On the divergence of certain integrals of the Wiener process

Lawrence A. Shepp; John R. Klauder; Hiroshi Ezawa

Annales de l'institut Fourier (1974)

  • Volume: 24, Issue: 2, page 189-193
  • ISSN: 0373-0956

Abstract

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Let f ( x ) be a nonnegative function with its only singularity at x = 0 , e.g. f ( x ) = | x | - α , α > 0 . We study the behavior of the Wiener process W ( t ) in left and right hand neighborhoods of level crossings by finding necessary and sufficient conditions on f for the integrals of f ( W ( t ) ) to be finite or infinite.

How to cite

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Shepp, Lawrence A., Klauder, John R., and Ezawa, Hiroshi. "On the divergence of certain integrals of the Wiener process." Annales de l'institut Fourier 24.2 (1974): 189-193. <http://eudml.org/doc/74171>.

@article{Shepp1974,
abstract = {Let $f(x)$ be a nonnegative function with its only singularity at $x=0$, e.g. $f(x)=\vert x\vert ^\{-\alpha \}$, $\alpha &gt;0$. We study the behavior of the Wiener process $W(t)$ in left and right hand neighborhoods of level crossings by finding necessary and sufficient conditions on $f$ for the integrals of $f(W(t))$ to be finite or infinite.},
author = {Shepp, Lawrence A., Klauder, John R., Ezawa, Hiroshi},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {2},
pages = {189-193},
publisher = {Association des Annales de l'Institut Fourier},
title = {On the divergence of certain integrals of the Wiener process},
url = {http://eudml.org/doc/74171},
volume = {24},
year = {1974},
}

TY - JOUR
AU - Shepp, Lawrence A.
AU - Klauder, John R.
AU - Ezawa, Hiroshi
TI - On the divergence of certain integrals of the Wiener process
JO - Annales de l'institut Fourier
PY - 1974
PB - Association des Annales de l'Institut Fourier
VL - 24
IS - 2
SP - 189
EP - 193
AB - Let $f(x)$ be a nonnegative function with its only singularity at $x=0$, e.g. $f(x)=\vert x\vert ^{-\alpha }$, $\alpha &gt;0$. We study the behavior of the Wiener process $W(t)$ in left and right hand neighborhoods of level crossings by finding necessary and sufficient conditions on $f$ for the integrals of $f(W(t))$ to be finite or infinite.
LA - eng
UR - http://eudml.org/doc/74171
ER -

References

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  1. [EKS] H. EZAWA, J. R. KLAUDER and L. A. SHEPP, Vestigial Effects of Singular Potentials in Diffusion Theory and Quantum Mechanics. J. Math. Phys., to appear. 
  2. [IMcK] K. ITO and H. P. McKEAN, Diffusion Processes and Their Sample Paths, Springer (1965). Zbl0127.09503MR33 #8031
  3. [KI] J. R. KLAUDER, Field Structure Through Model Studies: Aspects of Nonrenormalizable Theories, Acta Phys. Austriaca, Suppl. XI (1973), 341-387. 
  4. [Kn] F. B. KNIGHT, Existence of Small Oscillations at Zeros of Brownian Motion, manuscript. Zbl0305.60035
  5. [R] D. RAY, Sojourn Times of Diffusion Processes, Ill. J. Math., 7 (1963), 615-630. Zbl0118.13403MR27 #6306
  6. [S] L. A. SHEPP, Radon-Nikodym Derivatives of Gaussian Measures, Ann. Math. Stat., 37 (1966), 321-354. Zbl0142.13901MR32 #8408
  7. [V] D. E. VARBERG, Equivalent Gaussian Measures with a Particularly Simple Radon-Nikodym Derivative, Ann. Math. Stat., 38 (1967), 1027-1030. Zbl0171.15702MR35 #4981

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