A Tanaka formula for the derivative of intersection local time in $\reals^1$

Greg Markowsky

A Tanaka formula for the derivative of intersection local time in $\reals^1$

Probability 2007-05-23 v1

Authors: Greg Markowsky

Abstract

Let $B_t$ be a one dimensional Brownian motion, and let $\alpha'$ denote the derivative of the intersection local time of $B_t$ as defined in Jay Rosen's work (see references). The object of this paper is to prove the following formula $(1/2)\alpha'_t(x) + (1/2)sgn(x)t = \int_0^t L_s^{B_s - x}dB_s - \int_0^t sgn(B_t - B_u - x) du$ which was given as a formal identity by Rosen without proof.

Keywords

tanaka formula intersection local time brownian motion derivative one dimension

Cite

@article{arxiv.math/0609084,
  title  = {A Tanaka formula for the derivative of intersection local time in $\reals^1$},
  author = {Greg Markowsky},
  journal= {arXiv preprint arXiv:math/0609084},
  year   = {2007}
}

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