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Approximate discrete-time schemes for statistics of diffusion processes. (English) Zbl 0704.62072

Let \(X=(X_ t,t\geq 0)\) be a diffusion type process which is a solution of the stochastic differential equation \[ (*)\quad dX_ t=b(X_ t,\theta)dt+\sigma dw_ t. \] Here \(w=(w_ t,t\geq 0)\) is a standard Wiener process, b(\(\cdot)\) is a given nonlinear function and \(\theta\), \(\sigma\) are unknown parameters. Let \(\Delta_ n\), \(n\geq 1\), be a sequence of (positive) numbers converging to zero or remaining constant. It is assumed that we have in our disposal the discrete observations \(\{X(k\Delta_ k)\), \(k=1,...,n\}\) obtained from a suitable first-order approximation for (*). Thus the general problem is to use these observations in finding estimators for \(\theta\), \(\sigma^ 2\) and study their asymptotic behavior. More precisely, the likelihood function of \(\{X(k\Delta_ k)\}\) is used as a contrast function in order to construct the contrast estimators \(\theta_ n\) and \(\sigma^ 2_ n.\)
Under suitable conditions the consistency (in \(L^ 2\)-sense) as well as the asymptotic normality are established. The case of a discretization with a constant sampling interval \(\Delta\) is analyzed in more detail in a separate section. Several related topics are discussed. Finally, one can conclude that the present paper contains the answers to interesting questions from statistics of diffusion processes by using discrete observations.
Reviewer: J.M.Stoyanov

MSC:

62M05 Markov processes: estimation; hidden Markov models
60J60 Diffusion processes
62F12 Asymptotic properties of parametric estimators
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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