On the product of a random and a real measure

Abstract

The product of a random measure X and a real measure Y is defined as a random measure on X × Y . We obtain conditions under which the integral of a real function with respect to the product measure equals the iterated integrals of this function. Let (X,BX) and (Y,BY ) be measurable spaces, Z = X × Y , and BZ = BX ⊗ BY . By L0 = L0 (Ω,F ,P) we denote the set of all random variables defined on the probability space (Ω,F ,P) (to be more specific, L0 is the set of classes of equivalent random variables). The convergence in L0 is the convergence in probability. Definition 1. Any σ-additive mapping μ : BX → L0 is called a random measure on BX . Note that we do not assume that μ is nonnegative and we do not pose any moment condition. Here are some examples. If X(t), 0 ≤ t ≤ T , is a continuous square-integrable martingale, then μ(A) = ∫ T 0 IA(t) dX(t) is a random measure on Borel sets of [0, T ]. A fractional Brownian motion B(t) for H > 12 defines a random measure in a similar way (this follows from inequality (3.11) in [1]). Other examples as well as conditions for increments of a stochastic process to generate a random measure can be found in Chapters 7 and 8 of [2]. Further let μ be a random measure on BX , and m a finite nonnegative measure on BY . A set A ∈ BX is called μ-negligible if μ(B) = 0 a.s. for all B ∈ BX such that B ⊂ A. Let ξ be a random variable and put ‖ξ‖ = sup{δ : P{|ξ| > δ} > δ}. The integral ∫ A f dμ is defined and studied in [3] where f : X → R is a real measurable function and A ∈ BX . When constructing this integral one starts with simple functions and proceeds similarly to [2, Chapter 7] (see also [4]). In particular, any measurable bounded function f is integrable with respect to any measure μ. In this paper, we define the product of a random and a real measure and prove analogs of Fubini’s theorem for integrals of real functions. Theorem 1. There exists a unique random measure η on BZ such that η(A1 ×A2) = μ(A1)m(A2) 2000 Mathematics Subject Classification. Primary 60G57.