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 $\hat \theta _\tau$, based on observation in [0, T] is defined in terms of stochastic and ordinary integrals. So, in practice, to compute $\hat \theta _\tau$one 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 $\hat \theta _{N, T}$(resp. $\tilde \theta _{N, T}$) 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‖=δ_N goes to zero, $P - \lim \hat \theta _{N, T} = P - \lim \tilde \theta _{N, T} = \hat \theta _{\rm T} , \delta _N^{ - 1/2} (\hat \theta _{N, T} )$ and $\delta _N^{ - 1/2} (\tilde \theta _{N, T} - \hat \theta _T )$ both bounded in probability.