Existence and asymptotic behaviour of some time-inhomogeneous diffusions

Mihai Gradinaru; Yoann Offret

Annales de l'I.H.P. Probabilités et statistiques (2013)

  • Volume: 49, Issue: 1, page 182-207
  • ISSN: 0246-0203

Abstract

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Let us consider a solution of a one-dimensional stochastic differential equation driven by a standard Brownian motion with time-inhomogeneous drift coefficient ρ sgn ( x ) | x | α / t β . This process can be viewed as a Brownian motion evolving in a potential, possibly singular, depending on time. We prove results on the existence and uniqueness of solution, study its asymptotic behaviour and made a precise description, in terms of parameters ρ , α and β , of the recurrence, transience and convergence. More precisely, asymptotic distributions, iterated logarithm type laws and rates of transience and explosion are proved for such processes.

How to cite

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Gradinaru, Mihai, and Offret, Yoann. "Existence and asymptotic behaviour of some time-inhomogeneous diffusions." Annales de l'I.H.P. Probabilités et statistiques 49.1 (2013): 182-207. <http://eudml.org/doc/271961>.

@article{Gradinaru2013,
abstract = {Let us consider a solution of a one-dimensional stochastic differential equation driven by a standard Brownian motion with time-inhomogeneous drift coefficient $\rho \operatorname\{sgn\}(x)|x|^\{\alpha \}/t^\{\beta \}$. This process can be viewed as a Brownian motion evolving in a potential, possibly singular, depending on time. We prove results on the existence and uniqueness of solution, study its asymptotic behaviour and made a precise description, in terms of parameters $\rho $, $\alpha $ and $\beta $, of the recurrence, transience and convergence. More precisely, asymptotic distributions, iterated logarithm type laws and rates of transience and explosion are proved for such processes.},
author = {Gradinaru, Mihai, Offret, Yoann},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {time-inhomogeneous diffusions; time dependent potential; singular stochastic differential equations; explosion times; scaling transformations; change of time; recurrence and transience; iterated logarithm type laws; asymptotic distributions; scalar diffusion; time-dependent drift; transience; recurrence; asymptotic behavior},
language = {eng},
number = {1},
pages = {182-207},
publisher = {Gauthier-Villars},
title = {Existence and asymptotic behaviour of some time-inhomogeneous diffusions},
url = {http://eudml.org/doc/271961},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Gradinaru, Mihai
AU - Offret, Yoann
TI - Existence and asymptotic behaviour of some time-inhomogeneous diffusions
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 1
SP - 182
EP - 207
AB - Let us consider a solution of a one-dimensional stochastic differential equation driven by a standard Brownian motion with time-inhomogeneous drift coefficient $\rho \operatorname{sgn}(x)|x|^{\alpha }/t^{\beta }$. This process can be viewed as a Brownian motion evolving in a potential, possibly singular, depending on time. We prove results on the existence and uniqueness of solution, study its asymptotic behaviour and made a precise description, in terms of parameters $\rho $, $\alpha $ and $\beta $, of the recurrence, transience and convergence. More precisely, asymptotic distributions, iterated logarithm type laws and rates of transience and explosion are proved for such processes.
LA - eng
KW - time-inhomogeneous diffusions; time dependent potential; singular stochastic differential equations; explosion times; scaling transformations; change of time; recurrence and transience; iterated logarithm type laws; asymptotic distributions; scalar diffusion; time-dependent drift; transience; recurrence; asymptotic behavior
UR - http://eudml.org/doc/271961
ER -

References

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  1. [1] J. A. D. Appleby and D. Mackey. Polynomial asymptotic stability of damped stochastic differential equations. Electron. J. Qual. Theory Differ. Equ.2 (2004) 1–33. Zbl1067.60034MR2170470
  2. [2] J. A. D. Appleby and H. Wu. Solutions of stochastic differential equations obeying the law of the iterated logarithm, with applications to financial markets. Electron. J. Probab.14 (2009) 912–959. Zbl1191.60069MR2497457
  3. [3] R. N. Bhattacharya and S. Ramasubramanian. Recurrence and ergodicity of diffusions. J. Multivariate Anal.12 (1982) 95–122. Zbl0499.60084MR650932
  4. [4] A. S. Cherny and H.-J. Engelbert. Singular Stochastic Differential Equations. Lecture Notes in Mathematics 1858. Springer, Berlin, 2004. Zbl1071.60003MR2112227
  5. [5] L. E. Dubins and D. A. Freedman. A sharper form of the Borel–Cantelli lemma and the strong law. Ann. Math. Statist.36 (1965) 800–807. Zbl0168.16901MR182041
  6. [6] D. A. Freedman. Bernard Friedman’s urn. Ann. Math. Statist.36 (1965) 956–970. Zbl0138.12003MR177432
  7. [7] M. Gradinaru, B. Roynette, P. Vallois and M. Yor. Abel transform and integrals of Bessel local times. Ann. Inst. Henri Poincaré Probab. Stat.35 (1999) 531–572. Zbl0937.60080MR1702241
  8. [8] I. I. Gihman and A. V. Skorohod. Stochastic Differential Equations. Springer, New York, 1991. Zbl0242.60003MR346904
  9. [9] R. Z. Has’minskii. Stochastic Stability of Differential Equations. Sitjthoff & Noordhoff, Alphen aan den Rijn, 1980. Zbl0441.60060
  10. [10] N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam, 1981. Zbl0684.60040MR1011252
  11. [11] O. Kallenberg. Foundation of Modern Probability, 3rd edition. Springer, New York, 2001. Zbl0892.60001MR1464694
  12. [12] R. Mansuy. On a one-parameter generalisation of the Brownian bridge and associated quadratic functionals. J. Theoret. Probab.17 (2004) 1021–1029. Zbl1063.60049MR2105746
  13. [13] M. Menshikov and S. Volkov. Urn-related random walk with drift ρ x α / t β . Electron. J. Probab.13 (2008) 944–960. Zbl1191.60086MR2413290
  14. [14] M. Motoo. Proof of the law of iterated logarithm through diffusion equation. Ann. Inst. Statist. Math.10 (1959) 21–28. Zbl0084.35801MR97866
  15. [15] K. Narita. Remarks on non-explosion theorem for stochastic differential equations. Kodai Math. J.5 (1982) 395–401. Zbl0502.60044MR684796
  16. [16] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin, 1999. Zbl0917.60006MR1725357
  17. [17] D. W. Stroock and S. R. S. Varadhan. Multidimensional Diffusion Process. Springer, Berlin, 1979. Zbl1103.60005MR532498

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