Small Perturbations of Stochastic Differential Equations with Power Coefficients

Юрій Євгенович Приходько

Abstract


Background. Random perturbations of ordinary differential equations were considered by Bafico (1980), Bafico, Bal­di (1982), Delarue, Flandoli (2014), Delarue, Flandoli, Vincenzi (2014), Krykun, Makhno (2013), Pilipenko, Pro­ske (2015). Bafico, Baldi (1982) considered random perturbation of the differential equation that describes the Peano phenomenon. The coefficients of the initial differential equation are not Lipschitz continuous, so there may be no uniqueness of the solution. Then stochastic differential equation is considered instead of ordinary differential equation and the weak convergence of its solutions is proved.

Objective. The aim of this paper is to generalize the result of Bafico, Baldi (1982) to the case of stochastic differential equation dX(t)=a(X(t))dt+σ(X(t))dW(t) with power coefficients.

Methods. Small random perturbations of the initial equation dX(t)=a(X(t))dt+(ε+σ(X(t)))dW(t) are considered and the limit behaviour of its solutions is studied.

The methods used to prove the weak convergence of the solutions are based on the methods developed in Pilipenko, Prykhodko (2015 and 2016).

Results. The limit behaviour of the solutions of stochastic differential equations with perturbations is considered and the weak convergence of such solutions is proved.

Conclusions. The result of Bafico, Baldi (1982) is thus generalized to the case of stochastic differential equation with power coefficients.


Keywords


Stochastic differential equations; Stochastic differential equations with power coefficients; Stochastic differential equations with perturbations; Asymptotic behavior; Peano phenomena

References


A.S. Cherny and H.-J. Engelbert, Singular Stochastic Differential Equations, Ser. no. 1858. Springer Science & Business Media, 2005.

R. Bafico and P. Baldi, “Small random perturbations of Peano phenomena”, Stochastics, vol. 6, no. 3-4, pp. 279–292, 1982.

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes.Tokyo,Japan: Kodansha LTD, North-Holland Publishing Company, 1981.

R. Bafico, “On the convergence of the weak solutions of stochastic differential equations when the noise intensity goes to zero,” Boll. Unione Mat. Ital. Sez. B, vol. 17, pp. 308–324, 1980.

F. Delarue and F. Flandoli, “The transition point in the zero noise limit for a 1D Peano example”, Discrete and Continuous Dynamical Systems-Series A, vol. 34, pp. 4071–4084, 2014.

F. Delarue et al., “Noise prevents collapse of Vlasov-Poisson point charges”, Communications on Pure and Applied Mathematics, vol. 67, no. 10, pp. 1700–1736, 2014.

I.G. Krykun and S.Y. Makhno, “The Peano phenomenon for Itó equations”, J. Math. Sci., vol. 192, no. 4, pp. 441–458, 2013.

A. Pilipenko and F. N. Proske, “On a selection problem for small noise perturbation in multidimensional case”, arXiv:1510.00966, 2015.

A.Y. Pilipenko and Y.E. Prykhodko, “A limit theorem for singular stochastic differential equations”, arXiv:1609.01185, 2016.

A.Y. Pilipenko and Y.E. Prykhodko, “On the limit behavior of a sequence of Markov processes perturbed in a neighborhood of the singular point”, Ukrayins'kyy Matematychnyy Zhurnal, vol. 67, no. 4, pp. 564–583, 2015 (in Russian).

I.I. Gikhman and A.V. Skorokhod, Stochastic Differential Equations.Kyiv,USSR: Naukova Dumka, 1979 (in Russian).

M. Fedoryuk, The Saddle-Point Method. Moscow, USSR: Nauka 1977 (in Russian).


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DOI: https://doi.org/10.20535/1810-0546.2016.4.72068

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