On the eigenproblem for Gaussian bridges

Abstract

Spectral decomposition of the covariance operator is one of the main building blocks in the theory and applications of Gaussian processes. Unfortunately it is notoriously hard to derive in a closed form. In this paper we consider the eigenproblem for Gaussian bridges. Given a base process, its bridge is obtained by conditioning the trajectories to start and terminate at the given points. What can be said about the spectrum of a bridge, given the spectrum of its base process? We show how this question can be answered asymptotically for a family of processes, including the fractional Brownian motion.Various aspects of general Gaussian bridges are discussed in [11], [25]. Aside of being interesting mathematical objects, they are important ingredients in applications, such as statistical hypothesis testing [15], exact sampling of diffusions [3], etc.The covariance operator of the bridge with kernel (1.5) is a rank one perturbation of the covariance operator of its base process. This explains similarity between (1.3) and (1.4) and suggests that the spectra of the two processes must be closely related in general. This is indeed the case and one can find an exact expression for the Fredholm determinant of K in terms of the Fredholm determinant of K even for more general finite rank perturbations (see, e.g., [26], Ch. II, 4.6 in [13]). As mentioned above, the precise formulas for the eigenvalues and eigenfunctions of K are rarely known; however, the exact asymptotic approximations can be more tractable. This raises the following question:Can the exact asymptotics of the eigenvalues and the eigenfunctions for the bridge be deduced from that of the base process ?A rough answer to this question is given by the general perturbation theory [14], which implies that the eigenvalues of K and K agree in the leading asymptotic term, as it happens for (1.3) and (1.4) (see, e.g., the proof of Lemma 2 in [4]). More delicate spectral discrepancies are harder to exhibit and seem to be highly dependent on the perturbation structure. This is vividly demonstrated in the paper [21], where the kernels of the following form are considered, cf. (1.5):K Q (s, t) = K(s, t) + Qψ(s)ψ(t).(1.6)Here Q is a scalar real valued parameter and ψ is a function in the range of K. It turns out that for any Q greater than a certain critical value Q * , the spectrum of K Q coincides

Spectral asymptotics for a class of integro-differential equations arising in the theory of fractional Gaussian processes

“…In later papers [7], [8] similar results were obtained for some other fractional Gaussian processes.…”

“…In [7] this approach was applied to obtain the two-term spectral asymptotics for B H . Moreover, it was mentioned that the direct method developed in [6] for W H does not produce results quite as explicit as those in [6].…”

“…In the fractional setting, the Ornstein-Uhlenbeck process can be defined in a number of nonequivalent ways, see, e.g., [5]. Following [7], we consider the solution of the Langevin equation driven by the FBM:…”

Exact asymptotics in eigenproblems for fractional Brownian covariance operators

“…The analysis framework, set up in this paper, is applicable with some nontrivial adjustments to processes, related to the fBm, such as the corresponding bridge and the Ornstein-Uhlenbeck process, integrated fractional Brownian motion, etc. The results in this direction will be reported in the forthcoming work [16,17].…”

Exact spectral asymptotics of fractional processes

“…where β ∈ R is the drift parameter and X 0 ∼ N (0, σ 2 ) is the initial condition, independent of B. A nontrivial initial condition introduces a rank one perturbation to the covariance operator, which is inessential for our purposes (see [13] and [5]), and hereafter we will set X 0 = 0 for simplicity. In this case, the covariance function of X is given by the formula…”