Abstract
Strongly convex stochastic processes are introduced. Some well-known results concerning convex functions, like the Hermite–Hadamard inequality, Jensen inequality, Kuhn theorem and Bernstein–Doetsch theorem are extended to strongly convex stochastic processes.
Highlights
In 1980 Nikodem [7] considered convex stochastic processes
In 1995 Skowronski [9] obtained some further results on convex stochastic processes, which generalize some known properties of convex functions
The main subject of this paper is to extend some well-known results concerning convex functions to strongly convex stochastic processes
Summary
In 1980 Nikodem [7] considered convex stochastic processes. In 1995 Skowronski [9] obtained some further results on convex stochastic processes, which generalize some known properties of convex functions. Recall that a function f : I → R is called strongly convex with modulus c > 0, if f λx + (1 − λ)y λf (x) + (1 − λ)f (y) − cλ(1 − λ)(x − y) for any x, y ∈ I and λ ∈ [0, 1] (cf [2,8]). In this paper we propose the generalization of convexity of this kind for stochastic processes. Let (Ω, A, P ) be an arbitrary probability space. A function X : Ω → R is called a random variable, if it is A-measurable. A function X : I × Ω → R is called a stochastic process, if for every t ∈ I the function X(t, ·) is a random variable
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