Law of the iterated logarithm for solutions of stochastic equations. (English. Russian original) Zbl 1251.60027

Budkov, D. S.; Makhno, S. Ya.

Theory Probab. Math. Stat. 83, 47-57 (2011); translation from Teor. Jmovirn. Mat. Stat. 83, 39-48 (2010).

Summary: Strassen’s law of the iterated logarithm for a solution \( x(t)\) of Itô’s stochastic equation is considered in the paper. We obtain a result for small times in the uniform metric and for a more general normalizing function than the classical \( \sqrt { h\ln \ln \frac{1}{h}}\).

MSC:

60F17	Functional limit theorems; invariance principles
60F10	Large deviations

Keywords:

stochastic equation; large deviations; law of the iterated logarithm

References:

[1]	A. V. Bulinskiĭ, A new variant of the functional law of the iterated logarithm, Teor. Veroyatnost. i Primenen. 25 (1980), no. 3, 502 – 512 (Russian, with English summary). · Zbl 0436.60031
[2]	A. D. Wentzell, Limit theorems on large deviations for Markov stochastic processes, Mathematics and its Applications (Soviet Series), vol. 38, Kluwer Academic Publishers Group, Dordrecht, 1990. Translated from the Russian. · Zbl 0743.60029
[3]	M. I. Freidlin and A. D. Wentzell, Random perturbations of dynamical systems, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 260, Springer-Verlag, New York, 1984. Translated from the Russian by Joseph Szücs. · Zbl 0522.60055
[4]	Nina Gantert, An inversion of Strassen’s law of the iterated logarithm for small time, Ann. Probab. 21 (1993), no. 2, 1045 – 1049. · Zbl 0776.60099
[5]	Davar Khoshnevisan, Exact rates of convergence to Brownian local time, Ann. Probab. 22 (1994), no. 3, 1295 – 1330. · Zbl 0819.60067
[6]	Dmitrii S. Budkov and Sergey Ya. Makhno, Functional iterated logarithm law for a Wiener process, Theory Stoch. Process. 13 (2007), no. 3, 22 – 28. · Zbl 1199.60202
[7]	Стохастические дифференциал\(^{\приме}\)ные уравнения и их приложения, ”Наукова Думка”, Киев, 1982 (Руссиан). · Zbl 0169.48702