Limit Distribution of a Generalized Ornstein – Uhlenbeck Process

Let an Rd-valued random process ξ be the solution of an equation of the kind ξ(t) = ξ(0) + ∫ t 0 A(u)ξ(u)dι(u) + S (t), where ξ(0) is a random variable measurable w. r. t. some σ-algebra F (0), S is a random process with F (0)-conditionally independent increments, ι is a continuous numeral random process of locally bounded variation, and A is a matrix-valued random process such that for any t > 0 ∫ t 0 ∥A(s)∥ |dι(s)| < ∞. Conditions guaranteing existence of the limiting, as t → ∞, distribution of ξ(t) are found. The characteristic function of this distribution is written explicitly.


Introduction
The random processes under consideration are assumed given on a common probability space (Ω, F , P).It is assumed that a a filtration F = (F (t), t ∈ R + ) on F is given.We consider, without loss generality, that it is right-continuous and each F (t) contains all P-negligible sets from F (these are so called usual conditions -see (Gikhman & Skorokhod, 1982;Jacod & Shiryaev, 1987;Liptser & Shiryaev, 1989)).We will consider also the trivial filtration F 0 = (F 0 (t), t ∈ R + ), where F 0 (t) = F (0) for all t.Thus a random process is F 0 -adapted iff its value at any nonrandom time is an F (0)-measurable random variable.We introduce the notation: 0 is the class of all starting from zero F 0 -adapted continuous random processes; S + is the class of all nonnegative (in the spectral sense) symmetric d × d matrices with real entries; M + (C) is the class of all σ-finite measures on a σ-algebra C; l.i.p. signifies the limit in probability.In integrals with a discontinuous integrator, The quadratic characteristic of a locally square-integrable R d -valued martingale Z will be denoted by ⟨Z⟩.This is an increasing S + -valued random process.
The words "almost surely" are tacitly implied in relations between random variables, including the convergence relation unless it is explicitly written as the convergence in probability (denoted by The reference books for the notions and results of stochastic analysis used in this paper are (Gikhman & Skorokhod, 1982, 2009;Jacod & Shiryaev, 1987;Liptser & Shiryaev, 1989).A number of more specific definitions and statements relevant to the topic can be found in (Yurachkivsky, 2013).
The goal of the article is to find the limit, as t → ∞ (which will be tacitly meant in all asymptotic relations), of the one-dimensional distribution of the solution of the stochastic equation where S is a random process with F (0)-conditionally independent increments, ι ∈ V c 0 , and A is a matrix-valued F 0adapted random process such that for any t > 0 (2) We call thus defined ξ a generalized Ornstein -Uhlenbeck process.Recall that a classical Ornstein -Uhlenbeck process is the solution of (1) with d = 1, U = 0, ι(t) = t, constant A and a homogeneous Wiener process as S .For them, the problem is easy and solved long ago (the limit distribution exists iff A < 0).But even in a seemingly simple case when S is a homogeneous generalized (i.e., with random jumps) Poisson process the problem is nontrivial.It was solved first by Zakusilo in 1981 (this result is contained in (Anisimov, Zakusilo & Donchenko, 1987)).A more general theorem, but also only for the case of homogeneous S and ι(t) = t, was proved in (Sato & Yamazato, 1984).Unlike these authors, we focus on the described above model.This level of generality requires quite different technique that will be demonstrated below.The only known to the author work with ι(t) possibly other than t and nonhomogeneous S is (Ivanenko, 2009).But the proof therein relies on very specific assumptions (for example, A = −c1 I) and does not carry over to a more general situation.
The structure of the article is clear from the titles of its sections: General preliminaries, Special preliminaries and The main result.

General Preliminaries
The following statement is immediate from Lebesgue's dominated convergence theorem and Lévy's continuity theorem.
Proposition 2.1.Let ξ be an R d -valued random process.Suppose that there exists a C-valued random function J on (z) .Then there exists a random variable ξ ∞ such that z) and ξ(t Let us consider the equation The standard convention t entitles us to consider that the variables s and t independently range over R + .The integral on the r.h.s. of ( 22) being pathwise, the variable ω ∈ Ω performs in it as a parameter.Thus we may consider this equation deterministic.Its solution will be an important tool in our study.
Lemma 2.2 (Yurachkivsky, 2013;Lemma 3.10).Let ι be a numeral function of locally bounded variation, and A be a Borel matrix-valued function satisfying condition (2).Then the solution of equation (3) exists, is unique and has locally bounded variation in each argument.
In what follows, C is a σ-algebra of subsets of some set Θ.
We denote by K the class of F-adapted R d -valued random processes M such that, firstly, E 0 |M(t)| 2 < ∞ for all t ∈ R + , secondly, E(M(t) − M(s)|F (s)) = M(s) for all t > s ≥ 0 and, thirdly, the process E 0 |M| 2 is continuous (here E 0 is the extended conditional expectation, so we need not assume that E|M(t)| < ∞).This class is contained in the class of locally square integrable martingales (see, e. g., ( Yurachkivsky, 2013( Yurachkivsky, , 2014))).
From this time on we deal with the following particular case of (1): which may be written shortly in the notation of stochastic analysis (see (Gikhman & Skorokhod, 1982;Jacod & Shiryaev, 1987;Liptser & Shiryaev, 1989)) as To make it quite definite we impose the assumptions: 2. ι is an R-valued random process of class V c 0 .3. A is a matrix-valued F (0) ⊗ B + -measurable in (ω, t) random process satisfying, for all t, condition (2).
7. W is a starting from zero R d -valued continuous random process of class K with F 0 -adapted quadratic characteristic.
Lemma 2.5.Let µ be a finite measure on some σ-algebra X of subsets of a set X, and let for each t > 0 ζ t be a nonnegative measurable random function on X. Suppose the following: for any x ∈ X and for all t > 0 and x ∈ X Proof.In this and the subsequent proofs, ∫ stands for of which the inequality follows from (15), imply that for any positive ε and L By construction Z L → 0 as L → for all x ∈ X and ω ∈ Ω. Hence and from ( 14) we get by the dominated convergence theorem (DCT) applied at those ω ∈ Ω where ( 14) holds Condition ( 13) implies by the DCT that E(ζ t (x) ∧ L) → 0 for all L > 0 and x ∈ X. Hence by the DCT and due to finiteness by the Fubini -Tonelli theorem.This together with the preceding relation and ( 17) implies that for any positive ε and L It remains to make use of (18).
Lemma 2.6.Let µ be a σ-finite measure on some σ-algebra X of subsets of a set X, and let for each t > 0 ζ t be a nonnegative measurable random function on X. Suppose the following: for any x ∈ X relation (13) holds; there exists a nonrandom measurable function Z on X satisfying conditions (14), ( 15) and, for any r > 0, the condition Then relation ( 16) holds.
Proof.The set {x ∈ X : Z(x) > r} is nonrandom (since in this lemma so is Z), belongs to X (because Z is measurable) and has, due to (19), finite measure as r > 0. Then it follows from ( 13) -( 15) by Lemma 2.5 that for any r > 0 Consequently, for any positive r and ε Condition ( 15) implies that Z is nonnegative, whence with account of ( 14 Let n and d be natural numbers, P and S be n × d-matrices, and B be a d × d matrix.The identity In particular, for any B ∈ S + (so that ∥B∥ ≤ tr B) and Hence the following conclusion is immediate.
Lemma 2.9.Let K be the solution of equation (3) on R 2 + , where ι is a continuous numeral function of locally bounded variation, and A is a matrix-valued Borel function such that for any t > 0 (2) holds.Then for every Borel function L(•, •) that has locally bounded variation in the first argument the solution of the equation is given by the formula Proof.Without loss of generality s = 0. Then this statement is a particular case of Corollary 3.19 in (Yurachkivsky, 2013).
Throughout below, the prime does not signify differentiation.
Corollary 2.10.Let ι, A, K be as in Lemma 2.9, and K ′ be the solution of the equation on R 2 + , where A ′ is a matrix-valued measurable function such that for any t > 0 Then for all t ≥ s ≥ 0 Proof.To deduce this statement from Lemma 2.9 it suffices to write, on the basis of ( 3) and ( 23), Corollary 2.11.Let ι, A, A ′ , K and K ′ be as in Corollary 2.10.Suppose also that for some q ∈ R and all t, s ∈ R + .Then for all t ≥ s ≥ 0

Special Preliminaries
From now on, ∫ Θ will be written shortly as ∫ .
We impose two more assumptions: 12.There exists an M + (C)-valued random process π such that the equality holds for every nonnegative F (0) ⊗ B + ⊗ C-measurable random function χ on R + × Θ and all t > 0.
13.There exists an M + (C)-valued random processes ϖ such that the equality holds for every nonnegative F (0) ⊗ B + ⊗ C-measurable random function χ on R + × Θ and all t > 0.
For an M + (C)-valued random process κ we denote by E κ the class of all C-valued F (0 The next statement is obvious. Lemma 3.1.If assumption 12 is satisfied, then equality (26) holds for all χ ∈ E π and t > 0; under assumption 13, equality (27) holds for all χ ∈ E ϖ and t > 0.
We continue the list of assumptions: 14.There exist a nonrandom σ-finite measure Π on C and a positive random variable Ξ such that for all t ∈ R + and B ∈ C π(t, B) ≤ ΞΠ(B).
15.There exist a nonrandom σ-finite measure Σ on C and a positive random variable Ξ such that for all t ∈ R + and B ∈ C ϖ(t, B) ≤ ΞΣ(B).
16.There exists a nonrandom measurable function 17.There exists a nonrandom measurable function g on Θ such that (i) |g(s, θ)| ≤ g(θ) and (ii) In the next two assumptions and four statements, K 0 and K are continuous matrix-valued random function on R 2 + ; the assumption that K satisfies equation ( 3) is not used.We assume the following: 18.There exists a positive random variable κ such that for all t > s ∥K 0 (t, s)∥ ∨ ∥K(t, s)∥ ≤ e κ(s−t) .
19.There exists an increasing random process Λ such that for all t > s where κ is the same as in 18.
If assumption 19 is imposed and 18 is not, then κ will signify simply an F (0)-measurable positive random variable.
And this together with 16(ii) and ( 35) implies by the DCT that for any c ∈]0, 1] ∫ Π(dθ) Writing, for an arbitrary c ∈]0, 1[, and arguing as in the previous proof, we get from (32) and the assumption that Λ increases And this jointly with ( 40) and ( 38) implies that for all z ∈ R d * and ε > 0 Hence and from ( 39) relation ( 36) follows.
Lemma 3.5.Let assumptions 7, 18 and 19 be satisfied and condition (32) be fulfilled.Suppose also that there exists a positive random variable β such that for all t > s ≥ 0 Then for any t  43) proves the lemma.
Lemma 3.6.Let assumptions 4 and 19 be satisfied and condition (32) be fulfilled.Suppose also that there exists a positive random variable β such that for all t > s ≥ 0 var Proof.Writing on the basis of (44) and 19 we deduce the desired conclusion from (41) (emerging from (32)).

The Main Result
Theorem 4.1.Let assumptions 1 -8 and 11 -17 be satisfied.Suppose also that there exist: • an R d -valued random variable ϑ, • random matrices A 0 and Υ, • random σ-finite measures π 0 and ϖ 0 on C, • positive random variables κ, α and β, • R d -valued measurable random functions f 0 and g 0 on Θ, • a positive number a, • an increasing random process Λ with property (32) -such that: where Π, Σ and Ξ are from assumptions 14 and 15; for all u ∈ R + e uA 0 ≤ e −au ; ( 49) and inequalities (42) and (44) hold; for any θ ∈ Θ Then the distribution of ξ(t) weakly converges, as t → ∞, to the distribution with characteristic function where Rewriting condition (49) in the form and taking to account (50), (51) and the assumed properties of Λ, we see that K and K 0 satisfy assumptions 18 and 19 (with κ ∧ α as the former κ).Hence and from assumptions 14 -17 we get by Lemmas 3.3 and 3.4 where F t (z) and G t (z) are defined by equalities ( 28) and ( 29), respectively.
From (58) we have by Lemma 2.8, assumption 14 and the Fubini -Tonelli theorem On the strength of ( 60) where For convenience of the subsequent derivations, we set κ t (u, θ) = 0 as u > t.
Let us show that This will be done if we establish the relation where and π is defined for negative values of the temporal argument by zero.(The divider Ξ will enable us to construct a nonrandom majorant for ζ t and thereon to apply Lemma 2.6.) By construction, Lemma 2.8, assumption 14, condition (47) and formula (68) where By condition (54) for all u ∈ R + and z ∈ R d * .By construction Z(u, z) is nonrandom.Formula (83) and assumption 16(ii) imply that and (by the DCT) Z(u, z) → 0 as u → ∞, so that for any r > 0 the Lebesgue measure of the set {u : Z(u, z) > r} is finite.Now, (81) follows from ( 82) and ( 84) by Lemma 2.6.(87) This will be done if we establish relation ( 81), where this time (unlike item 4 • ) ∫ ( e izD(u)g 0 (θ) − 1 − izD(u)g 0 (θ) ) ϖ(t − u, dθ) − ∫ ( e izD(u)g 0 (θ) − 1 − izD(u)g 0 (θ) and ϖ is defined for negative values of the temporal argument by zero.
b] .The indicator of a set {• • • } is denoted by I{• • • }.All vectors are thought of, unless otherwise stated, as columns; all matrices are meant of size d × d, with real entries.The unit matrix is denoted by 1 I, and the space of all d-dimensional row vectors with real components by R d * .We use the Euclidean norm | • | of vectors and the operator norm ∥ • ∥ of matrices.For symmetric matrices A and B, the inequality A ≤ B means that B − A ∈ S + (so that one may speak about increasing S + -valued functions).