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What does it mean when in stochastic process, we say that the process has unit variance? What is its exact definition?

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Dunno, maybe that $\mathrm{var}(X_t)=1$ for every $t$.

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This link to Wolfram Mathworld seems to support this answer: "A standard normal distribution is a normal distribution with zero mean (mu=0) and unit variance (sigma^2=1)" mathworld.wolfram.com/StandardNormalDistribution.html – AfterWorkGuinness Aug 29 '15 at 20:32

"Did" has posted a guess. Here's another. Some processes $\{X_t\}_{t\ge0}$ satisfy $\operatorname{var}(X_t)=ct$ (Poisson processes, Wiener processes, many others). Possibly it could mean $c=1$.

(But if something like the Ornstein–Uhlenbeck process is being written about, probably "Did"'s guess is right.)

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