Smoothness for the collision local time of two multidimensional bifractional Brownian motions

Guangjun Shen; Litan Yan; Chao Chen

Czechoslovak Mathematical Journal (2012)

  • Volume: 62, Issue: 4, page 969-989
  • ISSN: 0011-4642

Abstract

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Let B H i , K i = { B t H i , K i , t ≥ 0 } , i = 1 , 2 be two independent, d -dimensional bifractional Brownian motions with respective indices H i ∈ ( 0 , 1 ) and K i ∈ ( 0 , 1 ] . Assume d ≥ 2 . One of the main motivations of this paper is to investigate smoothness of the collision local time ℓ T = ∫ 0 T δ ( B s H 1 , K 1 - B s H 2 , K 2 ) d s , T > 0 , where δ denotes the Dirac delta function. By an elementary method we show that ℓ T is smooth in the sense of Meyer-Watanabe if and only if min { H 1 K 1 , H 2 K 2 } < 1 / ( d + 2 ) .

How to cite

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Shen, Guangjun, Yan, Litan, and Chen, Chao. "Smoothness for the collision local time of two multidimensional bifractional Brownian motions." Czechoslovak Mathematical Journal 62.4 (2012): 969-989. <http://eudml.org/doc/246327>.

@article{Shen2012,
abstract = {Let $B^\{H_\{i\},K_i\}=\lbrace B^\{H_\{i\},K_i\}_t, t\ge 0 \rbrace $, $i=1,2$ be two independent, $d$-dimensional bifractional Brownian motions with respective indices $H_i\in (0,1)$ and $K_i\in (0,1]$. Assume $d\ge 2$. One of the main motivations of this paper is to investigate smoothness of the collision local time \[ \ell \_T=\int \_\{0\}^\{T\}\delta (B\_\{s\}^\{H\_\{1\},K\_1\}-B\_\{s\}^\{H\_\{2\},K\_2\}) \{\rm d\} s, \qquad T>0, \] where $\delta $ denotes the Dirac delta function. By an elementary method we show that $\ell _T$ is smooth in the sense of Meyer-Watanabe if and only if $\min \lbrace H_\{1\}K_1,H_\{2\}K_2\rbrace <\{1\}/\{(d+2)\}$.},
author = {Shen, Guangjun, Yan, Litan, Chen, Chao},
journal = {Czechoslovak Mathematical Journal},
keywords = {bifractional Brownian motion; collision local time; intersection local time; chaos expansion; bifractional Brownian motion; collision local time; intersection local time; chaos expansion},
language = {eng},
number = {4},
pages = {969-989},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Smoothness for the collision local time of two multidimensional bifractional Brownian motions},
url = {http://eudml.org/doc/246327},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Shen, Guangjun
AU - Yan, Litan
AU - Chen, Chao
TI - Smoothness for the collision local time of two multidimensional bifractional Brownian motions
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 4
SP - 969
EP - 989
AB - Let $B^{H_{i},K_i}=\lbrace B^{H_{i},K_i}_t, t\ge 0 \rbrace $, $i=1,2$ be two independent, $d$-dimensional bifractional Brownian motions with respective indices $H_i\in (0,1)$ and $K_i\in (0,1]$. Assume $d\ge 2$. One of the main motivations of this paper is to investigate smoothness of the collision local time \[ \ell _T=\int _{0}^{T}\delta (B_{s}^{H_{1},K_1}-B_{s}^{H_{2},K_2}) {\rm d} s, \qquad T>0, \] where $\delta $ denotes the Dirac delta function. By an elementary method we show that $\ell _T$ is smooth in the sense of Meyer-Watanabe if and only if $\min \lbrace H_{1}K_1,H_{2}K_2\rbrace <{1}/{(d+2)}$.
LA - eng
KW - bifractional Brownian motion; collision local time; intersection local time; chaos expansion; bifractional Brownian motion; collision local time; intersection local time; chaos expansion
UR - http://eudml.org/doc/246327
ER -

References

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