Integral representation of random variables with respect to Gaussian processes (Q5963505)

In the present paper, the author deals with integral representation theorems. More precisely, let \((\Omega, \mathcal{F}, \mathbb{P})\) be a complete probability space with left-continuous filtration \(\mathbb{F} = \{ \mathcal{F}_t \}_{t \in [0,T]}\). Furthermore, let \(\alpha \in (\frac{1}{2},1)\) be arbitrary, and let \(\mathcal{X}_T^{\alpha}\) be the class of Gaussian processes introduced in the paper (Definition 2.2). This class covers Gaussian processes with stationary increments as well as stationary Gaussian processes. Let \(X \in \mathcal{X}_T^{\alpha}\) be arbitrary. Under an additional condition on \(X\) (Assumption 2.6), the author shows in Theorem 3.7 that for every \(\mathcal{F}_T\)-measurable random variable \(\xi\) there exists an \(\mathbb{F}\)-adapted process \(\Psi_T\) such that NEWLINE\[NEWLINE \lim_{s \to T-} \int_0^s \Psi_T(s) \text{d} X_s = \xi \quad \text{\(\mathbb{P}\)-almost surely.} NEWLINE\]NEWLINE If, in addition, there exists a Hölder continuous process \(Z\) of order \(a > 1 - \alpha\) such that \(Z_T = \xi\), then the author shows in Theorem 3.11 that there even exists an \(\mathbb{F}\)-adapted process \(\Psi_T\) such that NEWLINE\[NEWLINE \int_0^T \Psi_T(s) \text{d} X_s = \xi \quad \text{\(\mathbb{P}\)-almost surely.} NEWLINE\]NEWLINE This extends results of \textit{Y. Mishura} et al. [Stochastic Processes Appl. 123, No. 6, 2353--2369 (2013; Zbl 1328.60131)], where a fractional Brownian as integrator \(X\) has been considered.NEWLINENEWLINEThe author discusses financial applications, the uniqueness of the integrand \(\Psi_T\) and the problem of zero integral, which refers to the question whether NEWLINE\[NEWLINE \int_0^T u_s \text{d} X_s = 0 \quad \text{\(\mathbb{P}\)-almost surely} NEWLINE\]NEWLINE implies NEWLINE\[NEWLINE u = 0 \quad \mathbb{P} \otimes \text{Leb}\big( [0,T] \big) \text{-almost everywhere.} NEWLINE\]