Consider the Itô stochastic differential equation (1) dξT(t)=aT(ξT(t))dt+dWT(t),t≥0,ξT(0)=x0, where T>0 0$]]> is a parameter, aT(x) , x∈R , are real-valued measurable functions such that for some constants LT>0 0$]]> and for all x∈R aT(x)≤LT , and WT={WT(t),t≥0} , T>0 0$]]>, is a family of standard Wiener processes defined on a complete probability space (Ω,ℑ,P) .

It is known from Theorem 4 in [19] that, for any T>0 0$]]> and x0∈R , equation (1) possesses a unique strong pathwise solution ξT={ξT(t),t≥0} , and this solution is a homogeneous strong Markov process.

We suppose that the drift coefficient aT(x) in equation (1) can have a very nonregular dependence on the parameter. For example, the drift coefficient can be of “δ”-type sequence at some points xk as T→∞ , or it can be equal to Tsin((x−xk)T) , or it can have degeneracies of some other types. Such a nonregular dependence of the coefficients in equation (1) first appeared in [5] and [4], where the limit behavior of the normalized unstable solution of Itô stochastic differential equation as t→∞ was investigated. In those papers, a special dependence of the coefficients aT(x)=Ta(xT) on the parameter T was considered in the case where a(x) is an absolutely integrable function on R . Assume that this is the case and let ∫Ra(x)dx=λ . The sufficiency of the condition λ=0 for the asymptotic equivalence of distributions ξT and WT(t) is established in [5], and the necessity of this condition is proved in [1]. If λ≠0 , then we can deduce from [4] that the distributions of the solution ξT of equation (1) weakly converge as T→∞ to the corresponding distributions of the Markov process ξˆ(t)=lζ(t) , where l(x)=c1x for x>0 0$]]> and l(x)=c2x for x≤0 ; ζ(t) is a strong solution of the Itô equation dζ(t)=σ¯(ζ(t))dW(t) , where σ¯(x)=σ1 for x>0 0$]]> and σ¯(x)=σ2 for x<0 , and ∫0tP{|ζ(s)|=0}ds=0 . The explicit form of the transition density of the process ξˆ(t) is obtained. Moreover, in [10], it is proved that ξˆ(t)=x0+β(t)+W(t), where β(t) is a certain functional of ζ(t) , and the necessity of the condition λ≠0 is established for the weak convergence as T→∞ of the solution ξT of equation (1) to the process ξˆ(t) .

Furthermore, in [5] and [4], a probabilistic method to study the “awkward” term T∫0ta(ξT(s)T)ds in equation (1) is developed. This method uses a representation of this “awkward” term through a family of continuous functions ΦT(x) of ξT(t) and a family of martingales ∫0tΦT′(ξT(s))dWT(s) , with the further application of the Itô formula. After the mentioned transformations, according to this method, we can apply Skorokhod’s convergent subsequence principle for ξTn(t) and WTn(t) (see [17], Chapter I, §6) in order to pass to the limit in the resulting representation.

Note that this method is also used in the present paper to study the asymptotic behavior of integral functionals.

It is known from [2], §16, that the asymptotic behavior of the solution ξT of equation (1) is closely related to the asymptotic behavior of harmonic functions, that is, functions satisfying the following ordinary differential equation almost everywhere (a.e.) with respect to the Lebesgue measure: fT′(x)aT(x)+12fT″(x)=0.

It is obvious that the functions fT(x) have the form (2) fT(x)=cT(1)∫0xexp{−2∫0uaT(v)dv}du+cT(2), where cT(1) and cT(2) are some families of constants.

The latter functions possess the continuous derivatives fT′(x) , and their second derivatives fT″(x) exist almost everywhere with respect to the Lebesgue measure and are locally integrable. Note that cT(1) are normalizing constants and cT(2) are centralizing constants in the limit theorems (see [18], §6). Further, for simplicity, we assume that in (2), cT(1)≡1 and cT(2)≡0 .

In this paper, we assume for the coefficient aT(x) of equation (1) that there exists a family of functions GT(x) , x∈R , with continuous derivatives GT′(x) and locally integrable second derivatives GT″(x) a.e. with respect to the Lebesgue measure such that, for all T>0 0$]]> and x∈R , the following inequalities hold: (A1)(GT′(x)aT(x)+12GT″(x))2+(GT′(x))2≤C(1+(GT(x))2),|GT(x0)|≤C.

Suppose additionally that the functions GT(x) , x∈R , introduced by condition (A1) satisfy the following assumptions:

Let GT be the class of the functions GT(x) , x∈R , satisfying conditions (A1) and (i)–(ii). The class of equations of the form (1) whose coefficients aT(x) admit GT(x) , x∈R , from the class GT will be denoted by KGT . It is easy to understand that class KGT does not depend on the constants cT(1) and cT(2) in representation (2).

It is clear that if there exist constants δ>0 0$]]> and C>0 0$]]> such that 0<δ≤fT′(x)≤C for all x∈R , T>0 0$]]>, then the corresponding equations (1) belong to the class KGT for GT(x)=fT(x) . We denote this subclass as K1 . Note that the class KGT contains in particular the equations for which, at some points xk , we have the convergence fT′(xk)→∞ or the convergence fT′(xk)→0 as T→∞ . For example, consider equation (1) with aT(x)=c0Tx1+x2T . It is easy to obtain that fT′(x)=11+x2Tc0 , and if c0>−12 -\frac{1}{2}$]]>, then such equations belong to the class KGT with GT(x)=x2 (here, at points x≠0 , we have fT′(x)→0 for c0>0 0$]]>, fT′(x)→∞ for −12<c0<0 , and fT′(x)≡1 for c0=0 ).

For the class of equations KGT , we study the asymptotic behavior as T→∞ of the distributions of the following functionals: βT(1)(t)=∫0tgT(ξT(s))ds,βT(2)(t)=∫0tgT(ξT(s))dWT(s),IT(t)=FT(ξT(t))+∫0tgT(ξT(s))dWT(s),βT(t)=∫0tgT(ξT(s))dξT(s), where the processes ξT(t) , WT(t) are related via equation (1), gT(x) is a family of measurable locally bounded real-valued functions, and FT(x) is a family of continuous real-valued functions.

This paper is a continuation of [13–15]. Note that the behavior of the distributions of functionals βT(1)(t) , βT(2)(t) for the solutions ξT of equations (1) from the class K1 is studied in [6] and [9]. The case where WT(t) is replaced with ηT(t) , where ηT(t) is a family of continuous martingales with the characteristics ⟨ηT⟩(t)→t as T→∞ , was studied in [8]. Paper [7] was devoted to a discrete analogue of the results from [8]. A similar problem for the functionals IT(t) in the case of equation (1) with aT(x)≡0 was considered in [18] and in [11], for the class K1 . In [13–15], the behavior of the distributions of the functionals βT(1)(t) , βT(2)(t) , and IT(t) with a special dependence of the drift coefficients aT(x)=Ta(xT) on the parameter T is considered, mainly in the case where xa(x)≤C for all x∈R . The behavior of the distributions of functionals βT(1)(t) was studied in [13], βT(2)(t) was studied in [14], and IT(t) was investigated in [15]. A more detailed review of the known results in this area is presented in [13–15]. Note that the functionals βT(1)(t) , βT(2)(t) , and βT(t) are particular cases of the functional IT(t) (see [15], Lemma 4.1).

The paper is organized as follows. Section 2 contains the statements of the main results. In Section 3, they are proved. Auxiliary results are collected in Section 4.

In what follows, we denote by C,L,N,CN any constants that do not depend on T and x. Assume that, for certain locally bounded functions qT(x) and any constant N>0 0$]]>, the following condition holds: (A3)limT→∞sup|x|≤NfT′(x)|∫0xqT(v)fT′(v)dv|=0, where fT′(x) is the derivative of the function fT(x) defined by Eq. (2).

The next theorem principally follows from [11]; however, we provide its proof for the reader’s convenience and completeness of the results. Theorem 2.6.

Let ξT be a solution of Eq. (1) from the class KGT for GT(x)=fT(x) , and let 0<δ≤fT′(x)≤C and fT(x0)→y0 as T→∞ . Also, let ζT(t)=fT(ξT(t)) , IT(t)=FT(ξT(t))+∫0tgT(ξT(s))dWT(s), FT(x) be continuous functions, gT(x) be locally square-integrable functions, and the processes ξT(t) and WT(t) be related via Eq. (1).

The two-dimensional process ζT(t),IT(t) weakly converges, as T→∞ , to the process ζ(t),I(t) , where I(t)=F0(ζ(t))+∫0tg0(ζ(s))dζ(s) , and ζ(t) is a weak solution of the Itô equation ζ(t)=y0+∫0tσ0(ζ(s))dW(s) , if and only if there exist constants cT(1) and cT(2) in (2) such that, as T→∞ : 1.

for all x, ∫0φT(x)[fT′(v)]2−σ02(fT(v))fT′(v)dv→0, where φT(x) is the inverse function of the function fT(x) ;

Proof of Theorem 2.1. Rewrite Eq. (3) as (7) ζT(t)=GT(x0)+∫0ta0(ζT(s))ds+αT(1)(t)+ηT(t), where αT(1)(t)=∫0tqT(1)(ξT(s))ds,qT(1)(x)=GT′(x)aT(x)+12GT″(x)−a0(GT(x)). The functions qT(1)(x) satisfy the conditions of Lemma 4.2. Thus, for any L>0 0$]]>, (8) sup0≤t≤L|αT(1)(t)|→P0 as T→∞ . It is clear that ηT(t) is a family of continuous martingales with quadratic characteristics (9) ⟨ηT⟩(t)=∫0t(GT′(ξT(s)))2ds=∫0tσ02(ζT(s))ds+αT(2)(t), where αT(2)(t)=∫0tqT(2)(ξT(s))ds,qT(2)(x)=(GT′(x))2−σ02(GT(x)).

The functions qT(2)(x) satisfy the conditions of Lemma 4.2. Thus, for any L>0 0$]]>, (10) sup0≤t≤L|αT(2)(t)|→P0 as T→∞ .

We have that relations (4) and (5) hold for the processes ζT(t) and ηT(t) , and, according to (8) and (10), these relations hold for the processes αT(k)(t) , k=1,2 , as well. This means that we can apply Skorokhod’s convergent subsequence principle (see [17], Chapter I, §6) for the process (ζT(t),ηT(t),αT(1)(t),αT(2)(t)) : given an arbitrary sequence Tn′→∞ , we can choose a subsequence Tn→∞ , a probability space (Ω˜,ℑ˜,P˜) , and a stochastic process (ζ˜Tn(t),η˜Tn(t),α˜Tn(1)(t),α˜Tn(2)(t)) defined on this space such that its finite-dimensional distributions coincide with those of the process (ζTn(t),ηTn(t),αTn(1)(t),αTn(2)(t)) and, moreover, ζ˜Tn(t)→P˜ζ˜(t),η˜Tn(t)→P˜η˜(t),α˜Tn(1)(t)→P˜α˜(1)(t),α˜Tn(2)(t)→P˜α˜(2)(t) for all 0≤t≤L , where ζ˜(t) , η˜(t) , α˜(1)(t) , α˜(2)(t) are some stochastic processes.

Evidently, relations (8) and (10) imply that α˜(k)(t)≡0 , k=1,2 , a.s. According to (5), the processes ζ˜(t) and η˜(t) are continuous. Moreover, applying Lemma 4.5 together with Eqs. (7) and (9), we obtain that (11) ζ˜Tn(t)=GTn(x0)+∫0ta0(ζ˜Tn(s))ds+α˜Tn(1)(t)+η˜Tn(t),⟨η˜Tn⟩(t)=∫0tσ02(ζ˜Tn(s))ds+α˜Tn(2)(t), where ζ˜Tn(t)→P˜ζ˜(t) , η˜Tn(t)→P˜η˜(t) , sup0≤t≤L|α˜Tn(k)(t)|→P˜0 , k=1,2 , as Tn→∞ . In addition, it is established in [12] that, for any constants L>0 0$]]> and ε>0 0$]]>, limh→0limTn→∞‾P˜{sup|t1−t2|≤h;ti≤L|ζ˜Tn(t2)−ζ˜Tn(t1)|>ε}=0. \varepsilon \Big\}=0.\]]]>

Using the latter convergence and (11), we conclude that, for any constants L>0 0$]]> and ε>0 0$]]>, limh→0limTn→∞‾P˜{sup|t1−t2|≤h;ti≤L|η˜Tn(t2)−η˜Tn(t1)|>ε}=0. \varepsilon \Big\}=0.\]]]>

Therefore, according to the well-known result of Prokhorov [16], we conclude that sup0≤t≤L|ζ˜Tn(t)−ζ˜(t)|→P˜0andsup0≤t≤L|η˜Tn(t)−η˜(t)|→P˜0 as Tn→∞ . According to Lemma 4.3, we can pass to the limit in (11) and obtain (12) ζ˜(t)=y0+∫0ta0(ζ˜(s))ds+η˜(t), where η˜(t) is a continuous martingale with the quadratic characteristic ⟨η˜⟩(t)=∫0tσ02(ζ˜(s))ds.