Parameter maximum likelihood estimation problem for time periodic modulated drift Ornstein Uhlenbeck processes

<Para ID="Par1">In this paper we investigate the large-sample behaviour of the maximum likelihood estimate (MLE) of the unknown parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\theta $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="italic">θ</mi> </math> </EquationSource> </InlineEquation> for processes following the model <Equation ID="Equ38"> <EquationSource Format="TEX">$$\begin{aligned} d\xi _{t}=\theta f(t)\xi _{t}\,dt+d\mathrm {B}_t, \end{aligned}$$</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 mathvariant="italic">ξ</mi> <mi>t</mi> </msub> <mo>=</mo> <mi mathvariant="italic">θ</mi> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi mathvariant="italic">ξ</mi> <mi>t</mi> </msub> <mspace width="0.166667em"/> <mi>d</mi> <mi>t</mi> <mo>+</mo> <mi>d</mi> <msub> <mi mathvariant="normal">B</mi> <mi>t</mi> </msub> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math> </EquationSource> </Equation>where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$f:{\mathbb {R}}\rightarrow {\mathbb {R}}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>f</mi> <mo>:</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math> </EquationSource> </InlineEquation> is a continuous function with period, say <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$P>0$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>P</mi> <mo>></mo> <mn>0</mn> </mrow> </math> </EquationSource> </InlineEquation>. Here the periodic function <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$f(\cdot )$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> is assumed known. We establish the consistency of the MLE and we point out its minimax optimality. These results comply with the well-established case of an Ornstein Uhlenbek process when the function <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$f(\cdot )$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> is constant. However the case when <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$\int _0^P f(t)dt=0$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>P</mi> </msubsup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </math> </EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$f(\cdot )$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> is not identically null presents some special features. For instance in this case whatever is the value of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$$\theta $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="italic">θ</mi> </math> </EquationSource> </InlineEquation>, the rate of convergence of the MLE is <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$$T$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>T</mi> </math> </EquationSource> </InlineEquation> as in the case when <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$$\theta =0$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi mathvariant="italic">θ</mi> <mo>=</mo> <mn>0</mn> </mrow> </math> </EquationSource> </InlineEquation> and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$$\int _0^Pf(t)dt\ne 0$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>P</mi> </msubsup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> <mi>t</mi> <mo>≠</mo> <mn>0</mn> </mrow> </math> </EquationSource> </InlineEquation>. Copyright Springer Science+Business Media Dordrecht 2015