Sample Path Properties of Self-Similar Processes with Stationary Increments

A real-valued process X = (X(t))ₜ is self-similar with exponent $H (H$-ss), if X(a·) =d aHX for all $a > 0$. Sample path properties of H-ss processes with stationary increments are investigated. The main result is that the sample paths have nowhere bounded variation if 0 < H ≤ 1, unless X(t) ≡ tX(1) and $H = 1$, and apart from this can have locally bounded variation only for $H > 1$, in which case they are singular. However, nowhere bounded variation may occur also for $H > 1$. Examples exhibiting this combination of properties are constructed, as well as many others. Most are obtained by subordination of random measures to point processes in R² that are Poincare, i.e., invariant in distribution for the transformations (t, x) ↦ (at + b, ax) of R². In a final section it is shown that the self-similarity and stationary increment properties are preserved under composition of independent processes: X₁ ∘ X₂ = (X₁(X₂(t)))t∈ R. Some interesting examples are obtained this way.