arXiv:1607.03661v1 [math.PR] 13 Jul 2016
Modern Stochastics: Theory and Applications 3 (2016) 191–208 DOI: 10.15559/16-VMSTA58
Asymptotic behavior of homogeneous additive functionals of the solutions of Itô stochastic differential equations with nonregular dependence on parameter Grigorij Kulinich, Svitlana Kushnirenko∗, Yuliia Mishura Taras Shevchenko National University of Kyiv, 64/13, Volodymyrska Street, 01601, Kyiv, Ukraine
[email protected] (G. Kulinich),
[email protected] (S. Kushnirenko),
[email protected] (Yu. Mishura) Received: 28 May 2016, Revised: 17 June 2016, Accepted: 17 June 2016, Published online: 4 July 2016
Abstract WeR study the asymptotic behavior of mixed functionals of the form IT (t) = t FT (ξT (t)) + 0 gT (ξT (s)) dξT (s), t ≥ 0, as T → ∞. Here ξT (t) is a strong solution of the stochastic differential equation dξT (t) = aT (ξT (t)) dt + dWT (t), T > 0 is a parameter, aT = aT (x) are measurable functions such that |aT (x)| ≤ CT for all x ∈ R, WT (t) are standard Wiener processes, FT = FT (x), x ∈ R, are continuous functions, gT = gT (x), x ∈ R, are locally bounded functions, and everything is real-valued. The explicit form of the limiting processes for IT (t) is established under very nonregular dependence of gT and aT on the parameter T . Keywords Diffusion-type processes, asymptotic behavior of additive functionals, nonregular dependence on the parameter 2010 MSC 60H10, 60J60
1
Introduction
Consider the Itô stochastic differential equation dξT (t) = aT ξT (t) dt + dWT (t), ∗ Corresponding
t ≥ 0, ξT (0) = x0 ,
author.
© 2016 The Author(s). Published by VTeX. Open access article under the CC BY license.
www.i-journals.org/vmsta
(1)
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where T > 0 is a parameter, aT (x), x ∈ R, are real-valued measurable functions such that for some constants LT > 0 and for all x ∈ R |aT (x)| ≤ LT , and WT = {WT (t), t ≥ 0}, T > 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 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 x as T → ∞, or it can be equal to T sin((x − k √ 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 coeffi√ cients aT (x) = T a(x T ) 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 R a(x) dx = λ. The sufficiency of the condition λ = 0 for the asymptotic equivR alence of distributions ξT and WT (t) is established in [5], and the necessity of this condition is proved in [1]. If λ 6= 0, then we can deduce from [4] that the distributions of the solution ξT of equation (1) weakly converge as T → ∞ to the corˆ = l (ζ(t)), where l(x) = c1 x responding distributions of the Markov process ξ(t) for x > 0 and l(x) = c2 x 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 and σ ¯ (x) = σ2 for x < 0, and Rt P {|ζ(s)| = 0} ds = 0. The explicit form of the transition density of the process 0 ˆ ξ(t) is obtained. Moreover, in [10], it is proved that ˆ = x0 + β(t) + W (t), ξ(t) where β(t) is a certain functional of ζ(t), and the necessity of the condition λ 6= 0 is established for the weak convergence as T → ∞ of the solution ξT of equation (1) to ˆ the process ξ(t). √ in [5] and [4], a probabilistic method to study the “awkward” term √ Furthermore, Rt T 0 a(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) Rt and a family of martingales 0 Φ′T (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) +
1 ′′ f (x) = 0. 2 T
Asymptotic behavior of homogeneous additive functionals
It is obvious that the functions fT (x) have the form ( ) Z x Z u (1) (2) fT (x) = cT exp − 2 aT (v) dv du + cT , 0
(1)
193
(2)
0
(2)
where cT and cT 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 (2) (1) are locally integrable. Note that cT are normalizing constants and cT are centralizing constants in the limit theorems (see [18], §6). Further, for simplicity, we assume (1) (2) that in (2), cT ≡ 1 and cT ≡ 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 G′T (x) and locally integrable second derivatives G′′T (x) a.e. with respect to the Lebesgue measure such that, for all T > 0 and x ∈ R, the following inequalities hold: 2 2 2 1 (A1 ) G′T (x)aT (x) + G′′T (x) + G′T (x) ≤ 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: (i) There exist constants C > 0 and α > 0 such that |GT (x)| ≥ C|x|α . (ii) 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 and for any measurable bounded set B, the following inequality holds: ! Z x Z u χB (GT (v)) ′ (A2 ) fT (u) dv du ≤ ψ λ(B) [1 + |x|m ] , ′ fT (v) 0 0 where χB (v) is the indicator function of a set B, λ(B) is the Lebesgue measure of B, and fT′ (x) is the derivative of the function fT (x) defined by equality (2).
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 K (GT ). It is easy (1) (2) to understand that class K (GT ) does not depend on the constants cT and cT in representation (2). It is clear that if there exist constants δ > 0 and C > 0 such that 0 < δ ≤ fT′ (x) ≤ C for all x ∈ R, T > 0, then the corresponding equations (1) belong to the class K (GT ) for GT (x) = fT (x). We denote this subclass as K1 . Note that the class K (GT ) 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 → ∞. c0 T x For example, consider equation (1) with aT (x) = 1+x 2 T . It is easy to obtain that 1 1 ′ fT (x) = (1+x2 T )c0 , and if c0 > − 2 , then such equations belong to the class K (GT ) with GT (x) = x2 (here, at points x 6= 0, we have fT′ (x) → 0 for c0 > 0, fT′ (x) → ∞ for − 21 < c0 < 0, and fT′ (x) ≡ 1 for c0 = 0).
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For the class of equations K (GT ), we study the asymptotic behavior as T → ∞ of the distributions of the following functionals: Z t Z t (2) (1) gT ξT (s) dWT (s), βT (t) = gT ξT (s) ds, βT (t) = 0 0 Z t Z t gT ξT (s) dξT (s), βT (t) = gT ξT (s) dWT (s), IT (t) = FT ξT (t) + 0
0
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 (2) (1) of functionals βT (t), βT (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 hηT i(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], (2) (1) the behavior of the distributions of the functionals βT √(t), βT√(t), and IT (t) with a special dependence of the drift coefficients aT (x) = T a(x T ) on the parameter T is considered, mainly in the case where |xa(x)| ≤ C for all x ∈ R. The behavior (1) (2) of the distributions of functionals βT (t) was studied in [13], βT (t) was studied in [14], and IT (t) was investigated in [15]. A more detailed review of the known results (1) (2) in this area is presented in [13–15]. Note that the functionals βT (t), βT (t), and βT (t) are particular cases of the functional IT (t) (see [15], Lemma 4.1). Remark 1.1. In this paper, we often apply the Itô formula to the process Φ(ξT (t)), where ξT (t) is a solution of equation (1), the derivative Φ′ (x) of the function Φ(x) is assumed to be continuous, and the second derivative Φ′′ (x) is assumed to exist a.e. with respect to the Lebesgue measure and to be locally integrable. Then it follows from [3] that with probability one, for all t ≥ 0, the following equality holds: Z t 1 Φ ξT (t) − Φ(x0 ) = Φ′ ξT (s) aT ξT (s) + Φ′′ ξT (s) ds 2 0 Z t + Φ′ ξT (s) dWT (s). 0
Remark 1.2. Let ξT be a solution of equation (1), and GT (x) be a family of functions satisfying condition (A1 ). Theorem 1 from [12] implies that the family of the processes {ζT (t) = GT (ξT (t)), t ≥ 0} is weakly compact. The proof of this result is based on the equality Z t 1 G′T ξT (s) aT ξT (s) + G′′T ξT (s) ds + ηT (t), (3) ζT (t) = GT (x0 ) + 2 0
where
ηT (t) =
Z
0
t
G′T ξT (s) dWT (s),
ζT (t) = GT ξT (t) .
Asymptotic behavior of homogeneous additive functionals
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In turn, the latter equality follows from Remark 1.1. In addition, it is established in the proof of Theorem 1 from [12] that for any constants L > 0 and ε > 0, lim lim sup P λT (t) > N = 0, N →∞ T →∞ 0≤t≤L
lim lim
sup
h→0 T →∞ |t1 −t2 |≤h; ti ≤L
P λT (t2 ) − λT (t1 ) > ε = 0,
and that, for any k > 1 and for certain constants Ck and C, k 4 E sup λT (t) ≤ Ck , E λT (t2 ) − λT (t1 ) ≤ C|t2 − t1 |2 ,
(4)
(5)
0≤t≤L
where λ = η or λ = ζ (see [2], §6, Theorem 4).
Remark 1.3. Here and throughout the paper, the weak convergence of the processes means the weak convergence in the uniform topology of the space of continuous functions C[0, L] for any L > 0. The processes that have continuous trajectories with probability 1 will be simply called continuous. 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. 2
Statement of the main results
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, the following condition holds: Z x q (v) T ′ dv = 0, (A3 ) lim sup fT (x) T →∞ |x|≤N 0 fT′ (v)
where fT′ (x) is the derivative of the function fT (x) defined by Eq. (2).
Theorem 2.1. Let ξT be a solution of Eq. (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: 1. the functions (1)
qT (x) = G′T (x)aT (x) + and
1 ′′ GT (x) − a0 GT (x) 2
2 (2) qT (x) = G′T (x) − σ02 GT (x)
satisfy assumption (A3 ); 2. the Itô equation
ζ(t) = y0 +
Z
0
has a unique weak solution.
t
a0 ζ(s) ds +
Z
0
t
ˆ (s) σ0 ζ(s) dW
(6)
Then the stochastic process ζT (t) = GT (ξT (t)) weakly converges, as T → ∞, to the solution ζ(t) of Eq. (6).
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Theorem 2.2. Let ξT be a solution of Eq. (1) from the class K (GT ), and let the assumptions of Theorem 2.1 hold. Assume that, for measurable 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) Rt (1) satisfies assumption (A3 ). Then the stochastic process βT (t) = 0 gT (ξT (s)) ds weakly converges, as T → ∞, to the process Z t (1) β (t) = g0 ζ(s) ds, 0
where ζ(t) is a solution of Eq. (6).
Theorem 2.3. Let ξT be a solution of Eq. (1) from the class K (GT ), and let the assumptions of Theorem 2.1 hold. Assume that, for measurable locally bounded functions gT (x), there exists a measurable locally bounded function g0 (x) such that Z x ′ gT (v) ′ G (x) G (x) dv−g (A4 ) lim sup fT (x) =0 T 0 T ′ T →∞ |x|≤N 0 fT (v) Rt (1) for all N > 0. Then the stochastic process βT (t) = 0 gT (ξT (s)) ds weakly converges, as T → ∞, to the process ! Z t Z ζ(t) (1) ˆ (s) , β˜ (t) = 2 g0 ζ(s) σ0 ζ(s) dW g0 (x) dx − y0
0
ˆ (t) are related via Eq. (6). where ζ(t) and the Wiener process W
Theorem 2.4. Let ξT be a solution of equation (1) from the class K (GT ), and let the assumptions of Theorem 2.1 hold. Assume that, for measurable locally bounded functions gT (x), there exists a measurable locally bounded function g0 (x) such that the function 2 qT (x) = gT (x) − g0 GT (x) G′T (x) Rt (2) satisfies assumption (A3 ). Then the stochastic process βT (t) = 0 gT (ξT (s)) dWT (s), where ξT (t) and WT (t) are related via Eq. (1), weakly converges, as T → ∞, to the process Z t Z t (2) β (t) = g0 ζ(s) dζ(s) − g0 ζ(s) a0 ζ(s) ds, 0
0
where ζ(t) is a solution of Eq. (6).
Theorem 2.5. Let ξT be a solution of Eq. (1) from the class K (GT ), and let the assumptions of Theorem 2.1 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, lim sup FT (x) − F0 GT (x) = 0, T →∞ |x|≤N
and let the functions gT (x) and g0 (x) satisfy the assumptions of Theorem 2.4. Then the stochastic process
Asymptotic behavior of homogeneous additive functionals
IT (t) = FT ξT (t) +
Z
197
t 0
gT ξT (s) dWT (s),
where ξT (t) and WT (t) are related via Eq. (1), weakly converges, as T → ∞, to the process Z t ˆ (s), g0 ζ(s) σ0 ζ(s) dW I0 (t) = F0 ζ(t) + 0
ˆ (t) are related via Eq. (6). where ζ(t) and the Wiener process W
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 K (GT ) 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)), Z t IT (t) = FT ξT (t) + gT ξT (s) dWT (s), 0
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 Rt process (ζ(t), I(t)), where I(t) = F0 (ζ(t)) + 0 g0 (ζ(s)) dζ(s), and ζ(t) is a weak Rt solution of the Itô equation ζ(t) = y0 + 0 σ0 (ζ(s)) dW (s), if and only if there exist (1) (2) constants cT and cT in (2) such that, as T → ∞: 1. for all x, Z
ϕT (x)
0
[fT′ (v)]2 − σ02 (fT (v)) dv → 0, fT′ (v)
where ϕT (x) is the inverse function of the function fT (x); 2. for all N > 0, sup FT (x) + fT (x) − F0 fT (x) → 0
|x|≤N
and Z
N
−N
3
|gT (x) − fT′ (x)[1 + g0 (fT (x))]|2 dx → 0. fT′ (x)
Proof of the main results
Proof of Theorem 2.1. Rewrite Eq. (3) as Z t (1) ζT (t) = GT (x0 ) + a0 ζT (s) ds + αT (t) + ηT (t),
(7)
0
where
(1) αT (t)
=
Z
0
t
(1)
qT
ξT (s) ds,
1 (1) qT (x) = G′T (x)aT (x)+ G′′T (x)−a0 GT (x) . 2
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The functions qT (x) satisfy the conditions of Lemma 4.2. Thus, for any L > 0, (1) P sup αT (t) → 0 (8) 0≤t≤L
as T → ∞. It is clear that ηT (t) is a family of continuous martingales with quadratic characteristics Z t Z t 2 (2) (9) ds = σ02 ζT (s) ds + αT (t), G′T ξT (s) hηT i(t) = 0
0
where
(2) αT (t)
=
Z
0
t
(2)
qT (2)
ξT (s) ds,
2 (2) qT (x) = G′T (x) − σ02 GT (x) .
The functions qT (x) satisfy the conditions of Lemma 4.2. Thus, for any L > 0, (2) P sup αT (t) → 0 (10) 0≤t≤L
as T → ∞. We have that relations (4) and (5) hold for the processes ζT (t) and ηT (t), and, (k) according to (8) and (10), these relations hold for the processes αT (t), k = 1, 2, as well. This means that we can apply Skorokhod’s convergent subsequence principle (1) (2) (see [17], Chapter I, §6) for the process (ζT (t), ηT (t), αT (t), αT (t)): given an arbi′ trary sequence Tn → ∞, we can choose a subsequence Tn → ∞, a probability space (2) (1) ˜ P), ˜ and a stochastic process (ζ˜T (t), η˜T (t), α ˜ ℑ, ˜ Tn (t)) defined on this ˜ Tn (t), α (Ω, n n space such that its finite-dimensional distributions coincide with those of the process (2) (1) (ζTn (t), ηTn (t), αTn (t), αTn (t)) and, moreover, ˜ P ˜ ζ˜Tn (t) → ζ(t),
˜ P
η˜Tn (t) → η˜(t),
(1)
˜ P
˜(1) (t), α ˜ Tn (t) → α
(2)
˜ P
˜ (2) (t) α ˜ Tn (t) → α
˜ 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 ˜ and η˜(t) are continuous. Moreover, applying Lemma 4.5 to (5), the processes ζ(t) together with Eqs. (7) and (9), we obtain that Z t (1) ˜ Tn (t) + η˜Tn (t), a0 ζ˜Tn (s) ds + α ζ˜Tn (t) = GTn (x0 ) + 0 (11) Z t (2) 2 ˜ ˜ Tn (t), σ0 ζTn (s) ds + α h˜ ηTn i(t) = 0
˜ ˜ P P (k) ˜ αTn (t)| → 0, k = 1, 2, as Tn → η˜Tn (t) → η˜(t), sup0≤t≤L |˜ where ζ˜Tn (t) → ζ(t), ∞. In addition, it is established in [12] that, for any constants L > 0 and ε > 0, o n ˜ ˜ ζTn (t2 ) − ζ˜Tn (t1 ) > ε = 0. lim lim P sup ˜ P
h→0 Tn →∞
|t1 −t2 |≤h; ti ≤L
Using the latter convergence and (11), we conclude that, for any constants L > 0 and ε > 0,
Asymptotic behavior of homogeneous additive functionals
˜ lim lim P
h→0 Tn →∞
n
sup |t1 −t2 |≤h; ti ≤L
199
o η˜Tn (t2 ) − η˜Tn (t1 ) > ε = 0.
Therefore, according to the well-known result of Prokhorov [16], we conclude that P˜ P˜ ˜ → 0 and sup η˜Tn (t) − η˜(t) → 0 ζTn (t) − ζ(t) sup ˜ 0≤t≤L
0≤t≤L
as Tn → ∞. According to Lemma 4.3, we can pass to the limit in (11) and obtain Z t ˜ ˜ = y0 + ds + η˜(t), (12) a0 ζ(s) ζ(t) 0
where η˜(t) is a continuous martingale with the quadratic characteristic Z t ˜ h˜ η i(t) = σ02 ζ(s) ds. 0
Now, it is well known that the latter representation provides the existence of a ˆ (t) such that Wiener process W Z t ˜ ˆ (s). dW (13) η˜(t) = σ0 ζ(s) 0
˜ ˆ (t)) satisfies Eq. (6), and the process ζ˜Tn (t) weakly conThus, the process (ζ(t), W ˜ verges, as Tn → ∞, to the process ζ(t). Since the subsequence Tn → ∞ is arbitrary and since a solution of Eq. (6) is weakly unique, the proof of the Theorem 2.1 is complete. Proof of Theorem 2.2. It is clear that, for all t > 0, with probability one, Z t (1) βT (t) = g0 ζT (s) ds + αT (t), 0
Rt
where αT (t) = 0 qT (ξT (s)) ds and qT (x) = gT (x) − g0 (GT (x)). The functions qT (x) satisfy the conditions of Lemma 4.2. Thus, for any L > 0, P sup αT (t) → 0 0≤t≤L
as T → ∞. Similarly to (11), we obtain the equality Z t (1) ˜ Tn (t), g0 ζ˜Tn (s) ds + α β˜Tn (t) =
(14)
0
˜ P
˜ P
˜ is ˜ and sup αTn (t)| → 0 as Tn → ∞. The process ζ(t) where ζ˜Tn (t) → ζ(t) 0≤t≤L |˜ a solution of Eq. (12), whereas by Lemma 4.5 the finite-dimensional distributions of (1) (1) the stochastic process βTn (t) coincide with those of the process β˜Tn (t). Using Lemma 4.3 and Eq. (14), we conclude that Z t P˜ ˜(1) ˜ sup βTn (t) − g0 ζ(s) ds → 0 0≤t≤L 0 (1)
as Tn → ∞. Thus, the process βTn (t) weakly converges, as Tn → ∞, to the process Rt β (1) (t) = 0 g0 (ζ(s)) ds, where ζ(t) is a solution of Eq. (6). Since the subsequence
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Tn → ∞ is arbitrary and since a solution ζ(t) of Eq. (6) is weakly unique, the proof of Theorem 2.2 is complete. Proof of Theorem 2.3. Consider the function ! Z x Z u gT (v) ′ ΦT (x) = 2 fT (u) dv du. ′ 0 0 fT (v) Applying the Itô formula to the process ΦT (ξT (t)), where ξT (t) is a solution of Eq. (1), we get that Z t (1) Φ′T ξT (s) dWT (s) βT (t) = ΦT ξT (t) − ΦT (x0 ) − 0
=2
Z
ξT (t)
x0
+2 =2
Z
Z
g0 GT (u) G′T (u) du − 2
ξT (t)
x0 ζT (t)
GT (x0 )
Z
qˆT (u) du − 2 g0 (u) du − 2
where
Z
0
t
0 t
Z
0
t
g0 ζT (s) G′T ξT (s) dWT (s)
qˆT ξT (s) dWT (s)
(2) (1) g0 ζT (s) dηT (s) + γT (t) − γT (t),
Z t Z ξT (t) (2) (1) qˆT ξT (s) dWT (s), qˆT (u) du, γT (t) = 2 γT (t) = 2 0 x 0 Z x g (v) T ′ dv − g G (x) G (x) . qˆT (x) = fT′ (x) 0 T T ′ 0 fT (v)
Denote PN T = P{sup0≤t≤L |ξT (t)| > N }. It is clear that, for any constants ε > 0, N > 0, and L > 0, we have the inequalities Z ξT (t) n o (1) 2 P sup γT (t) > ε ≤ PN T + E sup qˆT (u) du χ{|ξT (t)|≤N } ε 0≤t≤L x0 0≤t≤L Z N 4 2 qT (x)| |ˆ qT (u)| du ≤ PN T + N sup |ˆ ≤ PN T + ε −N ε |x|≤N and P
n
o (2) sup γT (t) > ε
0≤t≤L
≤ PN T
≤ PN T
Z t 2 4 + 2 E sup qˆT ξT (s) χ{|ξT (s))≤N |} dWT (s) ε 0≤t≤L 0 Z L 16 16 qˆT2 ξT (s) χ{|ξT (s))≤N |} ds ≤ PN T + 2 L sup qˆT2 (x) . + 2E ε ε |x|≤N 0
The inequality |GT (x)| ≥ C|x|α , α > 0, together with convergence (4), implies that lim lim PN T = 0. (15) N →∞ T →∞
Asymptotic behavior of homogeneous additive functionals
201
Thus, using the conditions of Theorem 2.3, we get the convergence (k) P sup γT (t) → 0, k = 1, 2, 0≤t≤L
as T → ∞. The same arguments as we used establishing (11) yield that Z ζ˜Tn (t) Z t (2) (1) (1) ηTn (s) + γ˜Tn (t) − γ˜Tn (t), (16) β˜Tn (t) = 2 g0 (u) du − 2 g0 ζ˜Tn (s) d˜ GTn (x0 )
0
where
P˜ ˜ → 0, ζTn (t) − ζ(t) sup ˜
˜ P
sup |˜ ηTn (t) − η˜(t)| → 0,
0≤t≤L
0≤t≤L
(k) P˜ sup γ˜Tn (t) → 0,
k = 1, 2,
0≤t≤L
as Tn → ∞ for all L > 0. According to (12) with η˜(t) defined in (13), the process ˜ ˆ (t)) satisfies Eq. (6). (ζ(t), W (1)
By Lemma 4.5 the finite-dimensional distributions of the stochastic process βTn (t) (1) coincide with those of the process β˜Tn (t). Using Lemma 4.3, we can pass to the limit as Tn → ∞ in (16) and obtain P˜ (1) sup β˜Tn (t) − β˜(1) (t) → 0 (17) 0≤t≤L
as Tn → ∞, where Z t Z ζ(t) ˜ (1) ˜ ˜ d˜ η (s) g0 (u) du − 2 g0 ζ(s) β (t) = 2 0
y0 ˜ ζ(t)
=2
Z
y0
g0 (u) du − 2
Z
t
0
˜ ˜ +2 dζ(s) g0 ζ(s)
Z
0
t
˜ ˜ ds, a0 ζ(s) g0 ζ(s)
˜ is a solution of Eq. (6). Therefore, we have that Theorem 2.3 holds for the and ζ(t) (1) process βTn (t) as Tn → ∞. Since the subsequence Tn → ∞ is arbitrary and since a solution ζ(t) of Eq. (6) is weakly unique, the proof of Theorem 2.3 is complete. Proof of Theorem 2.4. It is clear that Z t (2) (18) βT (t) = g0 ζT (s) dηT (s) + γT (t), 0
where
γT (t) =
Z
t 0
qT ξT (s) dWT (s),
qT (x) = gT (x) − g0 GT (x) G′T (x).
The process γT (t) is a continuous martingale with the quadratic characteristics Z t hγT i(t) = qT2 ξT (s) ds for all T > 0. 0
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According to the conditions of Theorem 2.4, the functions qT2 (x) satisfy the conP
ditions of Lemma 4.2. Thus, for any L > 0, we have the convergence hγT i(L) → 0 as T → ∞. The following inequality holds for any constants ε > 0 and δ > 0: n o P sup γT (t) > ε ≤ δ + P hγT i(L) > ε2 δ 0≤t≤L
(see [2], §3, Theorem 2), which implies the relation P sup γT (t) → 0
(19)
0≤t≤L
as T → ∞. ˜ P), ˜ ˜ ℑ, Then, similarly to representation (11), on a certain probability space (Ω, for an arbitrary subsequence Tn , we get the equality Z t (2) ηTn (s) + γ˜Tn (t), g0 ζ˜Tn (s) d˜ β˜Tn (t) = 0
where
P˜ ˜ → ζTn (t)−ζ(t) sup ˜ 0,
P˜ η (t) → 0, sup η˜Tn (t)−˜
0≤t≤L
0≤t≤L
P˜ sup γ˜Tn (t) → 0
0≤t≤L
˜ W ˆ (t)) satisfies Eq. (6), η˜(t) is as Tn → ∞ for any L > 0, where the process (ζ(t), (2) (2) defined in (13), and the processes β˜Tn (t) and βTn (t) are stochastically equivalent. Similarly to the proof of convergence (17), we obtain P˜ (2) sup β˜Tn (t) − β˜(2) (t) → 0
0≤t≤L
as Tn → ∞, where β˜(2) (t) =
Z
t
0
=
Z
t
0
Thus, the process
˜ ˜ − dζ(s) g0 ζ(s)
(2) β˜Tn (t)
Since the subsequence Tn (2)
˜ d˜ η (s) g0 ζ(s)
Z
t 0
˜ ˜ ds. a0 ζ(s) g0 ζ(s)
weakly converges, as Tn → ∞, to the process β˜(2) (t). (2) → ∞ is arbitrary and since the processes β˜ (t) and Tn
βTn (t) are stochastically equivalent, the proof of Theorem 2.4 is complete. Proof of Theorem 2.5. It is clear that Z t g0 ζT (s) dηT (s) + αT (t) + γT (t), IT (t) = F0 ζT (t) + 0
where
αT (t) = FT
ξT (t) − F0 ζT (t) ,
ηT (t) =
Z
0
t
G′T ξT (s) dWT (s),
Asymptotic behavior of homogeneous additive functionals
γT (t) =
Z
0
203
t
qT (x) = gT (x) − g0 GT (x) G′T (x).
qT ξT (s) dWT (s),
Denote, as before, PN T = P{sup0≤t≤L |ξT (t)| > N }. Since for any constants ε > 0, N > 0, and L > 0, we have the inequality o n P sup FT ξT (t) − F0 GT ξT (t) > ε 0≤t≤L
2 E sup FT ξT (t) − F0 GT ξT (t) χ{|ξT (t)|≤N } ε 0≤t≤L 2 + sup FT (x) − F0 GT (x) , ε |x|≤N
≤ PN T +
≤ PN T
we can apply conditions of Theorem 2.5 and convergence (15) to get that P sup αT (t) → 0
0≤t≤L
as T → ∞. The proof of the fact that, for γT (t), an analogue of convergence (19) holds is literally the same as in the proof of Theorem 2.4. Then, we can apply Skorokhod’s convergent subsequence principle to the process (ζT (t), ηT (t), αT (t), γT (t)) and, similarly to representation (11), obtain the following equality for an arbitrary ˜ P): ˜ ˜ ℑ, subsequence Tn in a certain probability space (Ω, Z t ˜ ˜ ˜ Tn (t) + γ˜Tn (t), ηTn (s) + α g0 ζ˜Tn (s) d˜ ITn (t) = F0 ζTn (t) + 0
where, as Tn → ∞, for any L > 0,
P˜ sup η˜Tn (t) − η˜(t) → 0,
P˜ ˜ → 0, ζTn (t) − ζ(t) sup ˜
0≤t≤L
0≤t≤L
P˜ sup α ˜Tn (t) → 0,
0≤t≤L
P˜ sup γ˜Tn (t) → 0.
0≤t≤L
To complete the proof of Theorem 2.5, we repeat the same arguments as in the proof of Theorem 2.4. Proof of Theorem 2.6. According to the Itô formula, the process ζT (t) = fT (ξT (t)) satisfies the equation dζT (t) = σ ˆT (ζT (t)) dWT (t), where σ ˆT (x) = fT′ (ϕT (x)), ϕT (x) is the inverse function of the function fT (x), and ζT (0) = fT (x0 ) → y0 as T → ∞. In addition, the following equality holds: Z t gˆT ζT (s) dζT (s), IT (t) = FˆT ζT (t) + 0
where FˆT (x) = FT (ϕT (x)) and gˆT (x) = gT (ϕT (x)) · σ ˆT−1 (x). It is easy to see that condition 1 of the present theorem implies that Z x Z x dv dv 2 (v) → 2 (v) σ ˆ σ 0 0 0 T
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as T → ∞ for all x, whereas condition 2 implies that (2) (1) sup FˆT (x) + cT + cT x − F0 (x) → 0 |x|≤N
and
Z
N
−N
gˆT (x) − c(1) − g0 (x) 2 dx → 0 T
as T → ∞ for any N > 0. This means that the necessary and sufficient conditions of weak convergence of the process (ζT (t), IT (t)) as T → ∞ to the process (ζ(t), I(t)) from [11] hold with (2) (1) bT = cT and aT = cT . 4
Auxiliary results
Lemma 4.1. Let ξT be a solution of Eq. (1) from the class K (GT ). Then, for any N > 0 and any Borel set B ⊂ [−N ; N ], there exists a constant CL such that Z L P GT ξT (s) ∈ B ds ≤ CL ψ λ(B) , 0
where λ(B) is the Lebesgue measure of B, and ψ (|x|) is a bounded function satisfying ψ (|x|) → 0 as |x| → 0. Proof. Consider the function Z ΦT (x) = 2
0
x
fT′ (u)
Z
0
u
! χB GT (v) dv du. fT′ (v)
The function ΦT (x) is continuous, the derivative Φ′T (x) of this function is continuous, and the second derivative Φ′′T (x) exists a.e. with respect to the Lebesgue measure and is locally bounded. Therefore, we can apply the Itô formula to the process ΦT (ξT (t)), where ξT (t) is a solution of Eq. (1). Furthermore, 1 Φ′T (x)aT (x) + Φ′′T (x) = χB (x) 2 a.e. with respect to the Lebesgue measure. Using the latter equality, we conclude that Z t Z t (20) Φ′T ξT (s) dWT (s) χB ζT (s) ds = ΦT ξT (t) − ΦT (x0 ) − 0
0
with probability one for all t ≥ 0, where ζT (t) = GT (ξT (t)). Hence, using the properties of stochastic integrals, we obtain that Z t (21) P ζT (s) ∈ B ds = E ΦT ξT (t) − ΦT (x0 ) . 0
According to condition (A2 ) and inequality |GT (x)| ≥ C|x|α , C > 0, α > 0, we have m ΦT (x) − ΦT (x0 ) ≤ 2ψ λ(B) [1 + |x|m ] ≤ 2ψ λ(B) 1 + C − m α G (x) α . T Hence, using inequality (5), we obtain that
Asymptotic behavior of homogeneous additive functionals
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E ΦT ξT (L) − ΦT (x0 ) ≤ CL ψ λ(B)
for some constant CL . The latter inequality and Eq. (21) prove Lemma 4.1. Lemma 4.2. Let ξT be a solution of Eq. (1) from the class K (GT ). If, for measurable locally bounded functions qT (x), condition (A3 ) holds, then, for any L > 0, Z t P sup qT ξT (s) ds → 0 0≤t≤L 0 as T → ∞.
Proof. Consider the function ΦT (x) = 2
Z
0
x
fT′ (u)
Z
u
0
! qT (v) dv du. fT′ (v)
The same arguments as used to obtain Eq. (20) yield that Z t Z t qT ξT (s) ds = ΦT ξT (t) − ΦT (x0 ) − Φ′T ξT (s) dWT (s). 0
(22)
0
It is clear that, for any constants ε > 0, N > 0, and L > 0, we have the inequali-
ties o 2 sup ΦT ξT (t) > ε ≤ PN T + ε 0≤t≤L ( ) Z t P sup Φ′T ξT (s) dWT (s) > ε
P
n
0≤t≤L
≤ PN T
≤ PN T ≤ PN T
Z
N
−N
Z fT′ (u)
u
0
qT (v) dv du, fT′ (v)
0
Z t 2 1 ′ Φ ξT (s) χ{|ξT (s))≤N |} dWT (s) + 2 E sup ε 0≤t≤L 0 T Z L 2 ′ 4 ΦT ξT (s) χ{|ξT (s))≤N |} ds + 2E ε 0 Z x 2 16 qT (v) ′ + 2 L sup fT (x) dv , ε f ′ (v) |x|≤N
0
T
where PN T = P{sup0≤t≤L |ξT (t)| > N }. Therefore, using convergence (15), we obtain P sup ΦT ξT (t) − ΦT (x0 ) → 0 0≤t≤L
and
Z t P sup Φ′T ξT (s) dWT (s) → 0 0≤t≤L 0
as T → ∞. Thus, Eq. (22) implies the statement of Lemma 4.2. Lemma 4.3. Let ξT be a solution of Eq. (1) from the class K (GT ), and let ζT (t) = P
GT (ξT (t)) → ζ(t) as T → ∞. Then for any measurable locally bounded function g(x), we have the convergence
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Z Z t t P sup g ζT (s) ds − g ζ(s) ds → 0 0≤t≤L 0 0
as T → ∞ for any constant L > 0.
Proof. Let ϕN (x) = 1 for |x| ≤ N , ϕN (x) = N + 1 − |x| for |x| ∈ [N, N + 1], and ϕN (x) = 0 for |x| > N + 1. Then, for all T > 0 and L > 0, ( ) Z t n o P sup g ζT (s) − g ζT (s) ϕN ζT (s) ds > 0 ≤ P sup ζT (t) > N , 0≤t≤L
0≤t≤L
0
(
) Z t P sup g ζ(s) − g ζ(s) ϕN ζ(s) ds > 0 0≤t≤L 0 n n o o sup ζT (t) > N . ≤ P sup ζ(t) > N ≤ lim P 0≤t≤L
T →∞
0≤t≤L
According to Theorem 2.1, convergence (4) holds for the process ζT (t). So, to complete the proof of Lemma 4.3, we need to establish that Z L P g ζT (s) ϕN ζT (s) − g ζ(s) ϕN ζ(s) ds → 0 (23) 0
as T → ∞. First, assume that the function g(x) is continuous. Then P g ζT (s) ϕN ζT (s) − g ζ(s) ϕN ζ(s) → 0
as T → ∞ for all 0 ≤ s ≤ L, and |g(x)ϕN (x)| ≤ CN for all x. Thus, by Lebesgue’s dominated convergence theorem we have convergence (23). Second, let the function g(x) be measurable and locally bounded. Then, using Luzin’s theorem, we conclude that, for any δ > 0, there exists a continuous function g δ (x) that coincides with g(x) for x ∈ / B δ , where B δ ⊂ [−N − 1, N + 1], and the Lebesgue measure satisfies the inequality λ(B δ ) < δ. Thus, for every δ > 0, convergence (23) holds for the function g δ (x). Since, for any ε > 0, (Z ) L δ g ζT (s) ϕN ζT (s) − g ζT (s) ϕN ζT (s) ds > ε P 0
≤
1 E ε
Z
L
Z
L
g ζT (s) ϕN ζT (s) − g δ ζT (s) ϕN ζT (s) χ{B δ } ζT (s) ds
0
Z CN L ≤ P ζT (s) ∈ B δ ds, ε 0 (Z ) L δ g ζ(s) ϕN ζ(s) − g ζ(s) ϕN ζ(s) ds > ε P 0
≤
CN ε
0
CN lim P ζ(s) ∈ B δ ds ≤ ε T →∞
Z
L
0
P ζT (s) ∈ B δ ds,
taking into account Lemma 4.1, we conclude that convergence (23) holds for such a function g(x) as well.
Asymptotic behavior of homogeneous additive functionals
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Lemma 4.4. Let ξT be a solution of Eq. (1) from the class K (GT ), and let ζT (t) = Rt P P GT (ξT (t)) → ζ(t) and ηT (t) = 0 G′T (ξT (s)) dWT (s) → η(t) as T → ∞. Then, for measurable locally bounded functions g(x), we have the convergence Z t Z t P sup g ζT (s) dηT (s) − g ζ(s) dη(s) → 0 0≤t≤L
0
0
as T → ∞ for any constant L > 0.
Proof. Similarly to the proof of Lemma 4.3, it suffices to obtain an analogue of convergence (23), that is, to get that, for any N > 0 and L > 0, Z t Z t P sup g ζ(s) ϕN ζ(s) dη(s) → 0 (24) g ζT (s) ϕN ζT (s) dηT (s) − 0≤t≤L
0
0
as T → ∞, where ϕN (x) is defined in the proof of Lemma 4.3. The proof of convergence (24) for a continuous function g(x) is similar to that of the corresponding theorem in [17], Chapter 2, §6. The explicit form of the quadratic characteristic hηT i(t) of the martingale ηT (t) and condition (A1 ) imply the inequality Z
0
L
2 ϕN ζT (t) dhηT i(t) ≤ CN L,
which is used for the proof of convergence (24). The extension of such a convergence to the class of measurable locally bounded functions is based on Lemma 4.1 and is provided similarly to the proof of Lemma 4.3. Lemma 4.5. Let ξT be a solution of Eq. (1) belonging to the class K (GT ), and let the stochastic process (ζT (t), ηT (t)), with ζT (t) = GT (ξT (t)) and ηT (t) = Rt ′ ˜ ˜T (t)). Then 0 GT (ξT (s)) dWT (s) be stochastically equivalent to the process (ζT (t), η the process Z t Z t g ζT (s) ds + q ζT (s) dηT (s), 0
0
where g(x) and q(x) are measurable locally bounded functions, is stochastically equivalent to the process Z
0
t
g ζ˜T (s) ds +
Z
0
t
q ζ˜T (s) d˜ ηT (s).
Proof. The proof is the same as that of Theorem 2 from [3]. References
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[3] Krylov, N.V.: On Ito’s stochastic integral equations. Theory Probab. Appl. 14(2), 330– 336 (1969). MR0270462 [4] Kulinich, G.L.: On the limit behavior of the distribution of the solution of a stochastic diffusion equation. Theory Probab. Appl. 12(3), 497–499 (1967). MR0215365 [5] Kulinich, G.L.: Limit behavior of the distribution of the solution of a stochastic diffusion equation. Ukr. Math. J. 19, 231–235 (1968). MR0215365 [6] Kulinich, G.L.: Limit distributions for functionals of integral type of unstable diffusion processes. Theory Probab. Math. Stat. 11, 82–86 (1976). [7] Kulinich, G.L.: Some limit theorems for a sequence of Markov chains. Theory Probab. Math. Stat. 12, 79–92 (1976). [8] Kulinich, G.L.: Limit theorems for one-dimensional stochastic differential equations with nonregular dependence of the coefficients on a parameter. Theory Probab. Math. Stat. 15, 101–115 (1978). [9] Kulinich, G.L.: On the asymptotic behavior of the solution of one dimensional stochastic diffusion equation. In: Lect. Notes Control Inf. Sci., vol. 25, pp. 334–343. SpringerVerlag (1980). MR0609199 [10] Kulinich, G.L.: On necessary and sufficient conditions for convergence of solutions to one-dimensional stochastic diffusion equations with a nonregular dependence of the coefficients on a parameter. Theory Probab. Appl. 27(4), 856–862 (1983). MR0681473 [11] Kulinich, G.L.: On necessary and sufficient conditions for convergence of homogeneous additive functionals of diffusion processes. In: Arato, M., Yadrenko, M., (eds.) Proceedings of the Second Ukrainian–Hungarian Conference: New Trends in Probability and Mathematical Statistics. TViMS, vol. 2, pp. 381–390 (1995). [12] Kulinich, G.L., Kaskun, E.P.: On the asymptotic behavior of solutions of one-dimensional Ito’s stochastic differential equations with singularity points. Theory Stoch. Process. 4(20), 189–197 (1998). MR2026628 [13] Kulinich, G.L., Kushnirenko, S.V., Mishura, Y.S.: Asymptotic behavior of the integral functionals for unstable solutions of one-dimensional Ito stochastic differential equations. Theory Probab. Math. Stat. 89, 93–105 (2013). MR3235178. doi:10.1090/s0094-90002015-00938-8 [14] Kulinich, G.L., Kushnirenko, S.V., Mishura, Y.S.: Asymptotic behavior of the martingale type integral functionals for unstable solutions to stochastic differential equations. Theory Probab. Math. Stat. 90, 115–126 (2015). MR3242024 [15] Kulinich, G.L., Kushnirenko, S.V., Mishura, Y.S.: Limit behavior of functionals of diffusion type processes. Theory Probab. Math. Stat. 92 (2016), to appear. [16] Prokhorov, Yu.V.: Convergence of random processes and limit theorems in probability theory. Theory Probab. Appl. 1(2), 157–214 (1956). MR0084896 [17] Skorokhod, A.V.: Studies in the Theory of Random Processes. Addison–Wesley Publ. Co., Inc., Reading, MA (1965). MR0185620 [18] Skorokhod, A.V., Slobodenyuk, N.P.: Limit Theorems for Random Walks. Naukova Dumka, Kiev (1970) (in Russian). MR0282419 [19] Veretennikov, A.Y.: On the strong solutions of stochastic differential equations. Theory Probab. Appl. 24(2), 354–366 (1979). MR0532447
