Abstract
In the present paper we study moving averages (also known as stochastic convolutions) driven by a Wiener process and with a deterministic kernel. Necessary and sufficient conditions on the kernel are provided for the moving average to be a semimartingale in its natural filtration. Our results are constructive - meaning that they provide a simple method to obtain kernels for which the moving average is a semimartingale or a Wiener process. Several examples are considered. In the last part of the paper we study general Gaussian processes with stationary increments. We provide necessary and sufficient conditions on spectral measure for the process to be a semimartingale.
Highlights
In this paper we study moving averages, that is processes (Xt)t∈R on the formR Xt = (φ(t − s) − ψ(−s)) dWs, t ∈, (1.1)R where (Wt)t∈R is a Wiener process and φ and ψ are two locally square integrable functions such that s → φ(t − s) − ψ(−s) ∈ L2R(λ) for all t ∈ (λ denotes the Lebesgue measure)
In the present paper we provide necessary and sufficient conditions on φ and ψ for (Xt)t≥0 to be an (FtX,∞)t≥0-semimartingale
We show that (Xt)t≥0 is an (FtX,∞)t≥0-semimartingale if and only if φ can be decomposed as t φ(t) = β + αf(t) + f h(s) ds, R λ-a.a. t ∈, (1.3)
Summary
R where (Wt)t∈R is a Wiener process and φ and ψ are two locally square integrable functions such that s → φ(t − s) − ψ(−s) ∈ L2R(λ) for all t ∈ (λ denotes the Lebesgue measure). Let (Xt)t≥0 be given by (1.1) and assume it is (FtW,∞)t≥0-adapted; it is easier for (Xt)t≥0 to be an (FtX,∞)t≥0-semimartingale than an (FtW,∞)t≥0-semimartingale and harder than being an (FtX )t≥0-semimartingale It follows from Basse [2008a, Theorem 4.8, iii] that when ψ equals 0 or φ and (Xt)t≥0 is an (FtX )t≥0-semimartingale with canonical decomposition Xt = X0 + Mt + At,. In the last part of the paper we are concerned with the spectral measure of (Xt)t∈R, where (Xt)t∈R is either a stationary Gaussian semimartingale or a Gaussian semimartingale with stationary increments and X0 = 0 In both cases we provide necessary and sufficient conditions on the spectral measure of (Xt)t∈R for (Xt)t≥0 to be an (FtX,∞)t≥0-semimartingale
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