Abstract
Consider a stochastic process {X t , 0 ≤ t ≤ T} governed by a stochastic differential equation given by
where \({\{W_t^H, 0 \leq t \leq T\}}\) is a standard fractional Brownian motion with known Hurst parameter \({H\in (1/2,1)}\) and the function S(.) is an unknown function. Suppose the process {X t , 0 ≤ t ≤ T} is observed over the interval [0, T]. We consider the problem of nonparametric estimation of trend function S t = S(x t ) by a kernel type estimator
and study the asymptotic behaviour of the estimator as \({\epsilon \rightarrow 0}\). Here x t is the solution of the differential equation given above when \({\epsilon =0}\).
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Mishra, M.N., Prakasa Rao, B.L.S. Nonparametric estimation of trend for stochastic differential equations driven by fractional Brownian motion. Stat Inference Stoch Process 14, 101–109 (2011). https://doi.org/10.1007/s11203-010-9051-x
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DOI: https://doi.org/10.1007/s11203-010-9051-x
Keywords
- Stochastic differential equation
- Trend
- Nonparametric estimation
- Kernel method
- Small noise
- Fractional Brownian motion