Existence and asymptotic behaviour of some time-inhomogeneous diffusions

Gradinaru, Mihai; Offret, Yoann

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 1, pp. 182-207.

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 ${\rho sgn\left(x\right)\left|x\right|}^{\alpha }/t^{\beta }$. 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 $\rho$, $\alpha$ et $\beta$, 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.

Let us consider a solution of a one-dimensional stochastic differential equation driven by a standard Brownian motion with time-inhomogeneous drift coefficient ${\rho sgn\left(x\right)\left|x\right|}^{\alpha }/t^{\beta }$. 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.

Classification : 60J60, 60H10, 60J65, 60G17, 60F15, 60F05
Mots clés : 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

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

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

[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 | Zbl

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

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

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

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

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

[13] M. Menshikov and S. Volkov. Urn-related random walk with drift $\rho x^{\alpha }/t^{\beta }$. Electron. J. Probab. 13 (2008) 944-960. | MR | Zbl

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

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

[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 | Zbl

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