Asymptotic behavior of integral functionals of unstable solutions of one-dimensional Itô stochastic differential equations

Asymptotic behavior of homogeneous additive functionals of the solutions of Itô stochastic differential equations with nonregular dependence on parameter

“…as T n → ∞ for all L > 0. According to (12) withη(t) defined in (13), the process (ζ(t),Ŵ (t)) satisfies Eq. (6).…”

“…as T n → ∞ for any L > 0, where the process (ζ(t),Ŵ (t)) satisfies Eq. (6),η(t) is defined in (13), and the processesβ Thus, the processβ Tn (t) weakly converges, as T n → ∞, to the processβ (2) (t).…”

“…where the processes ξ T (t), W T (t) are related via equation (1), g T (x) is a family of measurable locally bounded real-valued functions, and F T (x) is a family of continuous real-valued functions. This paper is a continuation of [13][14][15]. Note that the behavior of the distributions of functionals β…”

“…A similar problem for the functionals I T (t) in the case of equation (1) with a T (x) ≡ 0 was considered in [18] and in [11], for the class K 1 . In [13][14][15], the behavior of the distributions of the functionals β T (t) was studied in [13], β T (t) was studied in [14], and I T (t) was investigated in [15]. A more detailed review of the known results in this area is presented in [13][14][15].…”

“…In [13][14][15], the behavior of the distributions of the functionals β T (t) was studied in [13], β T (t) was studied in [14], and I T (t) was investigated in [15]. A more detailed review of the known results in this area is presented in [13][14][15]. Note that the functionals β T (t), and β T (t) are particular cases of the functional I T (t) (see [15], Lemma 4.1).…”