic scale, so its log-level is the non-stationary object.

3.3. A unit root, not a force

A random walk is the canonical unit-root process. In the recursion \(S_n = S_{n-1} + X_n\) the coefficient on the previous level is exactly one, and that single fact separates it from every mean-reverting process: a shock never decays. A step taken a thousand periods ago is still present in full in today's level, and no force pulls the process back toward a long-run average — the best forecast of the next level is simply the last observation.

Contrast the Ornstein-Uhlenbeck process — the mean-reverting building block behind the Vasicek and Hull-White short rates — where a restoring force proportional to the distance from the mean makes each shock's influence fade geometrically. In time-series language a random walk is \(I(1)\): the level must be differenced once to become stationary, and it is the differenced series — the increments, not the level — that estimation and modelling work with.

3.4. Already Markov, already no-memory

Because each step \(X_i\) is drawn independently of every earlier step, \(S_{n+1} = S_n + X_{n+1}\) depends only on the current position \(S_n\) (and the fresh draw \(X_{n+1}\)), never on how \(S_n\) was reached — the Markov property holds trivially for a random walk, and is easiest to see directly here, before any continuous-time subtlety is introduced.

3.5. The continuous-time limit

Rescale a simple random walk so that each step has size \(1/\sqrt{n}\) and steps occur every \(1/n\) of a unit of time, then let \(n \to \infty\): the rescaled walk converges to the Wiener process (Donsker's theorem, also called the functional central limit theorem). This is the precise sense in which the Wiener process is a continuous-time random walk: not merely an analogy, but a rigorously defined limit of the discrete object. Practically, it also runs the other way: simulating a Wiener process on a computer, which necessarily advances in discrete ticks, is already an approximating random walk that targets the continuous idealisation, not the continuous process itself.

3.6. First-passage problems

The question most often put to a random walk is not "where is the level at time \(n\)?" but "when does it first reach a given level?" The first-passage time is

\begin{equation} \tau = \inf\{\, n \ge 0 : S_n \in A \,\} \end{equation}

for a target set \(A\) — a barrier, a ruin level, a reliability threshold. First-passage questions differ from terminal-value questions in two ways. The distribution of \(\tau\) is long-tailed and skewed: crossings cluster neither around a typical time nor symmetrically about it, and in the symmetric \(\pm 1\) walk every level is reached with probability one while the expected time to reach it is infinite. And the crossing event, not the terminal level, is often what a decision or a payoff depends on — a barrier option pays out or expires at the moment of passage, not at maturity.

In ORE Studio the Generic Barrier Option and the wider barrier product family are the products that ask exactly this question; their valuation needs the distribution of the passage time of the underlying through the barrier, not only the distribution of the underlying at expiry.

4. See also

4.1. Further reading

  • Pearson, K. (1905). "The Problem of the Random Walk." Nature, 72(1865), 294. The paper that coined the term "random walk", posed as an open problem about a drunkard's path.
  • Donsker, M. D. (1951). "An Invariance Principle for Certain Probability Limit Theorems." Memoirs of the American Mathematical Society, 6, 1-12. The rigorous proof that a rescaled random walk converges to the Wiener process — the theorem this document's continuous-time limit section refers to.
  • Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1 (3rd ed.). Wiley. The standard classical treatment of random walks, including the simple \(\pm 1\) walk and its key combinatorial results.
  • Wikipedia: Random walk.
  • Tuncer, T. D. (2026). "A random walk is a discrete-time stochastic process driven by successive random increments." LinkedIn post, September 2026. Retrieved 10 September 2026. The post prompted the "Stationary steps, non-stationary level", "A unit root, not a force", and "First-passage problems" sections above; this document restates that material in its own terms and ties it to ORE Studio's processes.

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