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Small-diffusion asymptotics for discretely sampled stochastic differential equations. (English) Zbl 1043.60050

The paper deals with a family of \(d\)-dimensional diffusion processes defined by the stochastic differential equations \[ dX_t=b(X_t,\alpha)\,dt+ \varepsilon\sigma(X_t,\beta)dw_t,\;t\in [0,1],\;\varepsilon\in (0,1],\quad X_0=x_0, \] where \((\alpha,\beta)\) belong to open bounded convex subsets of \(R^d\) and \(R^q,\) respectively. It is assumed that \(b\) and \(\sigma\) are known apart from the parameters \(\alpha\) and \(\beta.\) The data are discrete observations of \(X\) at \(n\) points \(t_k=k/n\) on the fixed interval \([0,1],\) \((X_{t_k})_{0\leq k\leq n}.\) The purpose of the paper is to estimate \(\alpha\) and \(\beta\) based on these observations. The authors obtain a consistent, asymptotically normal and asymptotically efficient estimator of \((\alpha,\beta),\) when \(\varepsilon\to 0\) and \(n\to+\infty\) simultaneously.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60F05 Central limit and other weak theorems
62F12 Asymptotic properties of parametric estimators
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