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Estimation for the change point of the volatility in a stochastic differential equation

17 June 2009
S. Iacus
Nakahiro Yoshida
ArXiv (abs)PDFHTML
Abstract

We consider a multidimensional It\^o process Y=(Yt)t∈[0,T]Y=(Y_t)_{t\in[0,T]}Y=(Yt​)t∈[0,T]​ with some unknown drift coefficient process btb_tbt​ and volatility coefficient σ(Xt,θ)\sigma(X_t,\theta)σ(Xt​,θ) with covariate process X=(Xt)t∈[0,T]X=(X_t)_{t\in[0,T]}X=(Xt​)t∈[0,T]​, the function σ(x,θ)\sigma(x,\theta)σ(x,θ) being known up to θ∈Θ\theta\in\Thetaθ∈Θ. For this model we consider a change point problem for the parameter θ\thetaθ in the volatility component. The change is supposed to occur at some point t∗∈(0,T)t^*\in (0,T)t∗∈(0,T). Given discrete time observations from the process (X,Y)(X,Y)(X,Y), we propose quasi-maximum likelihood estimation of the change point. We present the rate of convergence of the change point estimator and the limit thereoms of aymptotically mixed type.

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