Depending on parameter integrals with respect to random measures. (Ukrainian, English) Zbl 1164.60371

Radchenko, V. M.

Teor. Jmovirn. Mat. Stat. 75, 140-143 (2006); translation in Theory Probab. Math. Stat. 75, 161-165 (2007).

Let \(X\) be an arbitrary set; let \(\mathcal B\) be the \(\sigma\)-algebra of subsets of \(X\); and let \((\Omega,{\mathcal F},P)\) be a complete probability space. Let us denote by \(L_0=L_0(\Omega,{\mathcal F},P)\) the set of all random variables. Convergence in \(L_0\) means convergence in probability. The \(\sigma\)-additive map \(\mu:\;{\mathcal B}\to L_0\) is called a random measure on \(\mathcal B\). Let \(T\) be an arbitrary set, the function \(f:\;T\times X\to \mathbb R\) is \(\mu\)-integrable for every fixed \(t\in T\). The author obtains sufficient conditions for the existence of a continuous modification of a random process \(\eta(t)=\int_{X}f(t,x)d\,\mu(x)\), \(t\in T\). The sufficient conditions for the existence of a random measure on the Borel sets in \([a,b]\) generated by the increments \(\eta(t)-\eta(s)\) are obtained.

Reviewer: Aleksandr D. Borisenko (Kyïv)

MSC:

60G57

Random measures

Keywords:

random measures; stochastic integral; continuity of trajectories of random processes