Asymptotic behavior of the martingale type integral functionals for unstable solutions to stochastic differential equations

Abstract

We consider functionals of the type t 0 g(ξ(s)) dW (s), t ≥ 0. Here g is a real valued and locally square integrable function, ξ is a unique strong solution of the Itô stochastic differential equation dξ(t) = a(ξ(t)) dt + dW (t), a is a measurable real valued bounded function such that |xa(x)| ≤ C. The behavior of these functionals is studied as t → ∞. The appropriate normalizing factor and the explicit form of the limit random variable are established.

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

“…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).…”

Asymptotic behavior of functionals of the solutions to inhomogeneous Itô stochastic differential equations with nonregular dependence on parameter

“…T (t)), where W * T (t) is a Wiener process. The same arguments as those used to get (9) in [7] yield that…”

Galerkin Finite Element Approximations for Stochastic Space-Time Fractional Wave Equations

“…For the practically physical system, the different stochastic perturbations are coming from many natural sources; sometimes, they can not be ignored and we need to incorporate them to the corresponding deterministic model; and then the stochastic differential equations (SDEs) are produced. In more recent decades, the initial and/or boundary value problems for SDEs have been extensively studied theoretically [8,15,20,23] and numerically [2,3,11,21,22,27,36]. However, it seems that there are less literatures related to the theoretical analysis or numerical approximation of stochastic wave equations with fractional derivative or driven by fractional noise.…”