What is meant by the drift of a stochastic process?

Question

Can someone please elaborate on what the drift of a stochastic process for eg. a Markov process mean? And what role does it play with respect to establishing the stability of that process ?

Answer 1

Intuitively, the drift is the rate at which the expected value of a process changes. For example, a random walk $y_n = \mu + y_{n-1} + \epsilon_n$ has drift $\mu$. The process can also be written as $ y_n = n\mu + \sum_{i=1}^n \epsilon_i$, where the expectation is $\mathbb{E}[y_n] = n\mu$. So as time passes, the expectation 'drifts' toward a value that gets larger and larger (assuming $\mu$ is positive).

If you are familiar with SDEs: for a process $X_t$ with Brownian Motion $W_t$, the SDE

$$ dX_t = \mu(X_t,t) dt + \sigma(X_t,t) dW_t $$

has drift $\mu(X_t,t)$: the change in $X_t$ is a function of drift (which is deterministically known at time $t$) and some random factor.

The drift describes the long-run behavior of the process; it tells us whether it tends to some steady-state, explodes, or neither (oscillation).