MSTA Modern Stochastics: Theory and Applications 2351-6054 2351-6046 2351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania MSTA83 10.15559/17-MSTA83 Research Article Asymptotic behavior of functionals of the solutions to inhomogeneous Itô stochastic differential equations with nonregular dependence on parameter KulinichGrigorijzag_mat@univ.kiev.ua KushnirenkoSvitlanabksv@univ.kiev.ua∗ Taras Shevchenko National University of Kyiv, 64/13, Volodymyrska Street, 01601 Kyiv, Ukraine Corresponding author. 2017 229201743199217 2952017 682017 1082017 © 2017 The Author(s). Published by VTeX2017 Open access article under the CC BY license.

The asymptotic behavior, as T→∞ , of some functionals of the form IT(t)=FT(ξT(t))+∫0tgT(ξT(s))dWT(s) , t≥0 is studied. Here ξT(t) is the solution to the time-inhomogeneous Itô stochastic differential equation dξT(t)=aT(t,ξT(t))dt+dWT(t),t≥0,ξT(0)=x0, T>0 0$]]> is a parameter, aT(t,x),x∈R are measurable functions, |aT(t,x)|≤CT for all x∈R and t≥0 , WT(t) are standard Wiener processes, FT(x),x∈R are continuous functions, gT(x),x∈R are measurable locally bounded functions, and everything is real-valued. The explicit form of the limiting processes for IT(t) is established under nonregular dependence of aT(t,x) and gT(x) on the parameter T.

 Diffusion-type processes asymptotic behavior of functionals nonregular dependence on the parameter 60H10 60F17 60J60 Introduction

Consider the time-inhomogeneous Itô stochastic differential equation dξT(t)=aT(t,ξT(t))dt+dWT(t),t≥0,ξT(0)=x0, where T>0 0$]]> is a parameter, aT(t,x),x∈R are real-valued measurable functions such that |aT(t,x)|≤LT for all (t,x) and some family of constants LT>0 0$]]>, 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 [12] that for any T>0 0$]]> and x0∈R equation (1) possesses a unique strong solution ξT={ξT(t),t≥0} .

In this paper, we study the weak convergence, as T→∞ , of the processes IT(t)=FT(ξT(t))+∫0tgT(ξT(s))dWT(s) , where ξT(t) is the solution to stochastic differential equation (1), FT(x) is a family of continuous real-valued functions, gT(x) is a family of measurable, locally bounded real-valued functions. All the results about asymptotic behavior are obtained under the condition which provides certain proximity of the coefficients aT(t,x) of equation (1) to some measurable functions aˆT(x) . In such situation, the limit processes, obtained under condition T→∞ , are some functionals of the limits of the solutions ξˆT(t) to the homogeneous stochastic differential equations dξˆT(t)=aˆT(ξˆT(t))dt+dWT(t).

The present paper generalizes similar results from [8] for the unique strong solutions ξˆT to homogeneous stochastic differential equations (2) to the case for the solutions ξT(t) to inhomogeneous equations (1). Some results about solutions ξˆT to homogeneous equations (2), which obtained in [8], have been extended to solutions ξT to inhomogeneous equations (1). Under the certain proposed conditions, which present a novelty in comparison with [8], we prove that the asymptotic behavior of the solutions and some functionals of the solutions to inhomogeneous Itô stochastic differential equations (1) is the same as that for the solutions to homogeneous Itô stochastic differential equations (2). The present paper also complements results of paper [9]. Moreover, we assume that the drift coefficient aT(t,x) in equation (1) can have nonregular dependence on the parameter T. For example, the drift coefficient aT(t,x) can tend, as T→∞ , to infinity at some points xk and at some points tk as well, or it can have degeneracies of some other types. Such a nonregular dependence on T of the coefficients in equation (1) appeared for the first time in [4] and [5], where the limit behavior of the normalized unstable solution of Itô stochastic differential equation, as t→∞ , was investigated for homogeneous equations. In those papers, a special dependence of the coefficients aT(x)=Ta(xT) on the parameter T was considered with a(x)∈L1(R) . The special dependence of the coefficients aT(t,x)=Ta(tT,xT) on the parameter T was considered in [6] for inhomogeneous stochastic differential equations (1).

A more detailed review of the known results in this area is presented, for example, in [7] and [8].

The paper is organized as follows. In Section 2, we set the notations and formulate basic definitions. Section 3 contains the statements of the main results. In Section 4, they are proved. Auxiliary results are collected in Section 5. Section 6 gives examples.

 Main definitions

In what follows we denote by C,L,N,CN,LN any constants that do not depend on T, x and t. To formulate and prove the main results we introduce the functions of the form fT(x)=∫0xexp{−2∫0uaˆT(v)dv}du,T>0. 0.\]]]>

Throughout the paper we use the following notations: 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), where ξT and WT are related via equation (1), gT(x) is a family of measurable, locally bounded real-valued functions, FT(x) is a family of continuous real-valued functions.

We say that a family of stochastic processes ζT={ζT(t),t≥0} weakly converges, as T→∞ , to a process ζ={ζ(t),t≥0} if, for any L>0 0$]]>, the measures μT[0,L] generated by the processes ζT on the interval [0,L] weakly converge to the measure μ[0,L] generated by the process ζ considered on the interval [0,L] .

To study the weak convergence, as T→∞ , of the processes IT(t)=FT(ξT(t))+∫0tgT(ξT(s))dWT(s) , where ξT is the solution to stochastic differential equation (1), we suppose additionally that the drift coefficients satisfy the following assumption: there exists a family of measurable, locally bounded functions aˆT(x) such that for any L>0 0$]]> limT→∞∫0Lsupx|aT(t,x)−aˆT(x)|dt=0.

Note, that due to condition ( A0 ), some results about solutions ξˆT to homogeneous equations (2), which are obtained in [8], have been extended to solutions ξT to inhomogeneous equations (1). Therefore, by analogy to the paper [8], consider equations (1) from the class K(GT) .

The class of equations of the form (1) will be denoted by K(GT) , if there exist families of functions aˆT(x) and GT(x) , x∈R , such that:

aˆT(x) are measurable locally bounded real-valued functions, satisfying condition ( A0 );

GT(x) have 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$]]>, x∈R and t≥0 , for some constant C>0 0$]]> the following inequalities hold: [GT′(x)aT(t,x)+12GT″(x)]2+[GT′(x)]2≤C[1+|GT(x)|2],|GT(x0)|≤C;

there exist constants C>0 0$]]> and α>0 0$]]> such that, for all x∈R , |GT(x)|≥C|x|α;

there exist a bounded function ψ(|x|) and a constant m≥0 such that ψ(|x|)→0 as |x|→0 and, for all x∈R and T>0 0$]]> and for any measurable bounded set B, the following inequality holds: |fT′(x)∫0xχB(GT(u))fT′(u)du|≤ψ(λ(B))[1+|x|m], where χB(x) is the indicator function of a set B, λ(B) is the Lebesgue measure of B, fT′(x) is the derivative of the function fT(x) defined by equality (3).

Assume that, for certain locally bounded functions qT(x) and any constant N>0 0$]]>, the following condition holds: limT→∞sup|x|≤N|fT′(x)∫0xqT(v)fT′(v)dv|=0.

Statement of the main results

We are in position to obtain the main result (Theorem 3.1) concerning the weak compactness of stochastic processes ζT={ζT(t)=GT(ξT(t)),t≥0} , and use it in further investigation of asymptotic behavior of the solutions (Theorem 3.2) and some functionals of the solutions (Theorems 3.3–3.7) to inhomogeneous Itô stochastic differential equations (1).

Let ξT be a solution to equation (1) and let there exist a family of continuous functions GT(x) , x∈R for which the derivative GT′(x) is continuous and the second derivative GT″(x) exists a.e. with respect to the Lebesgue measure and is locally integrable. Let the functions GT(x) satisfy assumption ( A1 ), for all T>0 0$]]>, t≥0 , x∈R . Then the family of the processes ζT={ζT(t)=GT(ξT(t)),t≥0} is weakly compact.

Let ξT be a solution to equation (1) from the class K(GT) and GT(x0)→y0 , as T→∞ . Assume that there exist measurable locally bounded functions a0(x) and σ0(x) such that:

the functions qT(1)(x)=GT′(x)aˆT(x)+12GT″(x)−a0(GT(x)),qT(2)(x)=[GT′(x)]2−σ02(GT(x)), satisfy assumption ( A3 );

the Itô equation ζ(t)=y0+∫0ta0(ζ(s))ds+∫0tσ0(ζ(s))dWˆ(s) has a unique weak solution (ζ,Wˆ) .

 Then the stochastic processes ζT=GT(ξT(t)) weakly converge, as T→∞ , to the solution ζ to equation (4).

Let ξT be a solution to equation (1) from the class K(GT) and let assumptions of Theorem 3.2 hold. Assume that for measurable and locally bounded functions gT(x) there exists measurable and locally bounded function g0(x) such that the function qT(x)=gT(x)−g0(GT(x)) satisfies assumption ( A3 ). Then the stochastic processes βT(1)(t)=∫0tgT(ξT(s))ds weakly converge, as T→∞ , to the process β(1)(t)=∫0tg0(ζ(s))ds, where ζ is a solution to equation (4).

Let ξT be a solution to equation (1) from the class K(GT) , and let the assumptions of Theorem 3.2 hold. Assume that, for measurable and locally bounded functions gT(x) , there exists a measurable locally bounded function g0(x) such that |fT′(x)∫0xgT(v)fT′(v)dv|χ|x|≤N≤CN,limT→∞sup|x|≤N|fT′(x)∫0xgT(v)fT′(v)dv−g0(GT(x))GT′(x)|=0 for all N>0 0$]]>. Then the stochastic processes βT(1)(t)=∫0tgT(ξT(s))ds weakly converge, as T→∞ , to the process β˜(1)(t)=2(∫y0ζ(t)g0(x)dx−∫0tg0(ζ(s))σ0(ζ(s))dWˆ(s)), where (ζ,Wˆ) is a solution to equation (4).

Let ξT be a solution to equation (1) from the class K(GT) , and let the assumptions of Theorem 3.2 hold. Suppose that the functions aˆT(x) satisfy assumption ( A3 ). Assume that, for measurable and locally bounded functions gT(x) , there exist two constants c0 and b0 such that for all N>0 0$]]> |fT′(x)∫0xgT(v)fT′(v)dv|χ|x|≤N≤CN,limT→∞sup|x|≤N|∫0x[fT′(u)∫0ugT(v)fT′(v)dv−c0]du|=0, and the functions QT(x)=[fT′(x)∫0xgT(v)fT′(v)dv−c0]2−b02 satisfy assumption ( A3 ). Then the stochastic processes βT(1)(t)=∫0tgT(ξT(s))ds weakly converge, as T→∞ , to the process 2b0W(t) , where {W(t),t≥0} is a Wiener process.

Let ξT be a solution to equation (1) from the class K(GT) and let assumptions of Theorem 3.2 hold. Assume that, for measurable and locally bounded functions gT(x) , there exists a measurable locally bounded function g0(x) such that the function qT(x)=[gT(x)−g0(GT(x))GT′(x)]2 satisfies assumption ( A3 ). Then the stochastic processes βT(2)(t)=∫0tgT(ξT(s))dWT(s), where ξT and WT are related via equation (1), weakly converge, as T→∞ , to the process β(2)(t)=∫0tg0(ζ(s))σ0(ζ(s))dWˆ(s), where (ζ,Wˆ) is a solution to equation (4).

Let ξT and WT be related via equation (1) from the class K(GT) and let the assumptions of Theorem 3.2 hold. Assume that, for continuous functions FT(x) and locally bounded measurable functions gT(x) , there exist a continuous function F0(x) and locally bounded measurable function g0(x) such that, for all N>0 0$]]> limT→∞sup|x|≤N|FT(x)−F0(GT(x))|=0, and let the functions gT(x) and g0(x) satisfy the assumptions of Theorem 3.6. Then the stochastic processes IT(t)=FT(ξT(t))+∫0tgT(ξT(s))dWT(s) weakly converge, as T→∞ , to the process I0(t)=F0(ζ(t))+∫0tg0(ζ(s))σ0(ζ(s))dWˆ(s), where (ζ,Wˆ) is a solution to equation (4).

 Proof of the main results

In the proofs of theorems, which are performed similarly to the proofs of the corresponding theorems in [8], we emphasize the differences associated with inhomogeneous equations. The proof of the Theorem 3.2 is given for a better understanding of the brief proofs of the other theorems.

The functions GT(x) have continuous derivatives GT′(x) for all T>0 0$]]>, their second derivatives GT″(x) exist a.e. with respect to the Lebesgue measure and are locally integrable. Therefore (see [3], Chap. II, §10), we can apply the Itô formula to the process ζT(t)=GT(ξT(t)) , and with probability one, for all t≥0 , we obtain ζT(t)=GT(x0)+∫0tLT(ξT(s))ds+∫0tGT′(ξT(s))dWT(s), where LT(x)=GT′(x)aT(t,x)+12GT″(x).

Let χN(t)=1,sup0≤s≤t|ζT(s)|≤N,0,sup0≤s≤t|ζT(s)|>N. N.\hspace{1em}\end{array}\right.\]]]>

It is clear that for s≤t we have χN(t)χN(s)=χN(t) with probability one. Thus, according to (5), the following equality holds with probability one: ζT(t)χN(t)=ζT(0)χN(t)+χN(t)∫0tLT(ξT(s))χN(s)ds+χN(t)∫0tGT′(ξT(s))χN(s)dWT(s).

Hence, using condition ( A1 ) and the properties of stochastic integrals, we obtain that EζT2(t)χN(t)≤3[EζT2(0)χN(t)+E(∫0tLT(ξT(s))χN(s)ds)2+E(∫0tGT′(ξT(s))χN(s)dWT(s))2]≤3[EζT2(0)χN(t)+t∫0tELT2(ξT(s))χN(s)ds+∫0tE[GT′(ξT(s))]2χN(s)ds]≤3[C+t∫0tC[1+EζT2(s)χN(s)]ds+C∫0t[1+EζT2(s)χN(s)]ds]≤CL(1)+CL(2)∫0tEζT2(s)χN(s)ds, where CL(1)=3C(1+t+t2) , CL(2)=3C(1+t) , C>0 0$]]> is a constant from condition ( A1 ), 0≤t≤L .

Using the Gronwall–Bellman inequality, we conclude that there exists a constant KL , which is independent of T, and for 0≤t≤L EζT2(t)χN(t)≤KL. Let N↑∞ , then ζT2(t)χN(t)↑ζT2(t) , and we get the inequality sup0≤t≤LEζT2(t)≤KL.

Similarly to (7), using (5) and the inequality Esup0≤t≤L[∫0tGT′(ξT(s))dWT(s)]2≤4∫0LE[GT′(ξT(s))]2ds, we conclude that Esup0≤t≤L|ζT(t)|2≤3[GT2(x0)+L∫0LC[1+EζT2(s)]ds+∫0LC[1+EζT2(s)]ds]≤C˜L(1)+C˜L(2)∫0LE[ζT(s)]2ds.

Therefore, considering (8), we obtain the inequality Esup0≤t≤L|ζT(t)|2≤K˜L for all L>0 0$]]>, where the constants K˜L are independent of T.

Using the inequalities for martingales and for stochastic integrals (see [2], Part I, §3, Theorem 6), we obtain that Esup0≤t≤L|∫0tGT′(ξT(s))χN(s)dWT(s)|2m≤(2m2m−1)2mE|∫0LGT′(ξT(s))χN(s)dWT(s)|2m≤(2m2m−1)2m[m(2m−1)]m−1Lm−1∫0LE[GT′(ξT(s))]2mχN(s)ds, for any natural number m. Therefore, similarly to (9) we have inequality Esup0≤t≤L|ζT(t)|2m≤KLm.

Furthermore, for all α>0 0$]]> there exists m∈N such that α≤2m and, for random variable η, we have E|η|α≤1+E|η|2m.

The last inequality together with (10) implies that Esup0≤t≤L|ζT(t)|α≤KLα for all α>0 0$]]> and L>0 0$]]>, where the constants KLα are independent of T.

Since for t1<t2≤L E[ζT(t2)−ζT(t1)]4≤8[E(∫t1t2LT(ξT(s))ds)4+E(∫t1t2GT′(ξT(s))dWT(s))4]≤8[(t2−t1)3∫t1t2E|LT(ξT(s))|4ds+36(t2−t1)∫t1t2E[GT′(ξT(s))]4ds], considering condition ( A1 ) and inequality (11), we get E[ζT(t2)−ζT(t1)]4≤CL|t2−t1|2, where the constants CL are independent of T.

According to (8) and (12), we have convergences limN→∞limT→∞‾sup0≤t≤LP{|ζT(t)|>N}=0,limh→0limT→∞‾sup|t1−t2|≤hti≤LP{|ζT(t2)−ζT(t1)|>ε}=0 N\big\}& \displaystyle =0,\\{} \displaystyle \underset{h\to 0}{\lim }\overline{\lim _{T\to \infty }}\underset{\begin{array}{c}|t_{1}-t_{2}|\le h\\{} t_{i}\le L\end{array}}{\sup }\operatorname{\mathsf{P}}\big\{\big|\zeta _{T}(t_{2})-\zeta _{T}(t_{1})\big|>\varepsilon \big\}& \displaystyle =0\end{array}\]]]> for any L>0 0$]]>, ε>0 0$]]>.

It means that we can apply Skorokhod’s convergent subsequence principle (see [11], Chapter I, §6) for the processes ζT(t) for all 0≤t≤L . According to this principle, given an arbitrary sequence Tn′→∞ , we can choose a subsequence Tn→∞ , a probability space (Ω˜,ℑ˜,P˜) , and stochastic processes ζ˜Tn(t) , ζ(t) defined on this space such that their finite-dimensional distributions coincide with those of the processes ζTn(t) , and, moreover, ζ˜Tn(t)→P˜ζ(t) , as Tn→∞ , for all 0≤t≤L . The processes ζ˜Tn(t) and ζ(t) can be considered separable.

Using (12), we have E[ζ˜Tn(t2)−ζ˜Tn(t1)]4≤CL|t2−t1|2 for all 0≤t1≤t2≤L .

By Fatou’s lemma, E[ζ(t2)−ζ(t1)]4≤CL|t2−t1|2. Thus, the processes ζ˜Tn(t) and ζ(t) are continuous with probability one. We have that finite-dimensional distributions of the processes ζ˜Tn(t) converge, as Tn→∞ , to the correspondent finite-dimensional distributions of the process ζ(t) . For a weak convergence of the processes ζTn(t) it is sufficient (see [1], Chapter IX, §2) to prove limh→0limTn→∞‾P{sup|t1−t2|≤hti≤L|ζTn(t2)−ζTn(t1)|>ε}=0 \varepsilon \Big\}=0\]]]> for any L>0 0$]]>, ε>0 0$]]>.

Due to inequalities (see [1], Chapter IX, §3) P{sup|t1−t2|≤hti≤L|ζTn(t2)−ζTn(t1)|>ε}≤∑kh≤LP{supkh≤t≤(k+1)h|ζTn(t)−ζTn(kh)|>ε4}≤(4ε)48∑kh≤L{Esupkh≤t≤(k+1)h(∫khtLTn(ξT(s))ds)4+Esupkh≤t≤(k+1)h(∫khtGTn′(ξT(s))dWT(s))4}≤(4ε)4KL∑kh≤L(h4+h2), \varepsilon \big\}\\{} & \displaystyle \hspace{1em}\le \sum \limits_{kh\le L}\operatorname{\mathsf{P}}\bigg\{\underset{kh\le t\le (k+1)h}{\sup }\big|\zeta _{T_{n}}(t)-\zeta _{T_{n}}(kh)\big|>\frac{\varepsilon }{4}\bigg\}\\{} & \displaystyle \hspace{1em}\le {\bigg(\frac{4}{\varepsilon }\bigg)}^{4}8\sum \limits_{kh\le L}\Bigg\{\mathsf{E}\underset{kh\le t\le (k+1)h}{\sup }{\Bigg(\hspace{0.2778em}{\int _{kh}^{t}}L_{T_{n}}\big(\xi _{T}(s)\big)\hspace{0.1667em}ds\Bigg)}^{4}\\{} & \displaystyle \hspace{2em}+\mathsf{E}\underset{kh\le t\le (k+1)h}{\sup }{\Bigg(\hspace{0.2778em}{\int _{kh}^{t}}{G^{\prime }_{T_{n}}}\big(\xi _{T}(s)\big)\hspace{0.1667em}dW_{T}(s)\Bigg)}^{4}\Bigg\}\le {\bigg(\frac{4}{\varepsilon }\bigg)}^{4}K_{L}\sum \limits_{kh\le L}\big({h}^{4}+{h}^{2}\big),\end{array}\]]]> where KL are independent of Tn , we obtain (14). The proof of Theorem 3.1 is complete.  □

Rewrite equality (5) as ζT(t)=GT(x0)+∫0ta0(ζT(s))ds+ηT(t)+αT(0)(t)+αT(1)(t), where ηT(t)=∫0tGT′(ξT(s))dWT(s),αT(0)(t)=∫0tGT′(ξT(s))ΔaT(s)ds,ΔaT(s)=aT(s,ξT(s))−aˆT(ξT(s)),αT(1)(t)=∫0tqT(1)(ξT(s))ds,qT(1)(x)=GT′(x)aˆT(x)+12GT″(x)−a0(GT(x)).

The conditions ( A0 ) and ( A1 ), together with inequality (11), imply that sup0≤t≤L|αT(0)(t)|≤∫0L|GT′(ξT(s))||ΔaT(s)|ds≤[C(1+sup0≤s≤L|ζT(s)|2)]12∫0Lsupx|aT(s,x)−aˆT(x)|ds→P0, as T→∞ for any L>0 0$]]>.

The functions qT(1)(x) satisfy conditions of Lemma 5.2. Thus, for any L>0 0$]]> sup0≤t≤L|αT(1)(t)|→P0, as T→∞ .

It is clear that ηT(t) is a family of continuous martingales with quadratic characteristics ⟨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 conditions of Lemma 5.2. Thus, for any L>0 0$]]> sup0≤t≤L|αT(2)(t)|→P0, as T→∞ .

According to Theorem 3.1, the family of the processes ζT(t) is weakly compact. It is easy to see that compactness conditions (14) are fulfilled for the processes ηT(t) . Using convergences (16), (17), (19), we have that relations (14) hold for the processes αT(k)(t) , k=0,1,2 as well. It means that we can apply Skorokhod’s convergent subsequence principle (see [11], Chapter I, §6) for the processes (ζT(t),ηT(t),αT(k)(t),k=0,1,2).

According to this principle, given an arbitrary sequence Tn′→∞ , we can choose a subsequence Tn→∞ , a probability space (Ω˜,ℑ˜,P˜) , and stochastic processes (ζ˜Tn(t),η˜Tn(t),α˜Tn(k)(t),k=0,1,2) defined on this space such that their finite-dimensional distributions coincide with those of the processes (ζTn(t),ηTn(t),αTn(k)(t),k=0,1,2), and, moreover, ζ˜Tn(t)→P˜ζ˜(t),η˜Tn(t)→P˜η˜(t),α˜Tn(k)(t)→P˜α˜(k)(t),k=0,1,2, as Tn→∞ , for all 0≤t≤L , where ζ˜(t) , η˜(t) , α˜(k)(t),k=0,1,2 are some stochastic processes.

Evidently, relations (16)–(19) imply that α˜(k)(t)≡0 , k=0,1,2 a.s. According to (12), the processes ζ˜(t) and η˜(t) are continuous with probability one. Moreover, applying Lemma 5.3 together with equalities (15) and (18), we obtain that ζ˜Tn(t)=GTn(x0)+∫0ta0(ζ˜Tn(s))ds+α˜Tn(0)(t)+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=0,1,2, as Tn→∞ .

An analogue of convergence (14) holds for the processes ζ˜Tn(t) and η˜Tn(t) . Therefore, according to the well-known result of Prokhorov (Lemma 1.11 in [10]), we conclude that for any L>0 0$]]> sup0≤t≤L|ζ˜Tn(t)−ζ˜(t)|→P˜0,sup0≤t≤L|η˜Tn(t)−η˜(t)|→P˜0, as Tn→∞ .

According to Lemma 5.4, we can pass to the limit in (20) and obtain ζ˜(t)=y0+∫0ta0(ζ˜(s))ds+η˜(t), where η˜(t) is the almost surely continuous martingale with the quadratic characteristic ⟨η˜⟩(t)=∫0tσ02(ζ˜(s))ds.

Now, it is well known, that the latter representation provides the existence of a Wiener process Wˆ(t) such that η˜(t)=∫0tσ0(ζ˜(s))dWˆ(s).

Thus, the process (ζ˜,Wˆ) satisfies equation (4), and the processes ζ˜Tn(t) weakly converge, as Tn→∞ , to the process ζ˜ . Since the sequence Tn′→∞ is arbitrary and since a solution to equation (4) is weakly unique, the proof of the Theorem 3.2 is complete.  □

The proof of Theorems 3.3–3.4, 3.6–3.7 is performed similarly to the proof of the corresponding theorems in [8] with some differences that we discuss below.

The proof of Theorem 3.4 differs from the proof of Theorem 2.3 in [8] by using other representation for the functional βT(1)(t)=∫0tgT(ξT(s))ds . In this case we have βT(1)(t)=2∫GT(x0)ζT(t)g0(u)du−2∫0tg0(ζT(s))dηT(s)+γT(1)(t)−γT(2)(t)−γT(0)(t), where γT(1)(t)=∫x0ξT(t)qˆT(u)du,γT(2)(t)=∫0tqˆT(ξT(s))dWT(s), γT(0)(t)=∫0tΦT′(ξT(s))[aT(s,ξT(s))−aˆT(ξT(s))]ds, ΦT(x)=2∫0xfT′(u)(∫0ugT(v)fT′(v)dv)du, qˆT(x)=ΦT′(x)−2g0(GT(x))GT′(x), fT′(x) is the derivative of the function fT(x) defined by equality (3).

The latter representation differs from the corresponding representation in [8] by the term γT(0)(t) . For any constants ε>0 0$]]>, N>0 0$]]> and L>0 0$]]>, we have the inequalities P{sup0≤t≤L|γT(0)(t)|>ε}≤PNT+2ε∫0LE|ΦT′(ξT(s))|×|aT(s,ξT(s))−aˆT(ξT(s))|χ|ξT(s)|≤Nds≤PNT+2εCN∫0Lsupx[aT(s,x)−aˆT(x)]ds, \varepsilon \Big\}& \displaystyle \le P_{\mathit{NT}}+\frac{2}{\varepsilon }{\int _{0}^{L}}\mathsf{E}\big|{\varPhi ^{\prime }_{T}}\big(\xi _{T}(s)\big)\big|\\{} & \displaystyle \hspace{1em}\times \big|a_{T}\big(s,\xi _{T}(s)\big)-\hat{a}_{T}\big(\xi _{T}(s)\big)\big|\chi _{|\xi _{T}(s)|\le N}\hspace{0.1667em}ds\\{} & \displaystyle \le P_{\mathit{NT}}+\frac{2}{\varepsilon }C_{N}{\int _{0}^{L}}\underset{x}{\sup }\big[a_{T}(s,x)-\hat{a}_{T}(x)\big]\hspace{0.1667em}ds,\end{array}\]]]> where PNT=P{sup0≤t≤L|ξT(t)|>N} N\}$]]>. Using condition 3) from Definition 2.2 and inequality (11), we obtain the convergence limN→∞limT→∞‾PNT=0 . Using the assumptions of Theorem 3.4, we conclude that sup0≤t≤L|γT(0)(t)|→P0 for any L>0 0$]]>, as T→∞ . The rest of the proof of Theorem 3.4 is the same as that of Theorem 2.3 in [8].

The proofs of Theorem 3.3 and Theorems 3.6–3.7 are literally the same as those of Theorem 2.2 and Theorems 2.4–2.5 from [8].

For the functional βT(1)(t)=∫0tgT(ξT(s))ds , with probability one, for all t≥0 , we have the representation βT(1)(t)=2c0∫0taˆT(ξT(s))ds+γT(t)−ηT(1)(t)−γT(0)(t)+γT(3)(t), where γT(t)=2∫x0ξT(t)[fT′(u)∫0ugT(v)fT′(v)dv−c0]du,ηT(1)(t)=∫0t[ΦT′(ξT(s))−2c0]dWT(s),γT(3)(t)=2c0∫0t[aT(s,ξT(s))−aˆT(ξT(s))]ds, γT(0)(t) and ΦT(x) are defined in Remark 4.1.

The functions aˆT(x) satisfy condition ( A3 ). Thus, using Lemma 5.2, we conclude that for any L>0 0$]]> sup0≤t≤L|∫0taˆT(ξT(s))ds|→P0, as T→∞ .

For any constants ε>0 0$]]>, N>0 0$]]> and L>0 0$]]>, we have the inequalities P{sup0≤t≤L|γT(t)|>ε}≤PNT+1εEsup0≤t≤L|∫x0ξT(t)[ΦT′(u)−2c0]du|χ{|ξT(t)|≤N}≤PNT+2εNsup|x|≤N|∫x0x[fT′(u)∫0ugT(v)fT′(v)dv−c0]du|, \varepsilon \Big\}& \displaystyle \le P_{\mathit{NT}}+\frac{1}{\varepsilon }\mathsf{E}\underset{0\le t\le L}{\sup }\bigg|{\int _{x_{0}}^{\xi _{T}(t)}}\big[{\varPhi ^{\prime }_{T}}(u)-2c_{0}\big]\hspace{0.1667em}du\bigg|\chi _{\{|\xi _{T}(t)|\le N\}}\\{} & \displaystyle \le P_{\mathit{NT}}+\frac{2}{\varepsilon }N\underset{|x|\le N}{\sup }\bigg|\hspace{0.2778em}\hspace{0.2778em}{\int _{x_{0}}^{x}}\bigg[{f^{\prime }_{T}}(u){\int _{0}^{u}}\frac{g_{T}(v)}{{f^{\prime }_{T}}(v)}\hspace{0.1667em}dv-c_{0}\bigg]\hspace{0.1667em}du\bigg|,\end{array}\]]]> where PNT is the same as that in Remark 4.1. Using the latter inequality and the assumptions of Theorem 3.5, we conclude that sup0≤t≤L|γT(t)|→P0, as T→∞ .

Since the term γT(0)(t) is the same as that in Remark 4.1, we have (22).

The inequality sup0≤t≤L|γT(3)(t)|≤2c0∫0Lsupx[aT(s,x)−aˆT(x)]ds implies that for any L>0 0$]]> sup0≤t≤L|γT(3)(t)|→P0, as T→∞ .

Thus, we have for any L>0 0$]]> sup0≤t≤L|βT(1)(t)+ηT(1)(t)|→P0, as T→∞ .

It is clear that ηT(1)(t) is the almost surely continuous martingale with the quadratic characteristic ⟨ηT(1)⟩(t)=4b02t+∫0tqT(ξT(s))ds, where qT(x)=[ΦT′(x)−2c0]2−4b02 . The functions qT(x) satisfy condition ( A3 ). Thus, using Lemma 5.2, we conclude that for any L>0 0$]]> sup0≤t≤L|⟨ηT(1)⟩(t)−4b02t|→P0, as T→∞ .

Then, using a random change of time (see [2], Part I, §4), we obtain ηT(1)(t)=WT∗(⟨ηT(1)⟩(t)) , where WT∗(t) is a Wiener process. The same arguments as those used to get (9) in [7] yield that sup0≤t≤L|βT(1)(t)−WT∗(4b02t)|→P0, as T→∞ . Thus, the processes βT(1)(t) weakly converge, as T→∞ , to the process 2b0W(t) .  □

 Auxiliary results

Let ξT be a solution to equation (1) from the class K(GT) . Then, for any N>0 0$]]>, L>0 0$]]> and any Borel set B⊂[−N;N] , there exists a constant CL such that ∫0LP{GT(ξT(s))∈B}ds≤CLψ(λ(B)), where λ(B) is the Lebesgue measure of the set B, ψ(|x|) is a certain bounded function satisfying ψ(|x|)→0 as |x|→0 .

Consider the function ΦT(x)=2∫0xfT′(u)(∫0uχB(GT(v))fT′(v)dv)du.

The function ΦT(x) is continuous, the derivative ΦT′(x) of this function is continuous and the second derivative ΦT″(x) exists a.e. with respect to the Lebesgue measure and is locally integrable. Therefore, we can apply the Itô formula to the process ΦT(ξT(t)) , where ξT(t) is a solution to equation (1).

Furthermore, ΦT′(x)aˆT(x)+12ΦT″(x)=χB(x), a.e. with respect to the Lebesgue measure. Using the Itô formula and the latter equality, we conclude that ∫0LχB(ζT(s))ds=ΦT(ξT(L))−ΦT(x0)−∫0LΦT′(ξT(s))dWT(s)−αT(L), with probability one for all t≥0 , where ζT(t)=GT(ξT(t)) , αT(t)=∫0tΦT′(ξT(s))[aT(s,ξT(s))−aˆT(ξT(s))]ds. Hence, using the properties of stochastic integrals, we obtain that ∫0LP{ζT(s)∈B}ds=E[ΦT(ξT(L))−ΦT(x0)]−EαT(L).

According to condition ( A2 ), inequalities |GT(x)|≥C|x|α , C>0 0$]]>, α>0 0$]]> and (11), we have |E[ΦT(ξT(L))−ΦT(x0)]|≤CL(1)ψ(λ(B)), for a certain constant CL(1) . Condition ( A0 ) implies that |EαT(L)|≤CL(2)ψ(λ(B)), for a certain constant CL(2) . Here the function ψ(λ(B)) is from condition ( A2 ). The latter inequalities and equality (23) prove Lemma 5.1.  □

Let ξT be a solution to equation (1) from the class K(GT) . If, for measurable locally bounded functions qT(x) , condition ( A3 ) holds, then, for any L>0 0$]]>, sup0≤t≤L|∫0tqT(ξT(s))ds|→P0, as T→∞ .

Consider the function ΦT(x)=2∫0xfT′(u)(∫0uqT(v)fT′(v)dv)du.

The function ΦT(x) is continuous, the derivative ΦT′(x) of this function is continuous and the second derivative ΦT″(x) exists a.e. with respect to the Lebesgue measure and is locally integrable. Therefore, we can apply the Itô formula to the process ΦT(ξT(t)) , where ξT(t) is a solution to equation (1).

Furthermore, ΦT′(x)aˆT(x)+12ΦT″(x)=qT(x), a. e. with respect to the Lebesgue measure. Using the latter equality, we conclude that with probability one for all t≥0 ∫0tqT(ξT(s))ds=ΦT(ξT(t))−ΦT(x0)−∫0tΦT′(ξT(s))dWT(s)−αT(t), where αT(t)=∫0tΦT′(ξT(s))[aT(s,ξT(s))−aˆT(ξT(s))]ds.

For any constants ε>0 0$]]>, N>0 0$]]> and L>0 0$]]>, we have P{sup0≤t≤L|αT(t)|>ε}≤PNT+2εsup|x|≤NfT′(x)|∫0xqT(v)fT′(v)dv|×∫0Lsupx[aT(s,x)−aˆT(x)]ds, \varepsilon \Big\}& \displaystyle \le P_{\mathit{NT}}+\frac{2}{\varepsilon }\underset{|x|\le N}{\sup }{f^{\prime }_{T}}(x)\bigg|{\int _{0}^{x}}\frac{q_{T}(v)}{{f^{\prime }_{T}}(v)}\hspace{0.1667em}dv\bigg|\\{} & \displaystyle \hspace{1em}\times {\int _{0}^{L}}\underset{x}{\sup }\big[a_{T}(s,x)-\hat{a}_{T}(x)\big]\hspace{0.1667em}ds,\end{array}\]]]> where PNT=P{sup0≤t≤L|ξT(t)|>N} N\}$]]>.

The same arguments as those used in [8] (see the proof of Lemma 4.2) and the assumptions of Lemma 5.2 yield that sup0≤t≤L|αT(t)|→P0,sup0≤t≤L|ΦT(ξT(t))−ΦT(x0)|→P0,sup0≤t≤L|∫0tΦT′(ξT(s))dWT(s)|→P0, as T→∞ . Thus, equality (24) implies the statement of Lemma 5.2.  □

The statements and the proofs of the following lemmas are the same as those of the corresponding lemmas from [8].

Let ξT be a solution to equation (1) belonging to the class K(GT) , and let the stochastic process (ζT,ηT) , with ζT(t)=GT(ξT(t)) and ηT(t)=∫0tGT′(ξT(s))dWT(s) be stochastically equivalent to the process (ζ˜T,η˜T) . Then the process ∫0tg(ζT(s))ds+∫0tq(ζT(s))dηT(s), where g(x) and q(x) are measurable locally bounded functions, is stochastically equivalent to the process ∫0tg(ζ˜T(s))ds+∫0tq(ζ˜T(s))dη˜T(s).

Let ξT be a solution to equation (1) from the class K(GT) , and let ζT(t)=GT(ξT(t))→Pζ(t) as T→∞ . Then for any measurable locally bounded function g(x) , we have the convergence sup0≤t≤L|∫0tg(ζT(s))ds−∫0tg(ζ(s))ds|→P0, as T→∞ for any constant L>0 0$]]>.

 Examples

We denote by bT the family of such constants that bT>1 1$]]> and bT↑∞ as T→∞ .

Consider equation (1) with the drift coefficient with nonregular dependence on the parameter T of the form aT(t,x)=bTγcos(xbT)+tbT1+t2bT2sin((x−1)bT),0≤γ<1.

The family of measurable locally bounded real-valued functions aˆT(x)=bTγcos(xbT) satisfies condition 1) from Definition 2.2: for any L>0 0$]]> limT→∞∫0Lsupx|aT(t,x)−aˆT(x)|dt≤limT→∞∫0LtbT1+t2bT2dt=0.

The rest of conditions from Definition 2.2 are fulfilled for the family of functions GT(x)=fT(x)=∫0xexp{−2∫0uaˆT(v)dv}du,T>0. 0.\]]]>

Since fT′(x)=exp{−2bTγbTsin(xbT)} , then there exist two constants c0 and δ0 such that, for all x∈R , we have 0<δ0≤fT′(x)≤c0 . Taking into account that GT(x)=∫0xfT′(v)dv , we obtain GT′(x)aˆT(x)+12GT″(x)≡0 . Therefore, the conditions of Definition 2.2 are fulfilled as follows:

[GT′(x)aT(t,x)+12GT″(x)]2+[GT′(x)]2=[GT′(x)tbT1+t2bT2sin((x−1)bT)]2+[GT′(x)]2≤2[GT′(x)]2≤2c02≤2c02[1+|GT(x)|2];|GT(x0)|=|∫0x0fT′(v)dv|≤c0·|x0|=C;

|GT(x)|=|∫0xfT′(v)dv|≥C|x|αwith C=δ0,α=1;

|∫0xfT′(u)(∫0uχB(GT(v))fT′(v)dv)du|≤C0δ0|∫0x∫0uχB(GT(v))dvdu|≤C1λ(B)|x|≤ψ(λ(B))[1+|x|m] with ψ(|x|)=C1|x| , m=1 .

Thus, equation (1) belongs to the class K(GT) . According to Theorem 3.1, the family of the processes ζT(t)=GT(ξT(t)) is weakly compact. We can find the form of the limit process using Theorem 3.2 with a0(x)≡0 , σ0(x)≡1 . According to Theorem 3.2, the stochastic processes ζT(t) weakly converge, as T→∞ , to the solution ζ(t) to equation (4) and the limit process is ζ(t)=x0+Wˆ(t) , where Wˆ(t) is a Wiener process.

Let the conditions of Example 6.1 hold. For the family of functions gT(x)=bTγ1+bT2x2,0≤γ<1, the assumptions of Theorem 3.3 hold with g0(x)≡0 .

According to Theorem 3.3, the stochastic processes βT(1)(t)=∫0tgT(ξT(s))ds=∫0tbTγ1+bT2ξT2(s)ds,0≤γ<1 weakly converge, as T→∞ , to the process β(1)(t)≡0 .

 Acknowledgments

The authors are deeply grateful to the referees for careful reading and valuable comments and suggestions, which helped to improve essentially the quality of the article.

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