Statistical inference for models driven by 𝑛-th order fractional Brownian motion

Abstract

We consider the following stochastic integral equation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X left-parenthesis t right-parenthesis equals mu t plus sigma integral Subscript 0 Superscript t Baseline phi left-parenthesis s right-parenthesis d upper B Subscript upper H Superscript n Baseline left-parenthesis s right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi> μ </mml:mi> <mml:mi>t</mml:mi> <mml:mo>+</mml:mo> <mml:mi> σ </mml:mi> <mml:msubsup> <mml:mo> ∫ </mml:mo> <mml:mn>0</mml:mn> <mml:mi>t</mml:mi> </mml:msubsup> <mml:mi> φ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>d</mml:mi> <mml:msubsup> <mml:mi>B</mml:mi> <mml:mi>H</mml:mi> <mml:mi>n</mml:mi> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">X(t)=\mu t + \sigma \int _0^t \varphi (s) dB_H^n(s)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t greater-than-or-equal-to 0"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">t\geq 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi"> <mml:semantics> <mml:mi> φ </mml:mi> <mml:annotation encoding="application/x-tex">\varphi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a known function and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Subscript upper H Superscript n"> <mml:semantics> <mml:msubsup> <mml:mi>B</mml:mi> <mml:mi>H</mml:mi> <mml:mi>n</mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">B^n_H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -th order fractional Brownian motion. We provide explicit maximum likelihood estimators for both <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu"> <mml:semantics> <mml:mi> μ </mml:mi> <mml:annotation encoding="application/x-tex">\mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma squared"> <mml:semantics> <mml:msup> <mml:mi> σ </mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\sigma ^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , then we formulate explicitly a least squares estimator for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu"> <mml:semantics> <mml:mi> μ </mml:mi> <mml:annotation encoding="application/x-tex">\mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and an estimator for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma squared"> <mml:semantics> <mml:msup> <mml:mi> σ </mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\sigma ^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by using power variations method. The consistency and asymptotic normality are established for those estimators when the number of observations or the time horizon is sufficiently large.