Stochastic process
Increment

An increment of a stochastic process is the difference between two random variables of the same stochastic process. For a stochastic process with an index set that can be interpreted as time, an increment is how much the stochastic process changes over a certain time period. For example, if { X (t): t ∈ T } {\displaystyle \{X(t):t\in T\}} is a stochastic process with state space S {\displaystyle S} and index set T = [ 0, ∞) {\displaystyle T=[0,\infty)}, then for any two non-negative numbers t 1 ∈ [ 0, ∞) {\displaystyle t_{1}\in [0,\infty)} and t 2 ∈ [ 0, ∞) {\displaystyle t_{2}\in [0,\infty)} such that t 1 ≤ t 2 {\displaystyle t_{1}\leq t_{2}}, the difference X t 2 − X t 1 {\displaystyle X_{t_{2}}-X_{t_{1}}} is a S {\displaystyle S} -valued random variable known as an increment. When interested in the increments, often the state space S {\displaystyle S} is the real line or the natural numbers, but it can be n {\displaystyle n} -dimensional Euclidean space or more abstract spaces such as Banach spaces.
keywords:existence, punctuation:no_comma, startend:quotation