Stochastic Global Smoothness Preservation

Abstract

Let (Ω, A,P) be a probability space and let C_Ω[a, b]denote the space of stochastically continuous stochastic processes with index set [a,b]. When C [a,b] ⊂ V ⊂ CΩ[a,b] and $\tilde L:V \to C_\Omega \left[ {a,b} \right]$ is an E(expectation)-commutative linear operator on V, sufficient conditions are given here for E-preservation of global smoothness of X ∈ V through $\tilde L$. Namely, it is given that

$$ {\omega _1}(E(\widetilde LX);\delta \leqslant \left\| L \right\|.{\widetilde \omega _1}\left( {EX;\frac{{c.\delta }}{{\left\| L \right\|}}} \right) \leqslant (\left\| L \right\| + c).{\omega _1}(EX;\delta ) $$

, where $L: = \tilde L|_{C\left[ {a,b} \right]}$ , and for 0 ≤ δ ≤ b-a, ω ₁ denotes the first order modulus of continuity with $\tilde \omega _1$ its least concave majorant and c a universal constant. Applications are given to different types of stochastic convolution operators defined through a kernel. Especially are studied extensively in this connection, stochastic operators defined through a bell-shaped trigonometric kernel. Another application of the above result is to stochastic discretely defined Kratz and Stadtmüller operators.