Let X(t) be a stochastic process whose parameter t runs over a finite or infinite n terval T. Let t
1
, t
2 ɛ T, t
1 〈 t2; the random variable X(t
2) – X(t
1) is called the increment of the process X(t) over the interval [t
1, t
2]. A process X(t) is said to be homogeneous if the distribution function of the increment X(t + τ) — X(t) depends only on the length τ of the interval but is independent of the endpoint t. Two intervals are said to be non-overlapping if they have no interior point in common. A process X(t) is called a process with independent increments if the increments over non-overlapping intervals are stochastically independent. A process X(t) is said to be continuous at the point t if plimτ→0 [X(t + τ) — X(t)] = 0, that is if for any ε > 0, limτ→0
P(| X(t + τ) — X(t) | > ε) = 0. A process is continuous in an interval [A, B] if it is continuous in every point of [A, B].
Multiple Integrals of a Homogeneous Process with Independent Increments