Stochastic integral representation of the $L^2$ modulus of Brownian local time and a central limit theorem

Yaozhong Hu (University of Kansas)
David Nualart (University of Kansas)


The purpose of this note is to prove a central limit theorem for the $L^2$-modulus of continuity of the Brownian local time obtained in [3], using techniques of stochastic analysis. The main ingredients of the proof are an asymptotic version of Knight's theorem and the Clark-Ocone formula for the $L^2$-modulus of the Brownian local time

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Pages: 529-539

Publication Date: November 13, 2009

DOI: 10.1214/ECP.v14-1511


  1. Barlow, M. T.; Yor, M. Semimartingale inequalities via the Garsia-Rodemich-Rumsey lemma, and applications to local times. J. Funct. Anal . 49 (1982), no. 2, 198--229. Math. Review MR0680660 (84f:60073)
  2. Borodin, A. N. Brownian local time Russian Math. Surveys 44 (1989), 1--51. Math. Review MR0998360 (90f:60138)
  3. Chen, X., Li, W., Marcus, M. B. and Rosen, J. A CLT for the $ L^{2}$ modulus of continiuty of Brownian local time.
  4. Nualart, D. The Malliavin Calculus and Related Topics. Second edition. Springer Verlag, Berlin, 2006. Math. Review MR2200233 (2006j:60004)
  5. Ocone, D. Malliavin calculus and stochastic integral representation of diffusion processes. Stochastics } 12 (1984), 161--185. Math. Review MR0749372 (85m:60101)
  6. Pitman, J.; Yor, M. Asymptotic laws of planar Brownian motion. Ann. Probab. 14 (1986), no. 3, 733--779. Math. Review MR0841582 (88a:60145)
  7. Revuz, D.; Yor, M. Continuous martingales and Brownian motion. Third edition. Springer-Verlag, Berlin, 1999. Math. Review MR1725357 (2000h:60050)
  8. Rosen, J. A stochastic calculus proof of the CLT for the $L^2$ modulus of continuity of local time.

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