Asymptotic Behavior of Solutions of Stochastic Differential Equations

Асимптотичні Властивості Процесів, Які Виникають У Задачах Оптимального Інвестування

“…The proof of Theorem 2.1 is based on the method developed in V. V. Buldygin, Klesov, and Tymoshenko (2018). For completeness, we demonstrate the proof of Theorem 2.1.…”

Bifurcations in asymptotically autonomous Hamiltonian systems under multiplicative noise

“…Since U 1 (x(t 0 ), y(t 0 ), t 0 ) ≤ (1+ǫ) 2 δ 2 /2, we can take δ = ε √ η(1−ǫ)/(1+ǫ) to obtain (15) with γ(t) ≡ 1.…”

“…It is well known that even weak random disturbances can lead to significant changes in the behavior of trajectories [11]. See, for example, [12][13][14][15][16][17][18][19], where the influence of autonomous stochastic perturbations on qualitative properties of solutions is discussed. Stochastic bifurcations associated with qualitative changes in the profile of stationary probability densities, in the Lyapunov spectrum function or in the dichotomy spectrum were investigated in [20][21][22][23][24][25] for systems of stochastic differential equations with time-independent coefficients.…”

On a selection problem for small noise perturbation in the multidimensional case

“…It is doubtfully that it has a discrete distribution or the uniform distribution. Moreover, if we take X ε (0) = xε α α+β−1 instead of X ε (0) = 0, then the answer may be different, see equality (13) for the model equation.…”