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The quadratic covariation for a weighted fractional Brownian motion

Xichao Sun

Department of Mathematics and Physics, Bengbu University, 1866 Caoshan Rd., Bengbu 233030, P. R. China

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Litan Yan

Department of Mathematics, Donghua University, 2999 North Renmin Rd., Songjiang, Shanghai 201620, P. R. China

E-mail Address: litanyan@dhu.edu.cn

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, and

Qinghua Zhang

Department of Mathematics, Donghua University, 2999 North Renmin Rd., Songjiang, Shanghai 201620, P. R. China

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https://doi.org/10.1142/S0219493717500290Cited by:3 (Source: Crossref)

Abstract

Let $B^{a, b}$ be a weighted fractional Brownian motion with indices $a$ and $b$ satisfying $a > - 1,$ $- 1 < b < 0,$ $| b | < 1 + a$ . In this paper, motivated by the asymptotic property

E [(B a, b s + 𝜀 - B a, b s) 2] = O (𝜀 1 + b) ≁ 𝜀 1 + a + b = E [(B a, b 𝜀) 2], 𝜀 \to 0 <math display="block" altimg="eq-00007.gif"><mrow><mi>E</mi><mo class="MathClass-open" stretchy="false">[</mo><msup><mrow><mo class="MathClass-open" stretchy="false">(</mo><msubsup><mrow><mi>B</mi></mrow><mrow><mi>s</mi><mo stretchy="false" class="MathClass-bin">+</mo><mi>𝜀</mi></mrow><mrow><mi>a</mi><mo class="MathClass-punc">,</mo><mi>b</mi></mrow></msubsup><mo stretchy="false" class="MathClass-bin">-</mo><msubsup><mrow><mi>B</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>a</mi><mo class="MathClass-punc">,</mo><mi>b</mi></mrow></msubsup><mo class="MathClass-close" stretchy="false">)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo class="MathClass-close" stretchy="false">]</mo><mo class="MathClass-rel">=</mo><mi>O</mi><mo class="MathClass-open" stretchy="false">(</mo><msup><mrow><mi>𝜀</mi></mrow><mrow><mn>1</mn><mo stretchy="false" class="MathClass-bin">+</mo><mi>b</mi></mrow></msup><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-rel">≁</mo><msup><mrow><mi>𝜀</mi></mrow><mrow><mn>1</mn><mo stretchy="false" class="MathClass-bin">+</mo><mi>a</mi><mo stretchy="false" class="MathClass-bin">+</mo><mi>b</mi></mrow></msup><mo class="MathClass-rel">=</mo><mi>E</mi><mo class="MathClass-open" stretchy="false">[</mo><msup><mrow><mo class="MathClass-open" stretchy="false">(</mo><msubsup><mrow><mi>B</mi></mrow><mrow><mi>𝜀</mi></mrow><mrow><mi>a</mi><mo class="MathClass-punc">,</mo><mi>b</mi></mrow></msubsup><mo class="MathClass-close" stretchy="false">)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo class="MathClass-close" stretchy="false">]</mo><mo class="MathClass-punc">,</mo><mspace width="1em" class="quad"></mspace><mi>𝜀</mi><mo class="MathClass-rel">\to</mo><mn>0</mn></mrow></math>

for all

$s > 0$ , we consider the generalized quadratic covariation

${[f (B^{a, b}), B^{a, b}]}^{(a, b)}$ defined by

[f(Ba,b),Ba,b](a,b)t=lim𝜀↓01+a+b𝜀1+b∫t+𝜀𝜀{f(Ba,bs+𝜀)−f(Ba,bs)}(Ba,bs+𝜀−Ba,bs)sbds,<math display="block" altimg="eq-00010.gif"><mrow><msubsup><mrow><mo class="MathClass-open" stretchy="false">[</mo><mi>f</mi><mo class="MathClass-open" stretchy="false">(</mo><msup><mrow><mi>B</mi></mrow><mrow><mi>a</mi><mo class="MathClass-punc">,</mo><mi>b</mi></mrow></msup><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-punc">,</mo><msup><mrow><mi>B</mi></mrow><mrow><mi>a</mi><mo class="MathClass-punc">,</mo><mi>b</mi></mrow></msup><mo class="MathClass-close" stretchy="false">]</mo></mrow><mrow><mi>t</mi></mrow><mrow><mo class="MathClass-open" stretchy="false">(</mo><mi>a</mi><mo class="MathClass-punc">,</mo><mi>b</mi><mo class="MathClass-close" stretchy="false">)</mo></mrow></msubsup><mo class="MathClass-rel">=</mo><munder class="msub"><mrow><mo class="qopname">lim</mo></mrow><mrow><mi>𝜀</mi><mi>↓</mi><mn>0</mn></mrow></munder><mfrac><mrow><mn>1</mn><mo stretchy="false" class="MathClass-bin">+</mo><mi>a</mi><mo stretchy="false" class="MathClass-bin">+</mo><mi>b</mi></mrow><mrow><msup><mrow><mi>𝜀</mi></mrow><mrow><mn>1</mn><mo stretchy="false" class="MathClass-bin">+</mo><mi>b</mi></mrow></msup></mrow></mfrac><msubsup><mrow><mo class="qopname">∫</mo></mrow><mrow><mi>𝜀</mi></mrow><mrow><mi>t</mi><mo stretchy="false" class="MathClass-bin">+</mo><mi>𝜀</mi></mrow></msubsup><mo class="MathClass-open" stretchy="false">{</mo><mi>f</mi><mo class="MathClass-open" stretchy="false">(</mo><msubsup><mrow><mi>B</mi></mrow><mrow><mi>s</mi><mo stretchy="false" class="MathClass-bin">+</mo><mi>𝜀</mi></mrow><mrow><mi>a</mi><mo class="MathClass-punc">,</mo><mi>b</mi></mrow></msubsup><mo class="MathClass-close" stretchy="false">)</mo><mo stretchy="false" class="MathClass-bin">−</mo><mi>f</mi><mo class="MathClass-open" stretchy="false">(</mo><msubsup><mrow><mi>B</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>a</mi><mo class="MathClass-punc">,</mo><mi>b</mi></mrow></msubsup><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-close" stretchy="false">}</mo><mo class="MathClass-open" stretchy="false">(</mo><msubsup><mrow><mi>B</mi></mrow><mrow><mi>s</mi><mo stretchy="false" class="MathClass-bin">+</mo><mi>𝜀</mi></mrow><mrow><mi>a</mi><mo class="MathClass-punc">,</mo><mi>b</mi></mrow></msubsup><mo stretchy="false" class="MathClass-bin">−</mo><msubsup><mrow><mi>B</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>a</mi><mo class="MathClass-punc">,</mo><mi>b</mi></mrow></msubsup><mo class="MathClass-close" stretchy="false">)</mo><msup><mrow><mi>s</mi></mrow><mrow><mi>b</mi></mrow></msup><mi>d</mi><mi>s</mi><mo class="MathClass-punc">,</mo></mrow></math>

provided the limit exists uniformly in probability. We construct a Banach space

$ℋ$ of measurable functions such that the generalized quadratic covariation exists in

$L^{2} (Ω)$ and the generalized Bouleau–Yor identity

[f(Ba,b),Ba,b](a,b)t=−1(1+b)𝔹(a+1,b+1)∫ℝf(x)ℒa,b(dx,t)<math display="block" altimg="eq-00013.gif"><mrow><msubsup><mrow><mo class="MathClass-open" stretchy="false">[</mo><mi>f</mi><mo class="MathClass-open" stretchy="false">(</mo><msup><mrow><mi>B</mi></mrow><mrow><mi>a</mi><mo class="MathClass-punc">,</mo><mi>b</mi></mrow></msup><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-punc">,</mo><msup><mrow><mi>B</mi></mrow><mrow><mi>a</mi><mo class="MathClass-punc">,</mo><mi>b</mi></mrow></msup><mo class="MathClass-close" stretchy="false">]</mo></mrow><mrow><mi>t</mi></mrow><mrow><mo class="MathClass-open" stretchy="false">(</mo><mi>a</mi><mo class="MathClass-punc">,</mo><mi>b</mi><mo class="MathClass-close" stretchy="false">)</mo></mrow></msubsup><mo class="MathClass-rel">=</mo><mo stretchy="false" class="MathClass-bin">−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mo class="MathClass-open" stretchy="false">(</mo><mn>1</mn><mo stretchy="false" class="MathClass-bin">+</mo><mi>b</mi><mo class="MathClass-close" stretchy="false">)</mo><mi>𝔹</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>a</mi><mo stretchy="false" class="MathClass-bin">+</mo><mn>1</mn><mo class="MathClass-punc">,</mo><mi>b</mi><mo stretchy="false" class="MathClass-bin">+</mo><mn>1</mn><mo class="MathClass-close" stretchy="false">)</mo></mrow></mfrac><msub><mrow><mo class="MathClass-op">∫</mo></mrow><mrow><mi>ℝ</mi></mrow></msub><mi>f</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>x</mi><mo class="MathClass-close" stretchy="false">)</mo><msup><mrow><mi mathvariant="script">ℒ</mi></mrow><mrow><mi>a</mi><mo class="MathClass-punc">,</mo><mi>b</mi></mrow></msup><mo class="MathClass-open" stretchy="false">(</mo><mi>d</mi><mi>x</mi><mo class="MathClass-punc">,</mo><mi>t</mi><mo class="MathClass-close" stretchy="false">)</mo></mrow></math>

holds for all

$f \in ℋ$ , where

$ℒ^{a, b} (x, t) = \int_{0}^{t} δ (B_{s}^{a, b} - x) d s^{1 + a + b}$ is the weighted local time of

$B^{a, b}$ and

$𝔹 (\cdot, \cdot)$ is the Beta function.

Keywords:

AMSC: 60G15, 60H05, 60G17

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