Adaptive prediction of non-Gaussian Ornstein-Uhlenbeck process.pdf Nowadays mathematical statistics along with economics, financial mathematics, engineering, biology and other fields of science that use mathematical tools for their benefits, are turned to development of predictive methods. Models allowing making predictions of high statistical quality are highly appreciated. Currently, one of the most popular continuous-time models that is extensively used in financial mathematics is a non-Gaussian Ornstein-Uhlenbeck process driven by the Levy process. Usually in practice the applied models depend on unknown parameters. Estimation problem of the unknown parameters of dynamic systems is a relevant one since the estimators of the dynamic parameters are to be used in various adaptive problems including the problem of adaptive prediction. The quality of adaptive predictors significantly depends on a choice of estimators of the model parameters. One of the most proper methods to solve this problem is the truncated estimation method proposed in [1]. It gives an opportunity to obtain the estimators of guaranteed quality by samples of fixed size under low level of a'priory information on system parameters. Adaptive prediction problem for discrete-time systems was solved in [2] on the basis of truncated estimators proposed in [1]. Later, the same problem for Gaussian Ornstein-Uhlenbeck process was solved in [3,4] on the basis of truncated estimators. In this paper we propose adaptive predictors of non-Gaussian Ornstein-Uhlenbeck process constructed on the basis of truncated estimators of dynamic parameters which are optimal in the sense of a special risk function. The risk function aims to optimize both the duration of observations and the predictive quality. The risk function of similar structure first appeared in [5] for the problems of parameter estimator's optimization. Later on, this idea was developed in [6, 7] etc. for optimization of predictors of dynamic discrete-time systems. (1) 1. Problem statement. Optimal prediction Consider the following regression model: dxt = axtdt + d^, t > 0 with zero mean initial condition x0, having all the moments. Here ^ - PjWt +p2Zt,^ ^ 0and p2 are some constants, (W' t > 0) is a standard Wiener process, given on a filtered probability space (Q,F,{FtP), adapted to a filtration {Ft}t>0, Zt - lYk is a compound Poisson process, where Yk, k > 0 are i.i.d. random zero k-i mean variables having all the moments and (Nt) is a Poisson process with the intensity X > 0, i.e. N -I%{Tj * 4 and Tj -± V j>i i-i Here (Tj) are jumps of the Poisson process (Nt )>0 and (rt )M are i.i.d. random variables that are exponentially distributed with the parameter X. It should be noted that for p2 - 0 the process (1) is a standard Ornstein-Uhlenbeck process. Suppose that the process (1) is stable, i.e. the parameter a < 0.Note that in this case for every m > 1 sup Exfm 0 The purpose is to construct a predictor for xt by observations xt-u - (xs )0 0 is a fixed time delay. The solution of the process x , obtained by the Ito formula, has the form t X - eatxti +J ea(t-z)d^, t > 0 0 and for given u > 0 we have the representation xt - bxt- u + u» t > u where t b - eau' 4+u =1 ea(t-s)d^s' E4tt_u - 0 and a2:- D^ - ± (p2 + Xp2EYi2 )[b2 -1]. ,a(t-s)^P -П „„Л _2 ._ _ K~2 , i „2rv2\r»,2 2a Optimal in the mean square sense predictor is the conditional mathematical expectation x° - bx-u' t > u 2. Adaptive prediction. Model parameter estimators As in practice the parameter a and, as follows, b are unknown, it is impossible to construct the optimal predictor for real processes. In order to solve the problem of prediction we define an adaptive predictor that is constructed by an estimator at of unknown parameter a. Define adaptive predictor as xt (t-u) - bt-uxt-u ' t > u (2) where bt_u - ea'-uu, t > u; at - proj0]at, at is the truncated estimator of the parameter a constructed similar to discrete-time case [2] on the basis of the least squares estimator t 1 xvdx. v ( t \ 1 x2dv > tlog-11 . (3) 0 a - -t-x 1 x2vdv 0 3. Risk functions and prediction criteria Denote the prediction errors of x(° and xt (t - u) as e - xt - x° - Vtt-u ' et (t - u) x' - xt (t - u) - (b - bt-u ) xt-u + 4,t-u ' t > u . Now we define the loss function A Ц = - e (t) +1, t > u, where 1 t e (t) = - J e2 (s - u)ds t u and the parameter A > 0 stands for the cost of prediction error. We also define the risk function R = ELt which has the following form R = jEe2(t) +1 and consider optimization problem R. ^ min. t For the optimal predictors x0 it is possible to optimize the corresponding risk function directly A a2 A R = E - (e (t))2 +1 =^ +1 ^ min, (4) where t 1 (e°(t ))2 = - J (e0)2 ds. tJ u In this case the optimal duration of observations T° and the corresponding value of R0O are respectively T0 = A1/2a, R0 = 2A1/2a, (5) TA where a However, since a and as follows, о are unknown and both T0 and R00 depend on a, the TA optimal predictor can not be used. Then we define the estimator TA of the optimal time T° as Ta= inf{t > tA: t > A1/2at), (6) where ^ := A/2 log 1A = o(A1 2). Here at := ^o2 is the estimator of unknown a, 1 t ~ 2 a2 ^-J( xs- btxs-u) ds. t - и J \ ' The estimator is defined like that since a2 = Ец^_и = E(xt - bxt_u )2. 4. Properties of parameter estimators and adaptive predictors Estimators at, bt and at that are used in construction of adaptive predictors have the properties given in Lemma below which can be proven similar to [3]. Compare to [3] this way of estimation of the variance a2 does not require the knowledge of parameters the model parameters. In this particular case it is not dependant on the true values of parameters pj, p2, EYX2X and their estimators. Moreover, the upper boundary for the moments of deviation of a2 is more accurate than in [3]. In what follows, C will denote a generic non-negative constant whose value is not critical (and not always the same). Lemma 1. Assume the model (1). Then for t -u > s0 := exp(2|a|) and some numbers C estimators ait and bt have the properties: 2 2 C E(a> -aГ 1. E(of-o2)2P < CP, p > 1. (8) a - a = - gt =- Proof. We prove the property (7) similar to [3]. By the definition (3) of the estimator at and using (1) let us find representation for the deviation of the estimator f xvd ^ i i x| f xldv > t log-11 - a • x| f xldv < t log-1t f xldv 0 Define 1 fxldv, g =(pl +p2ey;2x)>o, ft =1 \xvd^ t r, 2a t r, Then 2 P • xg > log-1]+ a2p • p[gt < log-11]=: I1 +12. A ^gt^ E(at - a )2p = E (9) Using the Cauchy-Schwarz-Bunyakovsky inequality for the first summand we get: T2p - ^r2p ( 2p , „ „ \k' 1t| = Eft •X[gt > log 11] {•x[gt > log 11] 1 + XC 11 = E t|< 2p 2p gt V k=1 k (g - gt r ft ^ у g gt .g t ggt C 1 < C + C • logtEff p\g - gt\ < C + C • logt • (Ef4 pE (g - gt )2}2 < C 1 1 < - + C-log t^-• (E (g - gt )2}2. Now we estimate the moments E (g - gt )2m. By the Ito formula for x] from [8] it is true that 2f xs-dxs +p12t + )2 = 0 0

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