### Sharp Large Deviations for the Ornstein - Uhlenbeck Process

Bernard Bercu, Bernard Bercu, Alain Rouault, Alain Rouault
2001 Teorija verojatnostej i ee primenenija
Доказывается точный принцип больших уклонений для хорошо известных случайных величин, ассоциированных с процессом Орнштейна-Уленбека, таких, как энергия, оценка максимального правдоподобия параметра сноса и логарифмическое отношение правдоподобия. Ключевые слова и фразы: большие уклонения, процесс Орнш тейна-Уленбека, оценка правдоподобия. 2) J o the maximum likelihood estimator of в Q_ _ fo x d x t = X\ -Г fiXldt 2fiX?dt' and the log-likelihood ratio .70 * Jo with 0o and 0i strictly negative.
more » ... t was already proven that a.s., as T goes to infinity Large deviations for the Ornstein-Uhlenbeck process 75 Fluctuations are also known . More recently, Bryc and Dembo  and Florens-Landais and Pham  have established large deviation principles for ST and 0т-Strictly speaking, Oj is not the maximum likelihood estimator of 0 since 0T may take nonnegative values whereas the parameter 0 is as sumed to be strictly negative. Nevertheless, we use this terminology through the paper. We present here sharp large deviation principles for the random variables 5т, 0т and Vr-As usual, we shall say that a family of real random variables (ZT) satisfies a Large Deviation Principle (LDP) with rate func tion /, if / is a lower semicontinuous function from R to [0, +oo] such that, for any closed set FcR, limsupT^oo T~L logP{ZT Gf}^ -inf^F I(%), while for any open set G С R -inf I(x) < lim inf ~ logP{Z T € G}. Moreover, J is a good rate function if its level sets are compact subsets of R. When J has an unique minimum m, which will always be the case here, an LDP for (ZT) gives the asymptotic behavior of P{ZT ^ c} or P{ZT ^ c} in a logarithmic scale whenever с > m or с < га, respectively. We shall say that a sequence of real random variables (ZT) satisfies a Sharp Large Deviation Principle (SLDP) if, for any real number c, it is possible to give asymptotic expansions of е т/ ( с )р{£т > с} or е т/ ( с )Р{£т < с} in powers of T" 1 . For the sake of simplicity, we present only the first one. In order to prove an LDP for (1.2), (1.3) or (1.4), the main tool is the normalized cumulant generating function (c.g.f.) of the pair (fiXtdXufJxfdt) %т(а,Ь) = i logE[exp(2f T (a,&))], (1.6) where, for any (a,b) € R 2 , 2f T (a, b) = a Г X t dX t + b Г X 2 t dt (1.7) J o Jo It is possible to establish SLDP for all linear combinations of / 0 T X t dX t and JQ X? dt. However, in order to improve the presentation of the paper, we prefer to focus our attention on the random variables (1.2), (1.3) and (1.4). In discrete time, the analog of (1.1) is the first order autoregressive process. It was studied, as a particular case, in  for LDP and  for SLDP. The proofs used Toeplitz matrices and their asymptotic spectral properties given by the first Szego theorem for LDP and the strong Szego theorem for SLDP. The covariance of the stationary Ornstein-Uhlenbeck process is a Wiener-Hopf operator and for some «quadratic forms», we can use a similar scheme . Here, we take advantage of the explicit expression of the c.g.f. An important point for proving an LDP is the determination of the domain Д of the limit S£ of the c.g.f. We shall say that % is steep if its