Nonparametric estimation of trend for stochastic differential equations driven by fractional Brownian motion

Author： Mishra M.  

Publisher： Springer Publishing Company

ISSN： 1387-0874

Source： Statistical Inference for Stochastic Processes, Vol.14, Iss.2, 2011-05, pp. : 101-109

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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 $${Hin (1/2,1)}$$ 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 estimatorand 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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