Large deviations for conditionally Gaussian processes: estimates of level crossing probability
Volume 5, Issue 4 (2018), pp. 483–499
Pub. online: 12 October 2018
Type: Research Article
Open Access
Open Access
Received
16 May 2018
16 May 2018
Revised
30 July 2018
30 July 2018
Accepted
1 October 2018
1 October 2018
Published
12 October 2018
12 October 2018
Abstract
The problem of (pathwise) large deviations for conditionally continuous Gaussian processes is investigated. The theory of large deviations for Gaussian processes is extended to the wider class of random processes – the conditionally Gaussian processes. The estimates of level crossing probability for such processes are given as an application.
References
Adler, R.J., Samorodnitsky, G., Gadrich, T.: The expected number of level crossings for stationary, harmonizable, symmetric, stable processes. Ann. Appl. Probab. 3, 553–575 (1993) MR1221165
Azencott, R.: Grande déviations et applications. In: École D’été de Probabilités de St. Flour VIII, L.N.M. Volume 774, (1980) MR0590626
Baldi, P.: Stochastic Calculus. Springer (2017) MR3726894. https://doi.org/10.1007/978-3-319-62226-2
Baldi, P., Pacchiarotti, B.: Explicit computation of second order moments of importance sampling estimators for fractional Brownian motion. Bernoulli 12(4), 663–688 (2006) MR2248232. https://doi.org/10.3150/bj/1155735931
Berlinet, A., Thomas-Agnan, C.: Reproducing Kernel Hilbert Spaces in Probability and Statistics. Kluwer Academic Publishers (2004) MR2239907. https://doi.org/10.1007/978-1-4419-9096-9
Borkar, S.V.: Probability Theory. Springer (1995) MR1367959. https://doi.org/10.1007/978-1-4612-0791-7
Caramellino, L., Pacchiarotti, B., Salvadei, S.: Large deviation approaches for the numerical computation of the hitting probability for Gaussian processes. Methodol. Comput. Appl. Probab. 17(2), 383–401 (2015) MR3343412. https://doi.org/10.1007/s11009-013-9364-5
Chaganty, N.R.: Large deviations for joint distributions and statistical applications. Sankhya 59(2), 147–166 (1997) MR1665683
Chiarini, A., Fischer, M.: On large deviations for small noise Itô processes. Adv. Appl. Probab. 46(4), 1126–1147 (2014) MR3290432. https://doi.org/10.1239/aap/1418396246
Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications. Jones and Bartlett, Boston, Ma (1998) MR1202429
Deuschel, J.D., Stroock, D.W.: Large Deviations. Academic Press, Boston, MA (1989) MR0997938
Giorgi, F., Pacchiarotti, B.: Large deviations for conditional Volterra processes. Stoch. Anal. Appl. 35(2), 191–210 (2017) MR3597612. https://doi.org/10.1080/07362994.2016.1237291
Gulisashvili, A.: Large deviation principle for Volterra type fractional stochastic volatility models. https://arxiv.org/abs/1710.10711 MR3858803. https://doi.org/10.1137/17M116344X
Hida, T., Hitsuda, M.: Gaussian Processes. AMS Translations (1993) MR1216518
Macci, C., Pacchiarotti, B.: Exponential tightness for Gaussian processes with applications to some sequences of weighted means. Stochastic 89(2), 469–484 (2017) MR3590430. https://doi.org/10.1080/17442508.2016.1248968
