Information that only grows: the filtration
Walk through this rung again in your mind. A stochastic process is a family of random variables X_0, X_1, X_2, ... unfolding in time, and a sample path is one whole trajectory you might actually observe. Watching it live, you learn more as the clock ticks: after step 3 you know X_0, X_1, X_2, X_3, but X_4 is still hidden. The first job of this guide is to make that creeping knowledge into a precise mathematical object, because almost every interesting question — "when should I stop?", "does the future depend on the past?" — is secretly a question about what you know when.
We package the information available at time n as a sigma-algebra F_n — the collection of all events you can decide yes-or-no using only what you have seen up to time n. The whole increasing chain F_0 ⊆ F_1 ⊆ F_2 ⊆ ... is called a filtration, and the inclusions are the heart of it: information never gets thrown away, it only accumulates. A process is adapted to a filtration if at each time n the value X_n is already decidable from F_n — that is, X_n is F_n-measurable. The natural filtration of a process is simply the smallest one that makes it adapted: F_n records exactly the history X_0, ..., X_n and nothing more. See filtration and adapted processes.
Deciding when to stop, without peeking
In the previous guides you watched a simple random walk wander and asked questions like "when does it first hit 0?" or, in gambler's ruin, "when does the fortune first reach 0 or N?". Each such question names a random time T — random because it depends on the path you happen to draw. But not every random time is fair to use. A legitimate rule for stopping must be decidable *as you go*: at each moment you should be able to say "stop now" or "keep going" using only what you have seen, never on a glimpse of the future.
This is exactly what a stopping time captures. A random time T is a stopping time with respect to a filtration if, for every n, the event {T = n} belongs to F_n — i.e. whether you have stopped by time n is decidable from the information available at time n. "First time the walk hits 0" is a stopping time: at each step you can check whether you have arrived yet. "The last time the walk is positive before time 100" is NOT a stopping time, because to know it is the *last* such time you must peek ahead and confirm the walk never returns — knowledge you do not have at the moment it happens.
stopping time T <=> for every n, {T = n} is in F_n
(equivalently, {T <= n} is in F_n)
first hitting time of a set A:
T = min { n >= 0 : X_n is in A }
-> always a stopping time, because deciding {T = n}
only needs X_0, X_1, ..., X_n (all in F_n)Why stopping times pay off: the optional stopping theorem
Stopping times are not just bookkeeping — they turn hard path questions into one-line computations. Recall a fair +/-1 walk is a martingale (a martingale is a process whose conditional expected next value, given the past, equals its current value: a perfectly fair game where E[M_(n+1) given F_n] = M_n). The optional stopping theorem says that, under the right conditions, the fairness survives even if you stop at a clever random time T: E[M_T] = E[M_0]. In words, no honest stopping rule can change the expected value of a fair game. See the optional stopping theorem and martingale relative to a filtration.
Watch it dissolve gambler's ruin in a single move. Start a fair walk at fortune k, with absorbing barriers at 0 and N, and let T be the first time it hits either — a stopping time. Optional stopping gives k = E[M_0] = E[M_T]. But at time T the fortune is exactly 0 or N, so E[M_T] = 0·P(ruin) + N·P(reach N) = N·P(reach N). Setting the two equal, P(reach N) = k/N. The clean answer you met in the gambler's-ruin guide drops out with no recursion, no algebra grind — just the conservation of fairness through an honest stopping rule.
Processes that forget: the Markov property
The third pillar is about *dependence on the past*. Some processes carry their whole history with them; others care only about where they are right now. A process has the Markov property if, given the present state, the future is independent of the past: knowing the entire history X_0, ..., X_n tells you nothing about X_(n+1) beyond what X_n alone already tells you. Symbolically, P(X_(n+1) = j given X_0, ..., X_n) = P(X_(n+1) = j given X_n). The present is a sufficient summary; the road you took to get here is irrelevant to where you go next. See the Markov property.
Our random walk is the friendliest example. Its next position is current position plus a fresh +/-1 step, and that step is independent of everything before — so the future depends on the past only through the current location. That is the Markov property in action, and it is exactly why we could analyze the walk with a transition matrix and call it a discrete-time Markov chain: the rule for moving depends only on the current state, never on the trail of states behind it. The same forgetting underlies the memorylessness of the exponential in continuous time — a process that, having waited any amount, starts its wait afresh.
Where the three meet: the strong Markov property
Now braid the strands together. The ordinary Markov property says the process restarts cleanly from a *fixed* time n. But the most powerful arguments need it to restart from a *random* time — and not just any random time, but a stopping time, one chosen without peeking. The strong Markov property states that a Markov process restarts afresh from a stopping time T just as it would from a fixed time: given that X_T = i, the future X_(T+1), X_(T+2), ... evolves as a brand-new copy of the process started at i, independent of the path that led up to T. See the strong Markov property.
Why does "strong" need a stopping time and not just any random time? Because if T were allowed to peek at the future, conditioning on X_T could smuggle information about what is coming, and the fresh-restart picture would break. The stopping-time condition is precisely the guarantee that, at the moment we restart, we know we have arrived but know nothing illegitimate about the path ahead. This is why all three ideas of this guide are one package: the filtration defines what "so far" means, the stopping time is an honest random instant inside it, and the strong Markov property is the reward — memorylessness that survives even random, history-respecting restarts.
- Fix a filtration F_n — the growing record of "what is known by time n" — and check your process is adapted to it.
- Identify a stopping time T (e.g. a first hitting time): confirm {T = n} is decidable from F_n, with no peek at the future.
- If the process is Markov, invoke the strong Markov property: from T onward it behaves like a fresh copy started at the state X_T.
- If it is also a martingale, check the optional-stopping conditions and read off E[M_T] = E[M_0] to answer the path question in one line.