Change point testing for the drift parameters of a periodic mean reversion process

In this paper we investigate the problem of detecting a change in the drift parameters of a generalized Ornstein–Uhlenbeck process which is defined as the solution of <Equation ID="Equ23"> <EquationSource Format="TEX">$$\begin{aligned} dX_t=(L(t)-\alpha X_t) dt + \sigma dB_t \end{aligned}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink" display="block"> <mrow> <mtable columnspacing="0.5ex"> <mtr> <mtd columnalign="right"> <mrow> <mi>d</mi> <msub> <mi>X</mi> <mi>t</mi> </msub> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mi>L</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <mi mathvariant="italic">α</mi> <msub> <mi>X</mi> <mi>t</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> <mi>t</mi> <mo>+</mo> <mi mathvariant="italic">σ</mi> <mi>d</mi> <msub> <mi>B</mi> <mi>t</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> </math> </EquationSource> </Equation>and which is observed in continuous time. We derive an explicit representation of the generalized likelihood ratio test statistic assuming that the mean reversion function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$L(t)$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>L</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> is a finite linear combination of known basis functions. In the case of a periodic mean reversion function, we determine the asymptotic distribution of the test statistic under the null hypothesis. Copyright Springer Science+Business Media Dordrecht 2014