On continuous and discrete sampling for parameter estimation in diffusion type processes

Abstract

This paper is concerned with the problem of estimation of the drift parameter ϑ in a linear stochastic differential equation, with constant coefficients. When continuous sampling of the solution process is available, the maximum likelihood estimate θˆτθˆτ{\textbackslash}textbackslashhat {\textbackslash}textbackslashtheta \_{\textbackslash}textbackslashtau, based on observation in [0, T] is defined in terms of stochastic and ordinary integrals. So, in practice, to compute θˆτθˆτ{\textbackslash}textbackslashhat {\textbackslash}textbackslashtheta \_{\textbackslash}textbackslashtauone has to approximate these integrals by appropriate finite sums which only depend on some discrete sampling in {t 0, t 1,…,t N } ⊂[0, T]. If θˆN,{TθˆN},T{\textbackslash}textbackslashhat {\textbackslash}textbackslashtheta _{N, T}(resp. θ{\textbackslash}{textasciitildeN},Tθ{\textbackslash}{textasciitildeN},T{\textbackslash}textbackslashtilde {\textbackslash}textbackslashtheta _{N, T}) denotes the resulting estimate (resp, maximum likelihood estimate based on observations at t 0,…,t N ), we show that, when Max‖t i+1−t i‖=δN goes to zero, P−{limθˆN},T=P−limθ{\textbackslash}{textasciitildeN},T=θˆT,δ−1/2N(θˆN,T)P−{limθˆN},T=P−limθ{\textbackslash}{textasciitildeN},T=θˆT,δN−1/2(θˆN,T)P - {\textbackslash}textbackslashlim {\textbackslash}textbackslashhat {\textbackslash}textbackslashtheta _{N, T} = P - {\textbackslash}textbackslashlim {\textbackslash}textbackslashtilde {\textbackslash}textbackslashtheta _{N, T} = {\textbackslash}textbackslashhat {\textbackslash}textbackslashtheta _\{{\textbackslash}textbackslashrm T} , {\textbackslash}textbackslashdelta _Nˆ{ - 1/2} ({\textbackslash}textbackslashhat {\textbackslash}textbackslashtheta _{N, T} ) and δ−1/2N(θ{\textbackslash}{textasciitildeN},T−θˆT)δN−1/2(θ{\textbackslash}{textasciitildeN},T−θˆT){\textbackslash}textbackslashdelta _Nˆ{ - 1/2} ({\textbackslash}textbackslashtilde {\textbackslash}textbackslashtheta _{N, T} - {\textbackslash}textbackslashhat {\textbackslash}textbackslashtheta _T ) both bounded in probability.