Some Large Deviations Principles for Time-Changed Gaussian Processes
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Lithuanian Mathematical Journal
Some large deviations principles for time-changed Gaussian processes Barbara Pacchiarotti Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica n. 1, 00133 Roma, Italy (e-mail: [email protected]) Received November 6, 2018; revised June 8, 2020
Abstract. Let X = (X(t))t0 (X(0) = 0) be a continuous centered Gaussian process on a probability space (Ω, F , P), and let (Yt )t∈[0,1] (Y0 = 0) be a continuous process (on the same probability space) with nondecreasing paths, independent of X. Define the time-changed Gaussian process Zt = X(Yt ), t ∈ [0, 1]. In this paper, we investigate a problem of finite-dimensional large deviations and a problem of pathwise large deviations for time-changed continuous Gaussian processes. As applications, we considered subordinated Gaussian processes. MSC: 60F10, 60G15, 60G52 Keywords: time-changed Gaussian processes, subordinated Gaussian processes, large deviations
1 Introduction Let (Zt )t∈[0,1] be a continuous time-changed Gaussian process, that is, Zt = X(Yt ) for t ∈ [0, 1], where X = (X(t))t0 (X0 = 0) is a continuous centered Gaussian process on a probability space (Ω, F, P), and (Yt )t∈[0,1] is a continuous process with nondecreasing paths, Y0 = 0, independent of X . In this paper, we study some large deviations principles for such processes. Large deviations theory is concerned with the probabilities of very “rare” events. There are events whose probabilities are very small; however, these events are of great importance; they may represent an atypical situation (i.e., a deviation from the average behavior), which may cause disastrous consequences: an insurance or a bank does bankrupt; a statistical estimator gives a wrong information; a physical or chemical system shows an atypical configuration. The aim of this paper is the extension of the theory of large deviations for Gaussian processes to a wider class of random processes, the time-changed Gaussian processes. Such processes were introduced in applications in finance, optimization, and control problems. See, for instance, [3, 10, 13]. The theory of large deviations for Gaussian processes and for conditioned Gaussian processes is already well developed. See, for instance, [6, Sect. 3.4] (and the references therein) for Gaussian processes and [9] and [16] for particular conditioned Gaussian processes. An extension of this theory is possible thanks to the results obtained by Chaganty [4]. We consider a family of processes (Y n , Z n )n∈N , Z n = X n ◦ Y n , on a probability space (Ω, F, P), where n (Y )n∈N is a family of processes with nondecreasing paths that satisfies a large deviation principle (LDP), and c 2020 Springer Science+Business Media, LLC 0363-1672/20/6003-0001
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(X n )n∈N is a family of Gaussian processes that also satisfies an LDP. We show a finite-dimensional LDP and a pathwise LDP for the family (Z n )n∈N . The paper is organized as follows. In Section 2, we recall some basic facts on large deviations theor
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