An Identification Problem for Systems with Additive Fractional Brownian Field
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AN IDENTIFICATION PROBLEM FOR SYSTEMS WITH ADDITIVE FRACTIONAL BROWNIAN FIELD O. M. Dåriyeva1† and S. P. Shpyga1‡
UDC 519.21
Abstract. The authors consider the problem of nonparametric estimation (identification) for a wide class of random fields on a plane satisfying solutions of stochastic partial differential equations with additive fractional Brownian field. The asymptotic properties of the estimate of drift parameter are analyzed with the use of the method of sieves. Keywords: identification problem, fractional Brownian field, drift parameter, method of sieves.
Problems of estimation of functions, which are coefficients of differential operators in parabolic and hyperbolic stochastic partial differential equations (SPDE) often arise in applied models in meteorology, physical chemistry, economics, oceanography, etc. The asymptotic properties of the estimates of SPDE parameters were studied, in particular, in [1–5]. To analyze regression models with semimartingales, Grenander used in 1981 the so-called sieve method [6], which implies maximizing the likelihood function on an increasing sequence of finite-dimensional subspaces. The sieve size tends to infinity as sample size grows, and the sequence of bounded maximum likelihood estimates of parameter is consistent and asymptotically normal. This method was used in [7] to analyze nonparametric estimates, in particular, functions of a Wiener process, estimate of drift coefficient of linear diffusion, mean value estimate for Gaussian process [7], and it was applied in [8] to estimate the infinite-dimensional parameter in the model of nonstationary linear diffusion. In the present paper, we will use the sieve method for the identification of SPDE drift parameter in a system with additive fractional Brownian field specified on a plane. Similar results were obtained by Rao in [9] for a one-dimensional system with additive fractional Brownian motion; a similar problem for SPDE with additive Brownian field is presented in [10]. Consider the SPDE
dX ( s, t ) = q( s, t ) X ( s, t ) dsdt + dB H ( s, t ), X (0, 0) = z, ( s, t ) Î[0, T ] 2 , where q( s, t ) Î L2 ([0, T ] 2 , dtds) , B H ( s, t ) is a fractional Brownian field with H = ( a, b ) , a, b Î (1 / 2, 1), z is a Gaussian random variable independent from B H ( s, t ) . In other words, X ( s, t ) satisfies the equation s t
X ( s, t ) = z + ò ò q( u, u ) X ( u, u ) dudu + B H ( s, t ) . 00
Let cq ( s, t ) = q( s, t ) X ( s, t ), ( s, t ) Î[0, T ] 2 , and assume that the trajectories cq are sufficiently smooth for the existence of random process 1
V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine, [email protected]; ‡[email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 3, May–June, 2016, pp. 182–190. Original article submitted December 3, 2015. †
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1060-0396/16/5203-0492 ©2016 Springer Science+Business Media New York
R q ( s, t ) = where
k a ( s, t ) =
d
d
dw sa dw tb
s t
ò ò k a (s, u)k b (t, u ) cq (u, u )dudu , 0 0
s1/ 2 - a ( t - s) 1/ 2
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