On some regularity properties of quadratic stochastic processes (Q789819)

Let (\(\Omega\),\({\mathfrak A},P)\) be a probability space. Let \({\mathbb{R}}\) be the real line. A function \(X:R\times \Omega \to R\) is called a quadratic stochastic process if it satisfies the equation \(X(u+v,.)+X(u- v,.)=2X(u,.)+2X(v,.)\) for all u,\(v\in {\mathbb{R}}\). It is proved in this paper that if \(X:{\mathbb{R}}\times \Omega \to {\mathbb{R}}\) is a quadratic stochastic process and \(\| X(t)\|_ p=(\int | X(t,w)|^ pdP(w))^{1/p}\) exists and is Lebesgue measurable for all \(t\in {\mathbb{R}}\) for some \(p\geq 1\), then \(\| X(t)\|_ p\) is bounded on every finite interval of \({\mathbb{R}}-\{0\}\) and \(X(t,\cdot)=t^ 2X(1,\cdot).\)