Geodesic
On Some Properties of the Fractional Derivative of the Brownian Local Time
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Noncommutative Analysis and Quantum Information Theory, Tome 324 (2024), pp. 109-123

Voir la notice de l'article provenant de la source Math-Net.Ru

We study the properties of the fractional derivative $D_\alpha l(t,x)$ of order $\alpha 1/2$ of the Brownian local time $l(t,x)$ with respect to the variable $x$. This derivative is understood as the convolution of the local time with the generalized function $|x|^{-1-\alpha }$. We show that $D_\alpha l(t,x)$ appears naturally in Itô's formula for the process $|w(t)|^{1-\alpha }$. Using the martingale technique, we also study the limit behavior of $D_\alpha l(t,x)$ as $t\to \infty $.
Keywords: stochastic processes, local time, fractional derivative. 
@article{TM_2024_324_a10,
     author = {I. A. Ibragimov and N. V. Smorodina and M. M. Faddeev},
     title = {On {Some} {Properties} of the {Fractional} {Derivative} of the {Brownian} {Local} {Time}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {109--123},
     publisher = {mathdoc},
     volume = {324},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2024_324_a10/}
}
TY  - JOUR
AU  - I. A. Ibragimov
AU  - N. V. Smorodina
AU  - M. M. Faddeev
TI  - On Some Properties of the Fractional Derivative of the Brownian Local Time
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2024
SP  - 109
EP  - 123
VL  - 324
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_2024_324_a10/
LA  - ru
ID  - TM_2024_324_a10
ER  - 
%0 Journal Article
%A I. A. Ibragimov
%A N. V. Smorodina
%A M. M. Faddeev
%T On Some Properties of the Fractional Derivative of the Brownian Local Time
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2024
%P 109-123
%V 324
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_2024_324_a10/
%G ru
%F TM_2024_324_a10
I. A. Ibragimov; N. V. Smorodina; M. M. Faddeev. On Some Properties of the Fractional Derivative of the Brownian Local Time. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Noncommutative Analysis and Quantum Information Theory, Tome 324 (2024), pp. 109-123. http://geodesic.mathdoc.fr/item/TM_2024_324_a10/