Stochastic integrals in additive processes and application to semi-lévy processes. Ken-iti Sato

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1 Research Report KSTS/RR-2/9 Oct. 15, 22 Stochastic integrals in additive processes and application to semi-lévy processes by Ken-iti Sato Ken-iti Sato Hachiman-yama Tenpaku-ku, Nagoya Japan Department of Mathematics Faculty of Science and Technology Keio University 22 KSTS Hiyoshi, Kohoku-ku, Yokohama, Japan

2 STOCHASTIC INTEGRALS IN ADDITIVE PROCESSES AND APPLICATION TO SEMI-LÉVY PROCESSES KEN-ITI SATO 1. Introduction A stochastic process {X t : t } on R d is called an additive process if it has independent increments and if it is continuous in probability with cadlag paths and X =. It is called a Lévy process if, in addition, it has stationary increments. Path behaviors and distributional properties of Lévy processes are deeply analyzed (see [1], [18]). Concerning additive processes, the Lévy Itô decomposition of paths is known in complete generality. ut, in order to get further results, we have to restrict our study to some special classes. Examples are the class of selfsimilar additive processes introduced in [17] and the class of semi-selfsimilar additive processes in [12]. Another interesting class is that of semi-lévy processes, that is, additive processes with semi-stationary (sometimes called periodically stationary) increments. In order to analyze distributional properties of processes of these classes, it is important to treat stochastic integrals (of nonrandom integrands) based on additive processes. Keeping in mind this application, we study in this paper stochastic integrals based on additive processes and their distributions. Our study in this paper does not depend on the cadlag property. We define additive processes in law, Lévy processes in law, and semi-lévy processes in law, dropping the cadlag requirement in their definitions but retaining the requirement of continuity in probability. We will call an additive process in law {X t : t } natural if the location parameter γ t in the generating triplet (A t, ν t, γ t ) of the distribution of X t is locally of bounded variation in t. An additive process is natural if and only if it is, at the same time, a semimartingale. This fact is essentially given in Jacod and Shiryaev [5]. Thus we can consider stochastic integrals for natural additive processes as a special case of semimartingale integrals of Kunita and Watanabe [9]. ut we will not rely on the theory of semimartingales, but directly define stochastic integrals (of nonrandom functions) and seek the representation of the characteristic functions of their distributions. This is in the same line as the study of independently scattered 1

3 random measures by Urbanik and Woyczynski [22] and Rajput and Rosinski [14]. We show that a natural additive process in law on R d induces an R d -valued independently scattered random measure, and vice versa. Thus our random measures are R d -valued, not R-valued as in [22] and [14]. Further, we are interested in construction of random measures on the same probability space as the original additive process in law is defined. For a natural additive process in law {X t : t } on R d we use a system of infinitely divisible distributions {ρ s : s } and a measure σ on [, ) such that Ee i z,x t = exp t log ρ s (z)σ(ds) for z R d, where ρ s (z) is the characteristic function of ρ s. We will call ({ρ s }, σ) a factoring of {X t }. In fact, existence of such a representation is a necessary and sufficient condition for naturalness. There is a canonical one among such pairs ({ρ s }, σ), which we call the canonical factoring of {X t }. For a class of l d matrix-valued functions F (s) including all locally bounded measurable functions, stochastic integrals F (s)dx s for bounded orel sets are defined and shown to satisfy [ ] E exp i z, F (s)dx s = exp log ρ s (F (s) z)σ(ds) for z R l, where F (s) is the transpose of F (s). ased on this formula we will study properties of stochastic integrals. Then we will treat the problem of the existence of stochastic integrals e sq dx s, where Q is a d d matrix all of whose eigenvalues have positive real parts and {X t : t } is a semi-lévy process in law. It will be shown that e sq dx s exists if and only if {X t } has finite log-moment. In a forthcoming paper joint with M. Maejima, these results will be applied to a study of the relationship of semi-lévy processes, semi-selfsimilar additive processes, and semi-stationary Ornstein Uhlenbeck type processes. This study will extend the theory of the representation of selfdecomposable distributions by Wolfe [23], Jurek and Vervaat [7], Sato and Yamazato [19], [2], Sato [17], and Jeanblanc, Pitman, and Yor [6] to the case of semi-selfdecomposable and (b, Q)-decomposable distributions. Natural additive processes in law and factorings are discussed in Section 2. Their relations to independently scattered random measures are studied in Section 3. Then stochastic integrals are treated in Section 4. Finally Section 5 contains the study of e sq dx t for semi-lévy processes in law. Our notation and definitions follow [18]. esides, we use the following: ID = ID(R d ) is the class of infinitely divisible distributions on R d ; (R d ) is the class of 2

4 orel sets in R d ; (R d ) is the class of (R d ) satisfying inf x x > ; J for an interval J is the class of orel sets in J; [, ) in [, ); p-lim stands for limit in probability; S + d is the class of bounded orel sets is the class of d d symmetric nonnegative-definite matrices; M l d is the class of l d real matrices; M d = M d d is the class of d d real matrices; tra is the trace of A S + d ; M+ d is the class of Q M d all of whose eigenvalues have positive (> ) real parts; I is the d d identity matrix. Recall that an element of the Euclidean space R d is understood to be a column vector with d components. For F M l d, F denotes the transpose of F. The norm of F M l d is F = sup x 61 F x. For b > and Q M d, b Q = n= (n!) 1 (log b) n Q n M d. The inner products in R d and R l are denoted by the same symbol. Thus we have z, F x = F z, x for x R d, z R l, and F M l d. A set or a function is called measurable if it is orel measurable. The characteristic function of a distribution µ is denoted by µ(z). Denote by L(X) the distribution of a random element X. When L(X) = L(Y ), we write X d = Y. For two stochastic processes {X t } and {Y t }, {X t } d = {Y t } means that they have an identical distribution as infinite-dimensional random elements, that is, have an identical system of finite-dimensional distributions, while X d t = Y t means that X t and Y t have an identical distribution for a fixed t. If the characteristic function µ(z) of a distribution µ on R d vanishes nowhere, then there is a unique continuous function f(z) on R d such that f() = and µ(z) = e f(z). This f(z) is called the distinguished logarithm of µ(z) and written as f(z) = log µ(z) ([18] p. 33). The word increase is used in the wide sense allowing flatness. 2. Natural additive processes in law and factorings When {X t : t } is an additive process in law on R d, we write µ t = L(X t ) ID. Let c(x) be a real-valued bounded measurable function satisfying { 1 + o( x ) as x, (2.1) c(x) = O( x 1 ) as x. Then we get the Lévy Khintchine representation of µ t of the form [ (2.2) µ t (z) = exp 1 ] 2 z, A tz + g R c (z, x)ν t (dx) + i z, γ t d with (2.3) g c (z, x) = e i z,x 1 i z, x c(x). 3

5 Here A t S + d, ν t is a measure on R d satisfying ν t ({}) = and (1 x 2 )ν t (dx) <, and γ t R d. They are called Gaussian covariance, Lévy measure, and location parameter, respectively. The triplet of A t, ν t, and γ t is denoted by (A t, ν t, γ t ) c. Here A t and ν t do not depend on the choice of c(x). See [18] Theorem 8.1 and Remark 8.4. Standard choice of c(x) is 1 { x 61}(x) or (1 + x 2 ) 1. The system {(A t, ν t, γ t ): t } satisfies the following: (1) A =, ν =, γ =, (2) A t A s S + d and ν t ν s for s t, (3) A s A t, ν s () ν t () for all (R d ), and γ s γ t as s t. Conversely, any system satisfying (1), (2), and (3) induces, uniquely in law, an additive process in law (Theorem 9.8 of [18]) and it has a modification which is an additive process (Theorem 11.5 of [18]). Definition. An additive process in law {X t } on R d is said to be natural if the location parameter γ t is locally of bounded variation in t (that is, of bounded variation on any finite subinterval of [, )). This definition does not depend on the choice of c(x) by the following assertion. Proposition 2.1. Let c 1 (x) and c 2 (x) be real-valued bounded measurable functions satisfying (2.1). Let {X t } be an additive process in law on R d with triplets (A t, ν t, γt 1 ) c1 and (A t, ν t, γt 2 ) c2. Then γt 1 is locally of bounded variation if and only if γt 2 is locally of bounded variation. Proof. We have γt 2 = γt 1 + f(t) with f(t) = R x(c d 2 (x) c 1 (x))ν t (dx). We can check that f(t) is locally of bounded variation. For example, any Lévy process in law {X t } is a natural additive process in law, since γ t = tγ 1. Proposition 2.2. If {X t } is an additive process in law on R d, then there is an R d - valued continuous function a(t) on [, ) such that {X t a(t)} is a natural additive process in law. Proof. Use an arbitrary c(x) satisfying (2.1) and choose a(t) = γ t. Proposition 2.3. Let {X t } be an additive process in law on R d. Suppose that x ν x 61 t(dx) < for all t. Let γ t be the drift of µ t. Then {X t } is natural if and only if γ t is locally of bounded variation. 4

6 Proof. Note that γ t = γ t xν x 61 t(dx) and that xν x 61 t(dx) is locally of bounded variation in t. For example, any additive process {X t } on R with increasing paths is a natural additive process. Henceforth we use (2.4) c(x) = (1 + x 2 ) 1, unless mentioned otherwise. Thus the triplet (A, ν, γ) of an infinitely divisible distribution stands for (A, ν, γ) c with c(x) of (2.4). The following fact is basic. Lemma 2.4. Let µ and µ n, n = 1, 2,..., be in ID(R d ) such that µ n µ as n. Let (A, ν, γ) and (A n, ν n, γ n ) be the triplets of µ and µ n, respectively. Then (2.5) and tra n + (1 x R 2 )ν n (dx) tra + d (1 x R 2 )ν(dx) d (2.6) γ n γ. Proof. Noting that c(x) of (2.4) is bounded and continuous, use Theorem 8.7 of [18]. Lemma 2.5. Let {X t } be an additive process in law on R d with triplet (A t, ν t, γ t ) of µ t = L(X t ). Then, for every [, ), there are a unique A S + d and a unique measure ν on R d such that A and ν (C) for any C (R d ) are countably additive in [, ) and that A [,t] = A t and ν [,t] = ν t. The components A jk (), j, k = 1,..., d, of A are absolutely continuous with respect to the measure tra on [, t ] for each t. If, moreover, {X t } is natural, then there is a unique γ such that γ is countably additive in [, ) and γ [,t] = γ t. Proof. Since tra t is increasing and continuous in t, it induces an atomless measure σ 1 on [, ). Let A jk (t) and γ j (t) be the components of A t and γ t. Since A jk (t) A jk (s) tra t tra s for s t, A jk (t) is locally of bounded variation and absolutely continuous with respect to σ 1. Thus A jk (t) induces a signed measure A jk (). We have j,k A jk()z j z k, since it is true when is an interval. Thus A = (A jk ()) S + d. If {X t} is natural, then the assertion on γ is proved similarly. Concerning ν, there is a unique measure ν on [, ) R d such that (2.7) ν([, t] C) = ν t (C) for C (R d ) and t, 5

7 as in [18] Remark 9.9. Then it suffices to let ν (C) = ν( C). Definition. Let {X t } be an additive process in law on R d. A pair ({ρ s : s }, σ) is called a factoring of {X t } if the following conditions are satisfied: (1) σ is a locally finite measure on [, ), that is, a measure on [, ) such that σ([, t]) < for all t [, ), (2) σ is continuous (that is, atomless), (3) ρ s ID(R d ) for all s [, ), (4) log ρ s (z) is measurable in s for each z R d, (5) t log ρ s(z) σ(ds) < for all t [, ) and z R d, (6) we have (2.8) µ t (z) = exp t log ρ s (z)σ(ds) for all t [, ) and z R d. For example, any Lévy process in law {X t } on R d has a factoring given by ρ s = L(X 1 ) for all s and by σ = Lebesgue. The following theorem is a main result of this section. Theorem 2.6. Let {X t : t } be an additive process in law on R d. Then, {X t } is natural if and only if {X t } has a factoring. Denote by (A ρ s, ν ρ s, γ ρ s ) the triplet of ρ s. Lemma 2.7. If ({ρ s : s }, σ) is a factoring of an additive process in law {X t } on R d, then (7) A ρ s, γ ρ s, and ν ρ s (C) for any C (R d ) are measurable in s, (8) we have t ( ) (2.9) tr(a ρ s) + (1 x R 2 )νs ρ (dx) + γs ρ σ(ds) < d (9) we have (2.1) A t = t A ρ sσ(ds), ν t () = t ν ρ s ()σ(ds), γ t = for all t [, ), t γ ρ s σ(ds) for (R d ), (1) t log ρ s(z) σ(ds) = log µ t (z), the distinguished logarithm of µ t (z). Proof. Since we can express A ρ s, γs ρ, and νs ρ (C) by using log ρ s (z) as in Section 8 of [18], assertion (7) is proved. To see (8), we use µ s (z) = exp t Re (log ρ s(z))σ(ds). Assertions (9) and (1) follow from (8). Details are omitted. 6

8 The if part of Theorem 2.6 is proved by this lemma. Indeed, if {X t } has a factoring ({ρ s }, σ), then γ t is locally of bounded variation by the expression in (2.1) and hence {X t } is natural. The only if part of the theorem will be proved in the form of Proposition 2.8 after we introduce the notions of canonical measures and canonical factorings. Definition. Let {X t } be a natural additive process in law on R d and let γ t be the variation function of γ t. Use the notation in Lemma 2.5. Denote by γ the measure such that γ [,t] = γ t. Then a measure σ on [, ) defined by (2.11) σ() = tra + (1 x R 2 ) ν (dx) + γ d is called the canonical measure of {X t }. A pair ({ρ s }, σ) is called a canonical factoring of {X t } if it is a factoring of {X t } and if σ is the canonical measure of {X t }. Proposition 2.8. Let {X t } be a natural additive process in law on R d. Then there exists a canonical factoring of {X t }. It is unique in the sense that, if ({ρ 1 s}, σ) and ({ρ 2 s}, σ) are canonical factorings of {X t }, then ρ 1 s = ρ 2 s for σ-a. e. s. If ({ρ s }, σ) is a canonical factoring of {X t }, then (2.12) esssup sup log ρ s (z) < s [, ) z 6a for any a (, ) and ( ) (2.13) esssup tr(a ρ s) + (1 x s [, ) R 2 )νs ρ (dx) + γs ρ <, d where the essential supremums are with respect to σ. Proof. Let σ 1, σ 2, and σ 3 be the measures defined by σ 1 () = tra, σ 2 () = R d(1 x 2 )ν (dx), and σ 3 () = γ. Let h l (t) be the Radon Nikodým derivative of σ l with respect to σ for l = 1, 2, 3. Let A jk (t) and γ j (t) be the Radon Nikodým derivatives of A jk () and γ j () with respect to σ 1 and σ 3, respectively. For the measure ν in the proof of Lemma 2.5, there are a measure σ on [, ) and measures ν s on R d such that σ is continuous and locally finite, ν s(c) is measurable in s for each C (R d ), R (1 x 2 )νs(dx) = 1, νs({}) =, and d ν( C) = σ (ds) νs(dx) for [, ), C (R d ). C The argument is similar to the construction of conditional distributions. Letting νs ρ (dx) = νs(dx)h 2 (s), A ρ s = (A ρ jk (s)) with Aρ jk (s) = A jk (s)h 1(s), and γs ρ = (γ ρ j (s)) with γ ρ j (s) = γ j (s)h 3(s), we can prove that ({ρ s }, σ) is a factoring of {X t } and that 7

9 (2.1) is satisfied. Properties (2.12) and (2.13) are proved easily. The uniqueness of a canonical factoring is proved by (9) and (1) in Lemma 2.7. Proposition 2.9. Let {X t } be an additive process in law on R d. Then it is natural if and only if µ t (z) is locally of bounded variation in t for each z R d. Proof. In order to see the only if part, notice that {X t } has a factoring by Theorem 2.6 and use (2.8). Let us prove the if part. Define Y t = X t γ t. Then {Y t } is a natural additive process in law and hence Ee i z,yt is locally of bounded variation in t. We have Ee i z,yt = e i z,γt µ t (z). Since µ t (z) is continuous, nonvanishing, and locally of bounded variation in t for each z, it follows that e i z,γt is locally of bounded variation in t for each z. Hence z, γ t is locally of bounded variation in t for each z. Hence, so is γ t. Proposition 2.1. Let {X t } be an additive process on R d. Then {X t } is natural if and only if {X t } is a semimartingale. Proof. This is a consequence of Proposition 2.9 combined with Jacod and Shiryaev [5], Chapter II, Theorem We add some facts on factorings. Proposition If {ρ s : s } and σ satisfy conditions (1), (2), (3), of the definition of a factoring and (7), (8) of Lemma 2.7, then ({ρ s }, σ) is a factoring of some additive process in law {X t : t } on R d. Proof. Define A t, ν t, and γ t by (2.1). Then (A t, ν t, γ t ) is the triplet of some µ t ID and satisfies conditions (1), (2), and (3) in the first paragraph of this section. Thus there is an additive process in law {X t } such that L(X t ) = µ t. Conditions (4) and (5) of the definition of a factoring follow from (7) and (8) and we can see that ({ρ s }, σ) is a factoring of {X t }. Proposition 2.12 (Time change). Let {X t : t } be a natural additive process in law on R d. Given an increasing continuous function τ(t) from [, ) into [, ) with τ() =, define Y t = X τ(t). Then {Y t : t } is a natural additive process in law on R d. If ({ρ s }, σ) is a factoring of {X t }, then ({ ρ s }, σ) defined by (2.14) ρ s = ρ τ(s) and σ([, s]) = σ([, τ(s)]). gives a factoring of {Y t }. If ({ρ s }, σ) is canonical, then ({ ρ s }, σ) is canonical. 8

10 Proof is elementary and omitted. Let us study conditions for naturalness in some classes of additive processes. In analogy to definitions in [17] and [18] we give the following definition. Definition. Let Q M + d. An Rd -valued stochastic process {X t : t } is called Q-selfsimilar if, for every a > 1 (hence for every a > ), (2.15) {X at : t } = d {a Q X t : t }. It is called Q-semi-selfsimilar if (2.15) holds for some a > 1. In this case a is called an epoch. A basic property we need of Q M + d c 1,..., c 4 such that is the following: there are positive constants (2.16) c 4 e c 2s x e sq x c 3 e c 1s x for s and x R d (see [21] p. 139 or [16]). We have c 1 c 2 and c 4 1 c 3 automatically. It follows that (2.17) c 1 3 e c 1s x e sq x c 1 4 e c 2s x for s and x R d. Theorem Let Q M + d. Let {X t} be a Q-semi-selfsimilar additive process in law on R d with epoch a. Then {X t } is natural if and only if γ t is of bounded variation on [1, a]. Proof. The only if part is evident. Let us prove the if part. Assume that γ t is of bounded variation on [1, a]. It follows from (2.15) that (2.18) γ at = a Q γ t + a R Q x r a (x)ν t (dx) d with 1 r a (x) = 1 + a Q x x = x 2 a Q x 2 2 (1 + a Q x 2 )(1 + x 2 ). Denote by γ [a1,a 2 ] the variation of γ t on [a 1, a 2 ]. Then γ [a n+1,a n+2 ] a Q γ [a n,a n+1 ] + a Q x r a (x) (ν a n+1 ν a n)(dx). Finiteness of the last integral follows from (2.17). Hence γ t is locally of bounded variation on [1, ). As n, a nq 1/n tends to max 16j6d a q j, where q 1,..., q d are the eigenvalues of Q ([8], p. 153). Since Q M + d, this limit is less than 1. Thus 9

11 we can choose an integer m 1 such that a mq < 1. Let b = a m and use b as an epoch of {X t }. We get (2.18) with a replaced by b. It follows that γ [b n 1,b n ] b Q γ [b n,b n+1 ] + x r b (x) (ν b n ν b n 1)(dx). Hence we obtain (1 b Q ) γ [,1] b Q γ [1,b] + x r b (x) ν 1 (dx). Now we see that γ [,1] <. Theorem Let Q M + d. Let {X t} be a Q-selfsimilar additive process in law on R d. Then {X t } is natural. Proof. It is enough to show that γ t is of bounded variation on [1, 2], since we can use the preceding theorem with a = 2. Since we have γ t = t Q γ 1 + t Q x r t (x)ν 1 (dx), we can prove that γ t has continuous derivative in t >. bounded variation on [1, 2]. It follows that γ t is of Remark. When Q = ci with c (, ), the Q-semi-selfsimilarity is the c-semiselfsimilarity studied in Maejima and Sato [12]. Theorems 7 and 1 of [12] show that, given a semi-selfdecomposable (see [18] Definition 15.1) distribution µ 1 on R d with span a c, there is a wide variety of choice of {µ t : 1 < t < a} in constructing a c-semiselfsimilar additive process in law {X t : t } with epoch a such that L(X t ) = µ t for t [1, a). Thus we can find a non-natural c-semi-selfsimilar additive process in law {X t : t } with epoch a satisfying L(X 1 ) = µ Independently scattered random measures Following Urbanik and Woyczynski [22] and Rajput and Rosinski [14] and extending the notion from real-valued to R d -valued, we give the following definition. Definition. A family of R d -valued random variables {M(): [, ) } is called an R d -valued independently scattered random measure (i. s. r. m.) if the following conditions are satisfied: (1) (countably additive) for any sequence 1, 2,... of disjoint sets in [, ) with n=1 n [, ), n=1 M( n) converges a. s. and equals M( n=1 n) a. s., (2) (independent increments) for any finite sequence 1,..., n of disjoint sets in [, ), M( 1),..., M( n ) are independent, (3) (atomless) M({a}) = a. s. for every one-point set {a}. 1

12 Note that, if 1, 2,... is an increasing sequence in [, ) with = n=1 n [, ), then M( n) M() a. s. This follows from property (1). Note also that property (1) implies that M( ) = a. s., where is the origin of R d. Lemma 3.1. If {M()} is an R d -valued i. s. r. m., then L(M()) ID for any [, ). Proof. Let [, ). Define Y t = M( [, t]). Then it follows from the defining properties of i. s. r. m. that {Y t : t } is an additive process in law. Hence L(Y t ) ID. Thus L(M()) ID, since M() = Y t for large t. Here is our main result in this section. Theorem 3.2. (i) Let {M(): [, ) } be an Rd -valued i. s. r. m. Define {X t : t } by (3.1) X t = M([, t]) for t [, ). Then {X t } is a natural additive process in law. (ii) Let {X t : t } be a natural additive process in law on R d. Then there is an R d -valued i. s. r. m. {M(): [, ) } such that (3.1) holds. This is unique in the sense that, if {M 1 ()} and {M 2 ()} both satisfy this condition, then M 1 () = M 2 () a. s. for every [, ). Denote µ = L(M()). Then A, ν, and γ in the triplet of µ coincides with those of Lemma 2.5. For any factoring ({ρ s }, σ) of {X t } and any [, ), (3.2) log µ (z) = log ρ s (z)σ(ds). Proof. (i) Using Lemma 3.1, denote the location parameter of L(M()) by γ. Then γ is countably additive in [, ), which follows from countable additivity of M() and (2.6) of Lemma 2.4. Hence γ t = γ [,t] is a function locally of bounded variation by Section 29 of [4]. (ii) Let {X t } be a natural additive process in law on R d with a factoring ({ρ s }, σ). We will define M() for [, ) in several steps. Step 1. If J is an empty set or a one-point set, then we define M(J) =. If J is a finite interval in [, ) with left end s and right end t, that is, J = (s, t), [s, t], (s, t], or [s, t), then we define M(J) = X t X s. If = n j=1 J j with disjoint finite intervals J 1,..., J n in [, ), then we define M() = n j=1 M(J j). This definition does not depend on the expression of. We see from (2.8) that (3.2) is true for this 11

13 . Finite additivity and independent increment property within the class of sets of this type are obvious. Step 2. Let G be a bounded open set in [, ). Then G is expressed uniquely (up to the order) as G = j J j, where J 1, J 2,... (finite or infinite sequence) are disjoint open intervals (possibly of the form [, t)). If it is a finite sequence, M(G) is defined in Step 1. So we assume that it is an infinite sequence. Let J j = (s j, t j ) and let Y n = n j=1 (X t j X sj ). Then, for m < n, Ee i z,yn Ym = exp log ρ s (z)σ(ds) 1 as m, n. S n j=m+1 J j In general, a sequence {Y n } of R d -valued random variables converges in probability if Ee i z,y n Y m 1 as m, n with m < n, because, for any ε >, 1 ε ε ε ( 1 Re Ee iz j ((Y n) j (Y m) j ) ) dz j P [ (Y n ) j (Y m ) j ε/2], where the subscript j denotes the jth component ([18] p. 43). This fact is often useful. Since the summands are independent, convergence in probability implies convergence a. s. Thus we define M(G) = j=1 M(J j) a. s. We can prove that this definition does not depend on the order of summation. Step 3. Let K be a compact set and K [, ). Choose t such that K [, t ) and let G = [, t ) \ K. Then G is open in [, ). Define M(K) by M(K) = M([, t )) M(G). If K is an interval, then this definition is consistent with that of Step 1. Let a = sup x K x. Express G = j J j, where J 1, J 2,... is a sequence of disjoint open intervals and J 1 = (a, t ). Let G n = n j=1 J j. Then M(K) = lim n M([, t ) \ G n ) a. s. We have an expression [, t ) \ G n = k n l=1 [s n,l, t n,l ], where [s n,1, t n,1 ],..., [s n,kn, t n,kn ] are disjoint closed intervals (possibly one-point sets). It follows that (3.3) M(K) = lim n k n l=1 (X tn,l X sn,l ) a. s. Obviously (3.2) is true for = K. Using the expression (3.3), we can prove that, if K 1 and K 2 are compact and K 1 K 2, then (3.4) Ee i z,m(k 1) M(K 2 ) = exp log ρ s (z)σ(ds). K 1 \K 2 We can also show that, if K 1,..(., K m are disjoint compact sets, then M(K 1 ),..., m ) M(K m ) are independent and M = m p=1 M(K p) a. s. p=1 K p 12

14 Step 4. Let [, ). y the regularity of the measure σ (see Section 52 of [4]), we can find an increasing sequence of compact sets K 1, K 2,... such that K p and lim p σ(k p ) = σ(). As p, q with q < p, Ee i z,m(k p) M(K q ) = exp log ρ s (z)σ(ds) 1 K p\k q by (3.4). Hence M(K p ) is convergent in probability as p. We define M() = p-lim p M(K p ). We can show that this definition does not depend on the choice of the sequence K p. Step 5. It follows from the definition in Step 4 that M has properties (1) and (2) of the definition of i. s. r. m. and also (3.2). Step 6. Proof of uniqueness of M. Let M 1 and M 2 be R d -valued i. s. r. m. satisfying (3.1). Fix t >. The class S of all [,t ] satisfying M 1 () = M 2 () is a λ- system and contains and all intervals of the form (s, t] in [, t ]. Hence S = [,t ] by Dynkin s π-λ theorem in [2] p. 37. Hence M 1 () = M 2 () a. s. for every. Let (A, ν, γ ) be its triplet. Then A, γ, and ν (C) for each C (R d ) are countably additive in [, ), which follows from countable additivity of M(), as in the proof of Theorem 9.8 of [18]. Hence they coincide with those of Lemma 2.5. Remark. Like in the proof of Proposition 2.1 (b) of [14], one can construct an R d -valued i. s. r. m. on the product measurable space (R d ) [, ), using Kolmogorov s extension theorem. Namely, let ω = (ω ) [, ) be a general element of this space. Start with µ ID(R d ) with triplet (A, ν, γ ) in Lemma 2.5 and define P ((ω 1,..., ω n ) D), D ((R d ) n ), as the product of µ 1,..., µ n if 1,..., n are disjoint. If 1,..., n are not disjoint, then express 1,..., n as unions of some of disjoint sets C 1..., C m and define P ((ω 1,..., ω n ) D) in the form derived from the product of µ C1,..., µ Cm and check the consistency. However, we cannot in this way construct {M()} in the same probability space that the given {X t } is defined. The case where the process {X t } in Theorem 3.2 is a Lévy process in law is important. An R d -valued i. s. r. m. {M()} is called homogeneous if M() d = M( + a) a. s. for any [, ) and a. Proposition 3.3. Let {M()} be an R d -valued i. s. r. m. Then the following statements are equivalent: (1) {M()} is homogeneous, 13

15 (2) M((s, t]) d = M((s + a, t + a]) a. s. if s < t < and a, (3) the process {X t : t } defined by X t = M([, t]) is a Lévy process in law. Proof is easy and omitted. Let {M()} be an R d -valued i. s. r. m. Then the canonical measure of the natural additive process in law {X t } defined by (3.1) is called the canonical measure of {M()}. Proposition 3.4. Let {M()} be an R d -valued i. s. r. m. and σ its canonical measure. Then, [, ) satisfies σ() = if and only if (3.5) M(C) = a. s. for all orel sets C satisfying C. Proof. If σ() =, then, for any orel set C, σ(c) = and hence M(C) = a. s. by formula (3.2). Conversely, if (3.5) holds, then A C =, ν C =, and γ C = for all orel sets C and we have σ() = by formula (2.11). Proposition 3.5. Let {M()} be an R d -valued i. s. r. m. If ({ρ s}, σ ) is a factoring of the natural additive process in law {X t } defined by (3.1), then the canonical measure σ is absolutely continuous with respect to σ. Proof. y Theorem 3.2, (3.2) is true both for ({ρ s }, σ) and for ({ρ s}, σ ). Let [, ). If σ () =, then σ (C) = for all C and thus M(C) = a. s. for all C, which implies σ() = by Proposition 3.4. The following useful result is by Urbanik and Woyczynski [22] when d = 1. Proposition 3.6. For n = 1, 2..., let {M n ()} be R d -valued i. s. r. m. Suppose that for each [, ) there is an Rd -valued random variable M() such that (3.6) p-lim M n () = M(). n Then, {M()} is an i. s. r. m. Proof. It is clear that M() is finitely additive and satisfies (2) and (3) of the definition of i. s. r. m. Since L(M n ()) ID and L(M n ()) L(M()), we have L(M()) ID for each. Let (A n, νn, γn ) and (A, ν, γ ) be the triplets of L(M n ()) and L(M()), respectively. Define τ n = tr(an ) + R (1 x 2 )ν n d (dx) and τ = tr(a ) + R (1 x 2 )ν d (dx). Then, for each, τ n τ and γ n γ by Lemma 2.4. Hence by the Nikodým Theorem (see Dunford and Schwartz [3] p. 16) τ and γ are countably additive in. We claim that, if 1, 2,... is a decreasing 14

16 sequence of bounded orel sets with k=1 k =, then p-lim k M( k ) =. Indeed, we have j=k C j = k for C j = j \ j+1 and hence j=k γ C j = γ k. This shows that γ k. Similarly, τ k. Since g c (z, x) const (1 x 2 ) for any fixed z, it follows that Ee i z,m( k) 1 as k. That is, p-lim M( k ) =. Hence M is countably additive. 4. Stochastic integrals based on natural additive processes in law In this section let {X t : t } be a natural additive process in law on R d and let ({ρ s }, σ) be its canonical factoring. y Theorem 3.2 {X t } induces a unique R d -valued independently scattered random measure {M(): [, ) }. We will define and study stochastic integrals of M l d -valued nonrandom functions based on this random measure. As is remarked in Proposition 2.1, the process {X t } is a semimartingale. Thus stochastic integrals based on {X t } are defined for some class of random integrands in L 2 -theory through random localization; see Kunita and Watanabe [9] and Jacod and Shiryaev [5]. ut we will define stochastic integrals of nonrandom integrands directly through convergence in probability and give a representation of the characteristic functions of the distributions of the integrals, as was done in [22] and [14]. In the case of Lévy processes, this was already done by Lukacs [1]; see also [15]. We prove some properties of the integrals including a Fubini type theorem. An M l d -valued function F (s) on [, ) is called a simple function if n (4.1) F (s) = 1 j (s)r j j=1 for some n, where 1,..., n are disjoint orel sets in [, ) and R,..., R n M l d. It is called a step function if, in addition, 1,..., n are intervals or one-point sets. The following definition of integrals and Proposition 4.1 follow [22] of the case d = 1. Definition. Let F (s) be an M l d -valued simple function on [, ) in (4.1) and let [, ). Define (4.2) F (s)dx s = F (s)m(ds) = n R j M( j ). We use F (s)dx s and F (s)m(ds) in the same meaning. The definition (4.2) does not depend (in the a. s. sense) on the choice of a representation (4.1) of F (s). j=1 15

17 Definition. An M l d -valued function F (s) on [, ) is said to be M-integrable or {X t }-integrable if it is measurable and if there is a sequence of simple functions F n (s), n = 1, 2,..., such that (1) F n (s) F (s) σ-a. e. and (2) for every [, ), the sequence F n(s)dx s is convergent in probability as n. Proposition 4.1. If F (s) is M-integrable and if F 1 n(s) and F 2 n(s) are sequences satisfying (1) and (2) above, then (4.3) p-lim n Fn(s)dX 1 s = p-lim n F 2 n(s)dx s a. s. for each [, ). Definition. For any M-integrable M l d -valued function F (s) on [, ), define (4.4) F (s)dx s = F (s)m(ds) = p-lim F n (s)dx s, n using the sequence F n (s) in the definition of M-integrability and using Proposition 4.1. Proposition 4.2. Let F 1 (s) and F 2 (s) be M-integrable M l d -valued functions on [, ). Then, for any a 1, a 2 R, a 1 F 1 (s) + a 2 F 2 (s) is M-integrable and (4.5) (a 1 F 1 (s) + a 2 F 2 (s))dx s = a 1 F 1 (s)dx s + a 2 F 2 (s)dx s a. s. for [, ). Proposition 4.3. Let F (s) be an M-integrable M l d -valued function on [, ). Let Λ() = F (s)dx s and λ = L(Λ()) for [, ). Then {Λ(): [, ) } is an R l -valued i. s. r. m., (4.6) t and (4.7) log λ (z) = log ρ s (F (s) z) σ(ds) < for t (, ), log ρ s (F (s) z) σ(ds) Here log ρ s (F (s) z) means (log ρ s (w)) w=f (s) z. for [, ). Proof of Propositions 4.1, 4.2, and 4.3. If F 1 (s) and F 2 (s) are simple functions, then a 1 F 1 (s)+a 2 F 2 (s) is simple and (4.5) is obvious. If F (s) is a simple function, then the statements in Proposition 4.3 are easily shown. Indeed, in this case, it follows 16

18 from (3.2) and (4.2) that Ee i z,λ() = E exp = n j=1 [ i z, ] n R j M( j ) = j=1 exp log ρ s (R jz)σ(ds) = exp j n j=1 log ρ s ( n j=1 Ee i z,r jm( j ) 1 j (s)r jz ) σ(ds), which gives (4.6) and (4.7). Let F 1 n(s) and F 2 n(s) be the sequences in the statement of Proposition 4.1. Define G n (s) = F 1 n(s) F 2 n(s), Λ n () = G n(s)dx s, and Λ() = p-lim n Λn (). Since Λ n is an i. s. r. m., Λ is also i. s. r. m. by Proposition 3.6. y Egoroff s theorem ([4] p. 88), for any t >, there are disjoint orel sets C 1, C 2,... in [, t ] such that lim n sup s Cl G n (s) = for each l and σ([, t ] \ C) =, where C = l=1 C l. Using (4.7) for Λ n and noting (2.12), we see that Ee i z, Λn( C e l ) 1 as n for every [, ) and l. Hence Λ( C l ) = a. s. Therefore, Λ( C) = l=1 Λ( C l ) = a. s. Moreover, Λ( \ C) = a. s. since Λ n ( \ C) = a. s. by (4.6) for Λ n. It follows that Λ() = a. s. for all [,t ] and thus for all [, ). This proves Proposition 4.1. Proposition 4.2 is now straightforward. Turning to proof of Proposition 4.3, let F n (s) be simple functions in the definition of M-integrability and let Λ n () = F n(s)dx s and λ n = L(Λ n()). This Λ n is an i. s. r. m. It follows from p-lim Λ n () = Λ() that Λ is an i. s. r. m. by Proposition 3.6 and that log λ n (z) log λ (z) by Lemma 7.7 of [18]. Fix t >. We see that log λ n (z) is countably additive in [,t ] and absolutely continuous with respect to σ, since it satisfies (4.7) with F n (s) replacing F (s). Hence log λ (z) is countably additive in [,t ] and absolutely continuous with respect to σ by the Vitali Hahn Saks theorem and the Nikodým theorem ([3] p ). Hence there is the Radon Nikodým derivative h(s, z) such that t h(s, z) σ(ds) < and log λ (z) = h(s, z)σ(ds). On the other hand, fix z and find that log ρ s(f n (s) z) log ρ s (F (s) z) for σ-a. e. s, since log ρ s (w) is continuous in w. A use of Egoroff s theorem as in the proof of Proposition 2.6 of [14] yields that log ρ s (F (s) z) = h(s, z) for σ-a. e. s in [, t ]. Hence (4.6) and (4.7) follow. 17

19 Corollary 4.4. Let ({ρ s}, σ ) be a (not necessarily canonical) factoring of {X t }. Then, in the situation of Proposition 4.3, t log ρ s(f (s) z) σ (ds) < for t (, ), log λ (z) = log ρ s(f (s) z) σ (ds) for [, ), and the additive process in law {Y t } defined by Y t = Λ([, t]) has a factoring ({ρ s}, σ ), where ρ s(z) = ρ s(f (s) z), z R l. Proof. y Proposition 3.5, the canonical measure σ is absolutely continuous with respect to σ. Thus there is a measurable function h(s) such that σ(ds) = h(s)σ (ds). Let C = {s: h(s) > }. We can prove that ρ s = (ρ s) 1/h(s) for σ-a. e. s and that ρ s = δ for σ -a. e. s in [, ) \ C, where δ is the unit mass at. Thus the assertion follows from (4.6) and (4.7). Details are omitted. Proposition 4.5. Let ({ρ s}, σ) be a factoring of {X t }. Let F (s) be an M l d -valued measurable function locally bounded on [, ). Then F (s) is M-integrable. If F n (s) is a sequence of simple functions on [, ) such that F n (s) F (s) σ -a. e. and, for any t >, F n (s) is uniformly bounded on [, t ], then p-lim F n (s)dx s = F (s)dx s for [, ). n Proof. We can find simple functions F n (s) such that F n (s) F (s) for all s and F n (s) is uniformly bounded on [, t ] for any t. Then [ ] E exp i z, F n (s)dx s F m (s)dx s = exp log ρ s ((F n (s) F m (s)) z)σ(ds), which tends to 1 as n, m, using (2.12). Hence, F n(s)dx s is convergent in probability. Hence F (s) is M-integrable. To prove the second half of the assertion we use Propositions 3.5, 4.1, and the argument above. Theorem 4.6. Let F (s) be an M l d -valued measurable function locally bounded on [, ). Define Λ() = F (s)dx s and Y t = Λ([, t]). Then, for any M m l -valued measurable function G(s) locally bounded on [, ) and for any [, ), (4.8) G(s)dY s = G(s)F (s)dx s a. s. Proof. Choose simple functions F n (s) and G k (s) such that F n (s) F (s) and G k (s) G(s) for all s and, for any t >, F n (s) and G k (s) are uniformly bounded 18

20 on [, t ]. Then G k(s)dy s = p-lim n G k(s)f n (s)dx s from the definitions and, using (4.7), we get G k(s)f n (s)dx s G k(s)f (s)dx s in probability as n. Then, letting k, we get (4.8). Remark. Let F (s) be an M-integrable M l d -valued function on [, ). Sometimes we write (4.9) t s F (u)dx (s,t] u for s < t < F (u)dx u = for s = t < F (u)dx (t,s] u for t < s <. y Theorem 11.5 of [18], there is an additive process modification {Ỹt : t } of the additive process in law {Y t : t } of Corollary 4.4. We understand t F (u)dx s u in the meaning that (4.1) without explicit mention. t s F (u)dx u = Ỹt Ỹs, Theorem 4.7. Let F (s) be M l d -valued and G(s) be M m l -valued, both locally bounded, measurable on [, ). Then, for t < t 1 <, t1 ( s ) t1 ( t1 ) (4.11) G(s) F (u)dx u ds = G(s)ds F (u)dx u a. s. t t Lemma 4.8. If G(s) is an M m l -valued bounded measurable function on [t, t 1 ], then there is a sequence G n (s) of uniformly bounded step functions on [t, t 1 ] such that G n (s) G(s) except on a set of Lebesgue measure. Proof. y Lusin s theorem ([4] p. 243), for each n, there is a closed set n [t, t 1 ] such that [t, t 1 ] \ n has Lebesgue measure < 2 n and the restriction of G(s) to n is continuous. Then, by Urysohn s theorem in general topology, there is an M m l -valued, uniformly bounded, continuous function G n on [t, t 1 ] such that G n = G on n. Now choose uniformly bounded step functions G n on [t, t 1 ] such that G n (s) G n(s) < 2 n. Outline of proof of Theorem 4.7. Define t1 ( s ) Y = G(s) F (u)dx u ds, Z = t Step 1. Show that (4.12) Ee i z,y = Ee i z,z = exp t t1 t t log ρ u u t1 ( t1 t u ( F (u) t1 u ) G(s)ds F (u)dx u. ) G(s) ds z σ(du). 19

21 The second equality in (4.12) is a consequence of (4.7). Calculation of Ee i z,y is done by approximation by Y n = ( t 1 ) τn (s) t G(s) t F (u)dx u ds, where τ n (s) = t n,k for t n,k 1 < s t n,k with t n,k = t + k2 n (t 1 t ). Step 2. Prove the identity t1 t sdx s = t 1 X t1 t X t t1 t X s ds a. s., by approximation of s by τ n (s) in Step 1. Then, using this, we can show that Y = Z a. s., under the assumption that F (s) and G(s) are step functions. Step 3. Let G(s) be a step function. If there are step functions F n (s) such that F n (s) F (s) σ-a. e., then we can show that Y = Z a. s. by using Step 2 for F n (s) and G(s) and then, for convergence, using (4.12) with F (u) F n (s) in place of F (u). Thus Y = Z a. s. is true if F (s) = 1 (s)r with an open set and R M l d. Then the case where F (s) = 1 (s)r with compact is treated and then the case with orel. measurable. Next we can show that Y = Z a. s. when F (s) is locally bounded and Step 4. Show that Y = Z a. s. when F (s) and G(s) satisfy the conditions in the theorem, using Lemma 4.8 and the result in Step 3. Corollary 4.9 (Integration-by-parts formula). Let F (s) be an M l d -valued function of class C 1 on [, ). Then, for t < t 1 <, (4.13) t1 t F (s)dx s = F (t 1 )X t1 F (t )X t t1 t df (s) ds X sds a. s., where X s in the integrand of the last integral is understood to be a Lévy process modification. Proof. Rewrite t 1 t F (s)dx s, using F (s) = F (t 1 ) t 1 (df (u)/du)du and then s apply Theorem 4.7. Theorem 4.1 (Time change). Let τ(t) be an increasing continuous mapping from [, ) into [, ) with τ() =. Define a natural additive process in law {Y t : t } on R d by Y t = X τ(t). Let F (s) be an M l d -valued measurable function locally bounded on [, ). Then, for any orel set satisfying [, t ] with some t < τ( ), (4.14) F (τ(s))dy s = F (s)dx s a. s. τ 1 () Proof. The process {Y t } is natural additive by Proposition Denote by {Λ()} the R d -valued i. s. r. m. induced by {Y t }. Then we can show that Λ(τ 1 ()) = 2

22 M() for any orel set satisfying [, t ] with some t < τ( ). Thus we can show (4.14) whenever F is a simple function. Then we can extend it to F in the theorem, using Propositions 2.8, 2.12, and Some stochastic integrals over unbounded sets In the preceding section we defined stochastic integrals F (s)dx s only for bounded orel sets in [, ). Now we consider unbounded orel sets. Definition. Let {X t : t } be a natural additive process in law on R d and let M be the R d -valued independently scattered random measure induced by {X t }. Let F (s) be an M-integrable M l d -valued function. Let be an unbounded orel set in [, ). We define F (s)dx s = F (s)m(ds) = p-lim F (s)dx s, t [,t] whenever this limit in probability exists. In this case we say that F (s)dx s is definable. When = [t, ) and F (s)dx s is definable, we sometimes write t F (s)dx s for F (s)dx s. When {X t } is a Lévy process on R d and F (s) = e sq with Q M + d, the following important facts are known (see [7], [15], [2], [23]). The integral e sq dx s is definable if and only if {X t } has finite log-moment, that is, E log + X t < for all t. If the integral e sq dx s is definable, then its distribution µ is Q-selfdecomposable, that is, for each b (, 1), there is a distribution (automatically infinitely divisible) ρ b such that µ(z) = µ(b Q z) ρ b (z). Conversely, any Q-selfdecomposable distribution can be expressed as L ( e sq dx s ) with a unique (in law) Lévy process with finite log-moment. We study a case where {X t } belongs to a class of additive processes in law more general than Lévy processes in law. Definition. A stochastic process {X t : t } on R d is called a semi-lévy process in law or additive process in law with semi-stationary increments on R d if it is an additive process in law on R d such that, for some p >, (5.1) X t X s d = X t+p X s+p for any choice of s < t <. 21

23 This p is called a period of the semi-lévy process in law. A cadlag modification of a semi-lévy process in law is called a semi-lévy process. An additive process in law {X t } on R d is said to have finite log-moment if E log + X t < for all t. Remark. Let {X t } be a semi-lévy process in law on R d with period p and let (A t, ν t, γ t ) be the triplet of X t. Then {X t } is natural if and only if γ t is of bounded variation on [, p]. There exist non-natural semi-lévy processes in law on R d. Proposition 5.1. Let {X t : t } be a natural additive process in law on R d. Then the following statements are equivalent: (1) {X t } is a semi-lévy process in law with period p, (2) the canonical factoring ({ρ s }, σ) of {X t } is periodic with period p in the sense that ρ s = ρ s+p for σ-a. e. s and σ() = σ( + p) for all ([, )), (3) the i. s. r. m. {M()} induced by {X t } is periodic with period p in the sense that M() d = M( + p) for all [, ). Using Proposition 2.8 and (2.11), proof of Proposition 5.1 is easy and omitted. Let us recall some classes of distributions defined in [13]. Let Q M + d and b (, 1). A probability measure µ on R d is said to be (b, Q)-decomposable if µ(z) = µ(b Q z) ρ(z) with some ρ ID(R d ). The class of all such probability measures is denoted by L (b, Q). In the terminology of [18], the class L (b, ci) with c > is the class of semi-selfdecomposable distributions with span b c. Theorem 5.2. Let {X t : t } be a natural semi-lévy process in law on R d with period p. Suppose that it has finite log-moment, that is, (5.2) E log + X p <. Then, for any Q M + d, the stochastic integral e sq dx s is definable and its distribution µ belongs to L (e p, Q). Moreover, for any a (, ), (5.3) and sup z 6a (5.4) log µ(z) = log ρ s (e sq z) σ(ds) < log ρ s (e sq z)σ(ds), where ({ρ s }, σ) is the periodic canonical factoring of {X t : t }. In particular, when Q = ci with c >, the stochastic integral e csi dx s is definable and has a semi-selfdecomposable distribution with span e cp, if condition (5.2) is satisfied. 22

24 The case without finite log-moment will be treated in Theorem 5.4. Remark. In a forthcoming paper jointly written with M. Maejima, it will be proved that, for any µ L (e p, Q), there exists a natural semi-lévy process in law {X t } with finite log-moment such that L ( e sq dx s ) = µ. Proof of Theorem 5.2 uses the following lemma. Lemma 5.3. Let {X t : t } be a semi-lévy process in law on R d with period p. Let ν t be the Lévy measure of X t and let ν be the unique measure on [, ) R d satisfying (2.7). Then, there are a measure ν on R d and measures σ x, x R d, on [, ) satisfying the following conditions: (1) ν ({}) = and R d (1 x 2 )ν (dx) <, (2) for any x R d, σ x is a periodic measure with period p and σ x((, p]) = σ x([, p]) = 1, (3) for any [, ), σ x() is measurable in x, (4) for any nonnegative function f(s, x) measurable in (s, x), (5.5) f(s, x) ν(d(s, x)) = ν [, ) R R (dx) f(s, x)σx(ds). d d [, ) If (ν, σx) and (ν, σx ) both satisfy these conditions, then ν = ν and σx = σx for ν -a. e. x. When {X t } in the lemma has a factoring ({ρ s }, σ), then (5.6) σ(ds) f(s, x)νs R ρ (dx) = d f(s, x) ν(d(s, x)) [, ) R d for all nonnegative measurable function f(s, x). If f(s, x) = 1 [,t] (s) 1 (x), then (5.6) holds by (2.1) and (2.7). From this (5.6) follows in general. Comparing (5.5) and (5.6), we see that ({ν ρ s }, σ) and (ν, {σ x}) are dual in a sense. Proof of Lemma 5.3. We have ν({t} R d ) = for all t. Fix a positive integer N. Applying the conditional distribution theorem to the probability measure m(c) = const C (1 x 2 ) ν(d(s, x)) on [, Np] R d, we get a measure ν on R d and measures σ x on [, Np] such that ν ({}) =, R d (1 x 2 )ν (dx) = (Na) 1, σx([, Np]) = N, σx() is measurable in x for each ([, Np]) and f(s, x)(1 x [,Np] R 2 ) ν(d(s, x)) = (1 x R 2 )ν (dx) f(s, x)σx(ds) d d [,Np] 23

25 for all nonnegative measurable f(s, x). Since ν p+s = ν p + ν s, we can show that ν (dx) σx(du) = ν (dx) σx(du) for (R d ). (p,p+s] Hence σ x((p, p+s]) = σ x((, s]) for ν -a. e. x. y right-continuity in s, the exceptional set of x can be chosen to be independent of s. Thus we can choose σ x satisfying property (2). y the uniqueness in the conditional distribution theorem, ν does not depend on N and σ x can be extended to a periodic measure on [, ). Proof of Theorem 5.2. Let ({ρ s }, σ) be the periodic canonical factoring of {X t }. Let us prove (5.3). First notice that Then, g c (z, x) C z x 2 (1 + x 2 ) 1 with C z = ( z z ) (4 + z ). log ρ s (e sq z) 1 2 (traρ s) e sq z 2 + z e sq γs ρ e sq x 2 + C z R 1 + e d sq x 2 νρ s (dx) + z e R sq x r s (x) νs ρ (dx), d where r s (x) = c(e sq x) c(x). Since the estimate (2.16) remains true if Q is replaced by Q and since σ is periodic, e sq z 2 σ(ds) const z 2 and e sq z σ(ds) const z. Note that tra ρ s and γs ρ are σ-essentially bounded (Proposition 2.8). Further, we will prove that (5.7) (5.8) (,s] e σ(ds) R sq x e d sq x 2 νρ s (dx) <, σ(ds) e R sq x r s (x) νs ρ (dx) <. d Write f(ξ) = ξ 2 /(1 + ξ 2 ). Then, by (5.5), (5.6), and (2.16), the iterated integral in (5.7) is ν R (dx) d Notice that, by (2) of Lemma 5.3, Hence, f(c 3 e c 1s x )σ x(ds) = a, say. (np,(n+1)p] f(c 3 e c 1s x )σ x(ds) f(c 3 e c 1np x ) 1 p a 1 p R d ν (dx) p np (n 1)p f(c 3 e c 1s x )ds. f(c 3 e c1s x )ds = 1 log(1 + c 2c 1 p R 2 3e 2c1p x 2 )ν (dx), d 24

26 which is finite by (1) of Lemma 5.3 and by log + x ν (dx) = log + x ν p (dx) <. This proves (5.7). Proof of (5.8) is similar, since the iterated integral in (5.8) is e const σ(ds) R sq x x 2 (1 + e d sq x 2 )(1 + x 2 ) νρ s (dx) and since (1 + ξ)/(1 + ξ 2 ) 2 for ξ. This finishes a proof of (5.3). Note that e sq is M-integrable by Proposition 4.5. Let t 1 < t 2. Then [ t2 ] t2 E exp i z, e sq dx s = exp log ρ s (e sq z)σ(ds) 1 t 1 as t 1, t 2, by using (4.7) and (5.3). Hence t e sq dx s is convergent in probability as t by the remark in Step 2 of the proof of Theorem 3.2 (ii). Let us prove that µ = L ( ) e sq dx s is in L (e p, Q). Let µ (p) = L ( p ) e sq dx s. Then µ (p) ID. Since p e sq dx s and e sq dx p s are independent and the latter has the same law as e pq e sq dx s by property (5.1), we get µ(z) = µ (p) (z) µ(e pq z). That is, µ is in L (e p, Q). Theorem 5.4. Let {X t : t } be a natural semi-lévy process in law on R d with period p. Assume that (5.9) E log + X p =. Then, for any Q M + d, ( e sq dx s is not definable. Moreover, for any sequence ) tn t n, L e sq dx s does not converge to any probability measure. ( ) tn Proof. Fix Q and a sequence t n. Denote L e sq dx s = µ (n). Suppose that µ (n) µ ( ) for some probability measure µ ( ). Then µ ( ) ID, since µ (n) ID. Let ν (n) and ν ( ) be the Lévy measures of µ (n) and µ ( ), respectively. Then, by [18] Theorem 8.7, (5.1) f(x)ν (n) (dx) t 1 f(x)ν ( ) (dx) for all bounded continuous functions f vanishing on a neighborhood of. We have, by (2.16), (5.5), and (5.6), tn ν (n) (dx) = σ(ds) x >1 = ν R (dx) d (,t n] 1 { e sq x >1}νs ρ (dx) 1 [,t n ] R {c4 e c 2 s x >1} ν(d(s, x)) d 1 {s<c 1 2 log c 4 x } σ x(ds), 1 { x >c 1 4 ec 2 s } σ x(ds) ν (dx) x >c 1 4 (,mp] 25

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