p-variation and Chung's LIL of sub-bifractional Brownian motion and applications

Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>S</m:mi> </m:mrow> <m:mrow> <m:mi>H</m:mi> <m:mo>,</m:mo> <m:mi>K</m:mi> </m:mrow> </m:msup> <m:mo>=</m:mo> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi>S</m:mi> </m:mrow> <m:mrow> <m:mi>H</m:mi> <m:mo>,</m:mo> <m:mi>K</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mi>t</m:mi> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mo>}</m:mo> </m:mrow> </m:math> {S}H,K=\{{S}H,K(t),t≥ 0\} be the sub-bifractional Brownian motion, with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>H</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mspace width="0.85358pt"/> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> H∈ (0,{0.85358pt}1) and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>K</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mspace width="0.85358pt"/> <m:mn>1</m:mn> </m:mrow> <m:mo>]</m:mo> </m:mrow> <m:mo>.</m:mo> </m:math> K∈ (0,{0.85358pt}1]. We investigate its <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> </m:math> p -variation and Chung’s law of the iterated logarithm. In addition, we give some applications of these properties.