Adaptive prediction of stochastic differential equations with unknown parameters.pdf Prediction is a momentous problem in modern world. Distinctive fields of science, for instance, economics, financial mathematics, engineering, biology etc. appeal to mathematical tools to get predictors of the real dynamic processes they explore. If a model allows making predictions of high statistical quality it is considered to be beneficial. Since models of dynamic systems often have unknown parameters, we have to deal with estimation problem in order to construct adaptive predictors. The quality of adaptive prediction significantly depends on a choice of estimators of the model parameters. Adaptive prediction problem for discrete-time systems was solved in [1] on the basis of truncated estimators proposed in [2]. In this paper we present analytically investigated predictors of continuous-time systems constructed on the basis of truncated estimators of dynamic parameters. 1. Prediction of Ornstein-Uhlenbeck process Assume the model dxt = axtdt + dwt, t > 0, (1) with an unknown parameter a, where x0 is zero mean random variable having finite moments of all order, wt is the standard Wiener process, x0 and wt are mutually independent. Suppose that the process (1) is stable, i.e. the parameter a < 0. Note that in this case for every m > 1 sup Exfm «». (2) t>0 The purpose is to construct a predictor for xt by observations xt-u = (xs )0 0 we have the representation x, =Xx, +£,,, , t > u, t t-u ~t,t-u ' ' t where X = eau, ^,t-u = J ea(t-s)dws, E^t_u = 0 and a2 := E\^ = 2-[e2au -1] . t-u Optimal in the mean square sense predictor is the conditional mathematical expectation xt =Xx, , t > u. t t-u ' As in practice the parameter a and, as follows, X 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 estimator at of unknown parameter а. Define adaptive predictor as xt (t - u) = Xt xt , t > u, (3) t \ / t-u t-u' ' V ' where Xt-u = ea,-uu, t > u; ait = proj(-M 0]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 v ft J xvdx, \ (4) at = - t t J xv2 dv > t log 11 J xv dv (5) (6) Lemma. Assume the model (1). Then for t - u > s0 := exp(2|a\)and some numbers C estimators at and Xt have the properties: Cp E ( - a))p tlog-11 - a -x| Jx)dv < tlog-11 a - a=- gt =■ J xv2 dv Define 1 Jx)dv,g=- )a,f = 7 Ь^. Then • x[gt > log-11] + a2p • P[gt log-11]=-(1+]T • g V k=1 2 p k л f (g -_g t У gk J x[ ft > log-11] + f- I1 = E < C C - < - + C • logt • Ef2 p •! g - gt\< - + C • logt • (Ef4 p • E (g - gt )2)^ < C 1 1 < - + C • logt • - • (E (g - gt )2)2. By the Ito formula for xf it is true that dxt = 2axt dt + 2xtdwt + dt and then 1 X1 1 t gt -g=--;--Jxtdwt■ 2a t at о Thus, by making use of the properties of the Ito integral and (2) for every m > 1 it holds с E(gt - gГ < 2. (8) tm Then I1 < C + C-^, t > u, (9) tP s 0 we have I2P \4 p , С 12 < a2p • P(gt - g\ > g - log-11)< a 1 • E(gt - g)4p < -C.. (10) ( - log t) t From (7), (9), (10) and definition of at the property (5) follows. The assertion (6) follows from the obvious inequality At -A u + exp (2L ). 2.1. One-parameter delay differential equation In this section we consider the differential equation with time-delay of a special structure. Assume the model dxt = bxt_rdt + dwt, t > 0, (12) with an unknown parameter b such that the process (12) is stable. Note it is stable when the parameter b e (-я /2,0), see [3]. Now we construct optimal and adaptive predictors for the process (12). Optimal in the mean square sense predictor is the conditional mathematical expectation z(k )(t - u ) = E (xt|xt-u) that satisfies the following equation [t-(u-r )^At t z(\t - u) = xt-u + b J xv-rdv + b J z{X (t - u)dv + t-u [t-(u-kr k-1 t + b Y J z()_{k_i)r (t - u), kr < u < (( + 1)r, t > u. (13) i=1t-r Here алр means the minimum between a and p. Since the parameter b in the definition of the optimal predictors z{k )(t -u)is unknown, we define the adaptive predictor by formula (13) replacing the unknown b with bt-u, where bt-u is the projection Lu = pro/j-^/2,o] bt-u (14) of the truncated estimator of the parameter b which is defined as follows t J xv-rdxv Г t \ bt = rt-Jx^-rdv > tlog-11 , t > max(u,r). J xldv ^r J r да Define the numbers c2 =J x2 (v)dv and s0 = max | r,exp (а-2 , where x0 (•) is the solution of the characteris- 0 tic equation x0 (v) = bx0 (v - r), v > 0 of the process (12) with x0 (0) = 1 and x0 (v) = 0, when v e [-r,0). For t > u + s0 estimators (14) have the properties E(bt - b)2P < ^, P > 1 which can be proven similarly to Lemma. Denote the adaptive prediction error and rewrite it in the form e(k)((-u):= xt -Z(k)(t-«) = e0(t-u)+ ё(к)(t-u), where et0((-u)= xt -E(xt | xt-u) and )((-u)= z(k)(t-u)-Z(k)(t-u). Then for every fixed k > 0 the following limit inequalities hold Шп t • (E (e(k) (t - u))2 - c02) < C. If it is known that b e[b0, b1 ], -я/2 < b0 < b1 < 0, then for t - u > s1 = max j r,exp (c-2 )|, where c° = ini^ ^c^ the non-asymptotic property is fulfilled 'K )(t - u ))2 E(вГ(t-u)) -С22 ^. 3. Simulations To confirm the convergence of the truncated estimators (4) and the properties of predictors (3) constructed on the basis of these estimators we made the simulations. For this purpose we used the software package MATLAB. In Tables I, II the average 1 Ю0 л 4 aT = - • 2 aT (к) 100 к=1 of estimators aT (к) constructed according to (4) by the к-th realization x(к) = (x^^ к = 1,...,100 of the process (1) and its quality characteristics (empirical mean square error): 2 1 100/ / 4 s2a(T) = - z(iT (к)- a)2 100 к=1 for different durations of observations T are given. Define the discretization step h and the number of discrete observations N = hrlT. T a b l e 1 Estimators aT and their quality characteristics S2(T) for h = 0,1 A N = 1000, T = 100 N = 2000, T = 200 N = 5000, T = 500 aT s2(T) aT s2a(T ) aj sO;(T) -0,3 -0,3135 0,0065 -0,3051 0,0030 -0,3070 0,0011 -0,5 -0,5113 0,0082 -0,5180 0,0059 -0,4969 0,0017 -0,8 -0,8283 0,0154 -0,8034 0,0074 -0,7949 0,0033 -1 -1,0126 0,0186 -0,9979 0,0111 -1,0017 0,0051 T a b l e 2 Estimators aT and their quality characteristics (T) for h = 0,15 A N = 1000, T = 150 N = 2000, T = 300 N = 5000, T = 750 aj s2(T) aj s2a(T ) aj sO;(T) -0,3 -0,3054 0,0050 -0,3098 0,0019 -0,3042 0,0001 -0,5 -0,5107 0,0053 -0,5081 0,0036 -0,5050 0,0014 -0,8 -0,8263 0,0124 -0,8114 0,0042 -0,8041 0,0020 -1 -1,0146 0,0111 -1,0165 0,0061 -1,0122 0,0028 According to the simulation results given in tables 1, 2 we can say that the empirical mean square error (MSE) s2 (t) of the estimators aT becomes less with growth of the sample size (see fig. 1 as well). It means that the estimator's value becomes closer to the true value of the parameter. This fact proves the effectiveness of the presented estimation procedure. There were also constructed predictors of the observed process by the obtained estimators of a = -0,5 with u = 1, h = 0,1: Xt (t-1) = ea'-1 xt-1, t > 1. The results are presented on the graphs below. The solid one identifies the real values of the observed process, the stipple one shows the predictor's value (see fig. 2). For the fig. 3 there was computed the empirical mean square prediction error that is equal to 1 100 2 _ sx2 (l ) = I (()- Xh\lh -1)), / = 10, N with T = 100, N = 1000, u = 1, h = 0,1. It shows that the prediction error converges to the value ст2 = 1 - e-1 « 0,6321 which corresponds to the obtained theoretical results. Fig. 1. Empirical MSE S2a(T) for a = -0,5; T = 100; h = 0,1; N = 1000 Fig. 2. Process xt (-) and adaptive predictor Xt (t-1) (. . . ) Fig. 3. Empirical MSE s2 (l) Conclusion Adaptive prediction problem of the Ornstein-Uhlenbeck process and one-parameter linear stochastic delay differential equation is solved. Non-asymptotic upper boundary of the prediction error is found. It is shown that this boundary is inverse proportional to the duration of observations. Non-asymptotic properties of adaptive predictors are obtained due to the usage of the truncated estimators [2] of the unknown parameters constructed by samples of fixed size. This method can be applied to various problems of parametric and non-parametric statistics. In this paper it is first employed to continuous-time systems. The obtained results can be used for the optimization problem of proposed predictors in the sense of special risk functions similar to discrete-time models [1].

Скачать электронную версию публикации