Existence and asymptotic behaviour of some time-inhomogeneous diffusions
Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 1, p. 182-207

Let us consider a solution of a one-dimensional stochastic differential equation driven by a standard Brownian motion with time-inhomogeneous drift coefficient ρsgn(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.

Nous considérons la solution d’une équation différentielle stochastique, dirigée par un mouvement brownien linéaire standard, dont le terme de dérive varie avec le temps ρsgn(x)|x| α /t β . Ce processus peut être vu comme un mouvement brownien évoluant dans un potentiel dépendant du temps, éventuellement singulier. Nous montrons des résultats d’existence et d’unicité et nous étudions le comportement asymptotique de la solution. Les propriétés de récurrence ou de transience de cette diffusion sont décrites en fonction des paramètres ρ, α et β, et nous donnons les vitesses de transience et d’explosion. Des résultats de convergence en loi et des lois de type logarithme itéré sont également obtenus.

DOI : https://doi.org/10.1214/11-AIHP469
Classification:  60J60,  60H10,  60J65,  60G17,  60F15,  60F05
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
@article{AIHPB_2013__49_1_182_0,
     author = {Gradinaru, Mihai and Offret, Yoann},
     title = {Existence and asymptotic behaviour of some time-inhomogeneous diffusions},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {49},
     number = {1},
     year = {2013},
     pages = {182-207},
     doi = {10.1214/11-AIHP469},
     zbl = {1267.60091},
     mrnumber = {3060153},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2013__49_1_182_0}
}
Gradinaru, Mihai; Offret, Yoann. Existence and asymptotic behaviour of some time-inhomogeneous diffusions. Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 1, pp. 182-207. doi : 10.1214/11-AIHP469. http://www.numdam.org/item/AIHPB_2013__49_1_182_0/

[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. | MR 2170470 | Zbl 1067.60034

[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. | MR 2497457 | Zbl 1191.60069

[3] R. N. Bhattacharya and S. Ramasubramanian. Recurrence and ergodicity of diffusions. J. Multivariate Anal. 12 (1982) 95-122. | MR 650932 | Zbl 0499.60084

[4] A. S. Cherny and H.-J. Engelbert. Singular Stochastic Differential Equations. Lecture Notes in Mathematics 1858. Springer, Berlin, 2004. | MR 2112227 | Zbl 1071.60003

[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. | MR 182041 | Zbl 0168.16901

[6] D. A. Freedman. Bernard Friedman's urn. Ann. Math. Statist. 36 (1965) 956-970. | MR 177432 | Zbl 0138.12003

[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. | Numdam | MR 1702241 | Zbl 0937.60080

[8] I. I. Gihman and A. V. Skorohod. Stochastic Differential Equations. Springer, New York, 1991. | MR 346904 | Zbl 0242.60003

[9] R. Z. Has'Minskii. Stochastic Stability of Differential Equations. Sitjthoff & Noordhoff, Alphen aan den Rijn, 1980. | Zbl 0441.60060

[10] N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam, 1981. | MR 1011252 | Zbl 0684.60040

[11] O. Kallenberg. Foundation of Modern Probability, 3rd edition. Springer, New York, 2001. | MR 1464694 | Zbl 0892.60001

[12] R. Mansuy. On a one-parameter generalisation of the Brownian bridge and associated quadratic functionals. J. Theoret. Probab. 17 (2004) 1021-1029. | MR 2105746 | Zbl 1063.60049

[13] M. Menshikov and S. Volkov. Urn-related random walk with drift ρx α /t β . Electron. J. Probab. 13 (2008) 944-960. | MR 2413290 | Zbl 1191.60086

[14] M. Motoo. Proof of the law of iterated logarithm through diffusion equation. Ann. Inst. Statist. Math. 10 (1959) 21-28. | MR 97866 | Zbl 0084.35801

[15] K. Narita. Remarks on non-explosion theorem for stochastic differential equations. Kodai Math. J. 5 (1982) 395-401. | MR 684796 | Zbl 0502.60044

[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. | MR 1725357 | Zbl 0917.60006

[17] D. W. Stroock and S. R. S. Varadhan. Multidimensional Diffusion Process. Springer, Berlin, 1979. | MR 532498 | Zbl 1103.60005