No abstract

We investigate large deviation properties of the maximum likelihood drift parameter estimator for Ornstein-Uhlenbeck process driven by mixed fractional Brownian motion.

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

We use the functional Itô calculus to prove that the solution of a BSDE with singular terminal condition is continuous at the terminal time. Hence we extend known results for a non-Markovian terminal condition.1 with the convention 0.∞ = 0. 2 French acronym for right continuous with left limits.