# On asymptotic independence of the exit moment and position from a small domain for diffusion processes

Open Mathematics (2003)

• Volume: 1, Issue: 1, page 86-96
• ISSN: 2391-5455

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## Abstract

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If ξ(t) is the solution of homogeneous SDE in R m, and T ∃ is the first exit moment of the process from a small domain D ∃, then the total expansion for the following functional showing independence of the exit time and exit place is $Eexp\left(-\lambda {T}_{\epsilon }\right)f\left(\frac{\xi \left({T}_{\epsilon }\right)}{\epsilon }\right)-Eexp\left(-\lambda {T}_{\epsilon }\right)Ef\left(\frac{\xi \left({T}_{\epsilon }\right)}{\epsilon }\right),\epsilon ↘0,\lambda >0.$

## How to cite

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Vitalii Gasanenko. "On asymptotic independence of the exit moment and position from a small domain for diffusion processes." Open Mathematics 1.1 (2003): 86-96. <http://eudml.org/doc/268812>.

@article{VitaliiGasanenko2003,
abstract = {If ξ(t) is the solution of homogeneous SDE in R m, and T ∃ is the first exit moment of the process from a small domain D ∃, then the total expansion for the following functional showing independence of the exit time and exit place is $Eexp( - \lambda T\_\varepsilon )f(\frac\{\{\xi (T\_\varepsilon )\}\}\{\varepsilon \}) - Eexp( - \lambda T\_\varepsilon )Ef(\frac\{\{\xi (T\_\varepsilon )\}\}\{\varepsilon \}),\varepsilon \searrow 0,\lambda > 0.$},
author = {Vitalii Gasanenko},
journal = {Open Mathematics},
keywords = {60 J 50},
language = {eng},
number = {1},
pages = {86-96},
title = {On asymptotic independence of the exit moment and position from a small domain for diffusion processes},
url = {http://eudml.org/doc/268812},
volume = {1},
year = {2003},
}

TY - JOUR
AU - Vitalii Gasanenko
TI - On asymptotic independence of the exit moment and position from a small domain for diffusion processes
JO - Open Mathematics
PY - 2003
VL - 1
IS - 1
SP - 86
EP - 96
AB - If ξ(t) is the solution of homogeneous SDE in R m, and T ∃ is the first exit moment of the process from a small domain D ∃, then the total expansion for the following functional showing independence of the exit time and exit place is $Eexp( - \lambda T_\varepsilon )f(\frac{{\xi (T_\varepsilon )}}{\varepsilon }) - Eexp( - \lambda T_\varepsilon )Ef(\frac{{\xi (T_\varepsilon )}}{\varepsilon }),\varepsilon \searrow 0,\lambda > 0.$
LA - eng
KW - 60 J 50
UR - http://eudml.org/doc/268812
ER -

## References

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1. [1] M. Liao, Hitting distributions of small geodesic spheres, Ann. Probab., 16 (1988), 1039–1050 Zbl0651.58037
2. [2] V.A. Gasanenko, A total expansion functional of exit time from a small ball for diffusion process, International Journal Istatistik, Vol. 3, Issue 3 (2000), 83–91
3. [3] O.A. Ladyjenskai and N.N. Ural’ceva, Linear and quasilinear equation of elliptic type, “Nauka”, Moscow (1973)
4. [4] I.I. Gikhman and A.V. Skorokhod, Introduction in the theory of random, “Nauka”, Moscow (1977) Zbl0429.60002

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