I feel a “random walk” and “random variable” are sometimes treated as interchangeable concepts. Watch out. Fundamentally different!
If a variable follows a stoch process (i.e. a type of random walk) then its Future [2] value at any Future time has a Probability distribution. If this PD is normal, then mean and stdev will depend on (characteristics of) that process, but also depend on the distance in time from the last Observation/revelation.
Let’s look at those characteristics — In many simple models, the drift/volatility of the Process are assumed unvarying[3]. I’m not familiar with the more complicated, real-world models, but suffice to say volatility of the Process is actually time-varying. It can even follow a stoch Process of its own.
Let’s look at the last Observation — an important point in the Process. Any uncertainty or randomness before that moment is irrelevant. The last Observation (with a value and its timestamp) is basically the diffusion-start or the random-walk-start. Recall Polya’s urn.
[2] Future is uncertain – probability. Statistics on the other hand is about past.
[3] and can be estimated using historical observations
Random walk isn’t always symmetrical — Suppose the random walk has an upward trend, then PD at a given future time won’t be a nice bell centered around the last observation. Now let’s compare 2 important random walks — Brownian Motion (BM) vs GBM.
F) BM – If the process is BM i.e. Wiener Process,
** then the variable at a future time has a Normal distribution, whose stdev is proportional to sqrt(t)
** Important scenario for theoretical study, but how useful is this model in practice? Not sure.
G) GBM – If the process is GBM,
** then the variable at a future time has a Lognormal distribution
** this model is extremely important in practice.
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