Diffusion Process with Evolution and its Parameter Estimation
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DIFFUSION PROCESS WITH EVOLUTION AND ITS PARAMETER ESTIMATION V. S. Koroliuk,1 D. Koroliouk,2† and S. O. Dovgyi2‡
UDC 519.24
Abstract. A discrete Markov process in an asymptotic diffusion environment with a uniformly ergodic embedded Markov chain can be approximated by an Ornstein–Uhlenbeck process with evolution. The drift parameter estimation is obtained using the stationarity of the Gaussian limit process. Keywords: discrete Markov process, diffusion approximation, asymptotic diffusion environment, Ornstein–Uhlenbeck process, phase merging, drift parameter estimation. We consider a random evolution z( t ) , t ³ 0 , that depends on a random environment Y ( t ) , t ³ 0 , which in turn, is switched by an embedded Markov chain X k , k ³ 0 . Below, we will explain the relation between the continuous-time t ³ 0 and the discrete-time k ³ 0 . The purpose of this study is to prove the convergence (in distribution) of the process z( t ) , t ³ 0 , to the Ornstein–Uhlenbeck process under some scaling of the process and its time parameter. The limit will be considered by a small series parameter e > 0 , e ® 0 . ASYMPTOTIC DIFFUSION ENVIRONMENT Consider a discrete Markov process in a semi-Markov asymptotic diffusion environment, defined by the solution of the following scaled difference stochastic equation:
z e ( t ne+ 1 ) = - e 2V (Y ne ) z e ( t ne ) + es (Y ne ) D m e ( t ne+ 1 ) ,
(1)
where t ne := ne 2 ; hence t ne+ 1 = t ne + e 2 , n > 0 , e > 0 , for the process increments Dz e ( t ne+ 1 ): = z e ( t ne+ 1 ) - z e ( t ne ) , n ³ 0 . The asymptotic diffusion environment Y ne , n ³ 0 , is also a random evolution process generated by the solution of the following scaled difference evolutionary equation:
DY e ( t ne+ 1 ) = eA0 (Y ne ; X ne ) + e 2 A (Y ne ; X ne ) , n ³ 0 ,
(2)
with the embedded Markov chain X ne : = X ( t ne ) , n ³ 0 . The terms A0 ( y; x ) and A ( y; x ) are Lipschitz functions, together with the first derivative A0¢ y ( y; x ) . Here, the predictable evolutional component in (1) is defined by the following conditional expectation [1]:
V (Y ne ) z e ( t ne ) : = E [ D z e ( t ne+ 1 ) | Y ne , z e ( t ne )] z e ( t ne ) , where the drift regression function V ( z ) is assumed to be positive: V ( z ) > 0 "z . 1
Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine. 2Institute of Telecommunications and Global Information Space, National Academy of Sciences of Ukraine, Kyiv, Ukraine, † [email protected]; ‡[email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 5, September–October, 2020, pp. 55–62. Original article submitted March 22, 2020. 732
1060-0396/20/5605-0732 ©2020 Springer Science+Business Media, LLC
e e The martingale difference D m e ( t n+1 ) , n ³ 1 , generated by the process D z e ( t n+1 ) , n ³ 1 , is defined by the
following conditional second moment:
- e 2 s 2 (Y ne ) : = E [( D z e ( t ne+ 1 ) + e 2V (Y ne ) z e ( t ne )) 2 | Y ne ]. The embedded Markov chain X ne : = X ( t ne ) , t ne := ne 2 , n ³ 0 , is supposed to be
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