Densities for Ornstein-Uhlenbeck processes with jumps

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1 Densities for Ornstein-Uhlenbeck processes with jumps 11 april 28 Enrico Priola 1 Dipartimento di Matematica, Università di Torino, via Carlo Alberto 1, 1123, Torino, Italy. enrico.priola@unito.it Jerzy Zabczyk 2 Instytut Matematyczny, Polskiej Akademii Nauk, ul. Sniadeckich 8, -95, Warszawa, Poland. zabczyk@impan.gov.pl Mathematics Subject Classification 2): 6H1, 6J75, 47D7. Key words: Ornstein-Uhlenbeck processes, absolute continuity, Lévy processes. Abstract: We consider an Ornstein-Uhlenbeck process with values in R n driven by a Lévy process Z t ) taking values in R d with d possibly smaller than n. The Lévy noise can have a degenerate or even vanishing Gaussian component. Under a controllability rank condition and a mild assumption on the Lévy measure of Z t ), we prove that the law of the Ornstein-Uhlenbeck process at any time t > has a density on R n. Moreover, when the Lévy process is of α-stable type, α, 2), we show that such density is a C -function. 1 Supported by the Italian National Project MURST Equazioni di Kolmogorov and by the Polish Ministry of Science and Education project 1PO 3A Stochastic evolution equations with Lévy noise. 2 Supported by the Polish Ministry of Science and Education project 1PO 3A Stochastic evolution equations with Lévy noise. 1

2 1 Introduction and statement of the main results We study absolute continuity of the laws of a n-dimensional Ornstein-Uhlenbeck process X x t ), which solves the stochastic differential equation dx t = AX t dt + BdZ t, X = x R n. 1.1) Here Z t ) is a given Lévy process, with values in R d, defined on some stochastic basis Ω, F,F t ) t, P). The dimension d might be different and also smaller than n. Let us recall that Z t ) is a stochastic process having independent, time-homogeneous increments and càdlàg trajectories, starting from see [1]). Moreover A is a real n n matrix and B a real n d matrix. Ornstein-Uhlenbeck processes appear in many areas of science, for instance in physics see [1] and the references therein) and in mathematical finance see [2], [5], [6] and the references therein). Ornstein-Uhlenbeck processes with jumps have recently received much attention see [26], [24], [23] and [19]). In the paper we present two main results: one on existence of densities of Xt x ), and the other on the regularity of such densities. Both theorems assume the following controllability) rank condition Rank [B, AB,..., A n 1 B] = n. 1.2) Here [B, AB,..., A n 1 B] denotes the n nd matrix, composed of matrices B,..., A n 1 B, which corresponds to the linear mapping: u,..., u n 1 ) Bu A n 1 Bu n 1, from R nd into R n. An interesting example of an Ornstein- Uhlenbeck process with degenerate noise satisfying the rank condition with d = 1 and n = 2) is a solution of the equation Xt 1 = Z t, X 1 = x 1, Xt 2 = x 1 t + Z s ds + x 2, t, x = x 1, x 2 ) R ) It is a generalization of a famous example due to Kolmogorov, in which Z t ) was a real Wiener process. In [14] Kolmogorov showed that the law of the random variable X 1 t, X 2 t ) is absolutely continuous with respect to the Lebesgue measure, 2

3 for any t >, x R 2, and, in fact, its density is a C -function on R 2. This Gaussian example has also been considered by Hörmander in [11]. When the process Z t ) is a standard d-dimensional Wiener process, it is known that Xt x has a density if and only if the rank condition holds see, e.g., [7] and [8]). Moreover under 1.2) the random variables Xt x, t >, x R n, have C - densities. This regularity result can be easily extended to the case when the Lévy process Z t ) is given by a non-degenerate d-dimensional Wiener process with a drift plus an independent pure jump process see Section 2). Indeed, in such case, Xt x has two independent components, one of which is a Gaussian Ornstein- Uhlenbeck process at time t having a C -density. Note that convolution of two Borel probability measures has a density as long as at least one of the two measures has a density see [23, Lemma 27.1]). However, the situation is less clear if the Gaussian component of Z t ) degenerates or vanishes. In this paper we consider such case. Indeed, we formulate our mild assumptions for absolute continuity only in terms of the Lévy measure of Z t ). Our main first theorem is the following one. Theorem 1.1. Assume the rank condition 1.2). Assume also that the Lévy measure ν of Z t ) is infinite and that there exists r > such that ν restricted to the ball {x R d : x r} has a density with respect to the Lebesgue measure. Then, for any t > and x R n, the law of X x t is absolutely continuous. It also turns out see Proposition 2.1) that under the assumptions of the theorem, the Ornstein-Uhlenbeck process Xt x ) is strong Feller. There are a number of papers dealing with the absolute continuity of laws of degenerate diffusion processes with jumps see [4], [16], [18], [15] and [13]). They apply appropriate extensions of Malliavin calculus for jump processes assuming also the well-known Hörmander condition on commutators which becomes the rank condition 1.2) for Ornstein-Uhlenbeck processes). In [4] it is assumed that the Lévy measure of Z t ) has a sufficiently smooth density. In [16], [15], [18] and [13] α-stable type Lévy processes Z t ) are considered. The very weak sufficient conditions for the absolute continuity of the laws of degenerate diffusions with jumps, formulated in Theorem 1.1, are new. Moreover, in the proof, we use analytical methods as 3

4 well as control theoretic arguments. To formulate our second theorem, concerned with existence of regular densities, we need a new hypothesis on the Lévy measure ν. Hypothesis 1.2. There exist C > and α, 2), such that, for sufficiently small r >, the following estimate holds: z, h 2 νdz) C r 2 α, h R d, with h = ) {z R d : z,h r} This condition was introduced in [18]. However, both [18] and [13] prove C - regularity of densities of solutions of SDEs with jumps assuming a strictly stronger version of Hypothesis 1.2 in which the integral with respect to ν is taken over the smaller set {z R d : z r}. An interesting example of measure ν for which 1.4) holds but the stronger hypothesis is not verified is given in [18, Remark 1]. Clearly, if Z t ) is a d-dimensional α-stable process which is rotation invariant i.e., ψh) = c α h α, for h R d, α, 2), where c α is a positive constant) then 1.4) holds. Thus our next theorem generalizes the Kolmogorov regularity result concerning 1.3) to the case when Z t ) is a Lévy process of α-stable type. Theorem 1.3. Assume the rank condition 1.2) and Hypothesis 1.2. Then, at any time t >, x R n, the Ornstein-Uhlenbeck process Xt x ) has a C -density with all bounded derivatives. Moreover, for any t >, x R n, f : R n R Borel and bounded, E[fX x t )] = = 1 2π) n fe ta x + y) e i y,h exp R n R n 1 2π) n fz) e i z,h e i eta h,x exp R n R n 1.5) ) ) ψb e sa h)ds dh dy ) ) ψb e sa h)ds dh dz. 2 Existence of densities Consider the Ornstein-Uhlenbeck process introduced in 1.1). It is well known that this is given by X x t = e ta x + e t s)a BdZ s = e ta x + Y t, t, x R n, 2.1) 4

5 where the stochastic convolution Y t can be defined as a limit in probability of Riemann sums see, for instance, [23, Section 17] and [24]). The law µ x t of X x t has the characteristic function or Fourier transform) ˆµ x t, ˆµ x t h) = e i eta x,h ˆµ t h) = e i eta h,x exp where µ t denotes the law of Y t and ψ is the exponent of Z t ), E[e i u,zt ] = e tψu), u R d. ) ψb e sa h)ds, h R n, 2.2) By, and we indicate the inner product and the Euclidean norm in R k, k N, respectively. Moreover B denotes the adjoint or transposed) matrix of B. Recall the Lévy-Khintchine representation for ψ, ψs) = 1 ) 2 Qs, s i a, s e i s,y 1 i s, y I D y) νdy), s R d, 2.3) R d where I D is the indicator function of the ball D = {x R d : x 1}, Q is a symmetric d d non-negative definite matrix, a R d, and ν is the Lévy measure of Z t ). Thus ν is a σ-finite measure on R d, such that ν{}) =, 1 y 2 ) νdy) <. R d The triplet Q, a, ν) which gives 2.3) is unique. According to 2.3), the process Z t ) can be represented by the Lévy-Itô decomposition as Z t = at + RW t + Z t, t, 2.4) where R is a d d matrix such that RR = Q, W t ) is a standard R d -valued Wiener process and Z t ) is a Lévy jump process see [1]). The processes W t ) and Z t ) are independent. Let P t ) be the transition semigroup determined by X x t ), i.e., P t fx) = E[fX x t )], t, x R n, f B b R n ), where B b R n ) denotes the space of all real Borel and bounded functions on R n. The semigroup P t ) or the process X x t )) is called strong Feller if P t f is a continuous function, for any t > and for any f B b R n ). 5

6 Applying a result due to Hawkes see [12]) we show now that the strong Feller property for P t ) is equivalent to the existence of a density for the law of Xt x, for any t >, x R n. This result holds for any Ornstein-Uhlenbeck process defined in 1.1) without requiring the rank condition 1.2)). For related results in infinite dimensions, see [22] and [2]. Proposition 2.1. The semigroup P t ) is strong Feller if and only if, for each t >, x R n, the law µ x t of Xt x is absolutely continuous with respect to the Lebesgue measure. Proof. Fix t > and let µ t be the law of Y t see 2.1)). Since µ x t = δ e ta x µ t where δ a denotes the Dirac measure concentrated in a R n ) µ x t is absolutely continuous, for any x R n, if and only if µ t has the same property. We write, for any f B b R n ), x R n, P t fx) = fe ta x + y)µ t dy) = f e ta )x + e ta y)µ t dy) = R n R n f e ta )x + z)e ta µ t )dz), R n where e ta µ t ) is the image of the probability measure µ t under e ta. Applying [12, Lemma 2.1], we know that the Markov operator T t gx) = R gx+z)e ta n µ t )dz), x R n, maps Borel and bounded functions into continuous ones if and only if e ta µ t ) is absolutely continuous with respect to the Lebesgue measure. Hence P t f is continuous, for any f B b R n ), if and only if e ta µ t ) is absolutely continuous. This gives the assertion, since e ta is an isomorphism. The proof of Theorem 1.1 requires two lemmas. The first one is of independent interest. Lemma 2.2. Assume the rank condition 1.2). Then there exists T > depending on the dimension n and on the eigenvalues of A) such that for any integer m n + 1, for any s 1 <... < s m T, the linear transformations l s1,...,s m : R dm R n, m l s1,...,s m y 1,..., y m ) = e sja By j, are onto. 2.5) j=1 6

7 Proof. The proof is divided into two parts. I Part. We define T >. Let λ j ) be the distinct complex eigenvalues of A, j = 1,..., k with k n). Consider the following complex polynomial: pλ) = k j=1 λ λ j) n, λ C, and the corresponding ordinary linear differential operator pd) of order n, pd)yt) = k D λ j ) n) yt) = y n) t) + a 1 y n 1) t) a n, t R, j=1 where y C n R), a i C, and y i) denotes the i-derivative of y, i = 1,..., n. By a result due to Nehari see [17]) we know, in particular, that there exists T > depending on n and on the coefficients a 1,..., a n ) such that any nontrivial solution yt) to the equation pd)y = has at most n zeros on [ T, T ]. By this theorem, we deduce that the following quasi-polynomials yt) = k n 1 c rj e λjt t r, 2.6) j=1 r= which are solutions for pd)y = see, for instance, [3, Chapter 3]), have always at most n zeros on [ T, T ] no matter what are the complex coefficients c rj except the trivial case in which all c rj are zero). II Part. We prove the assertion. Introduce the following linear and bounded operators depending on t > ) L t : L 2 [, t]; R d ) R n, L t u = e sa Bus)ds, u L 2 [, t]; R d ). The controllability condition is equivalent to the fact that each L t is onto, t > see, for instance, [27, Chapter 1]). Hence, in particular, ImL T ) = R n T > is defined in the first part of the proof). To prove the assertion it is enough to show that ImL T ) Im l s1,...,s m ) 2.7) for any s 1 <... < s m T, and m n + 1. We fix m n + 1 and take s 1,..., s m ) with s 1 <... < s m T. Let v R n, v, be orthogonal to Iml s1,...,s m ). Assertion 2.7) follows if we prove that v, L T u =, for any u L 2 [, T ]; R d ). 2.8) 7

8 To this purpose, note that the orthogonality of v to Iml s1,...,s m ) is equivalent to B e s ja v =, for j = 1,..., m, i.e., B e s ja v, e k =, j = 1,..., m, k = 1,..., d, 2.9) where e k ) is the canonical basis in R d. Note that each mapping s B e sa v, e k, k = 1,..., d, is a quasi-polynomial like 2.6). Since m n + 1, condition 2.9) implies that each mapping B e sa v, e k is identically zero on [, T ] by the first part of the proof. It follows that L T u, v = T us), B e sa v ds =, for any u L 2 [, T ]; R d ). This implies that v is orthogonal to ImL T ) and so 2.7) holds. The proof is complete. Lemma 2.3. Let L : R p R q, p q, be an onto linear transformation. Let γ be a probability measure on R p having a density h with respect to the Lebesgue measure). Then the probability measure L γ, image of γ under L, has a density on R q. Proof. Since the result is clear when p = q, let us assume that p > q. We identify L with a q p matrix with respect to the canonical bases f i ) 1 i p in R p and e i ) 1 i q in R q. Consider the transposed matrix L and complete the system of vectors L e 1,..., L e q with vectors f i1,... f ip q in order to get a basis in R p. Define an invertible p p matrix S having the vectors L e 1,..., L e q, f i1,... f ip q as rows. If π : R p R q is the projection on the first q coordinates, we have that L = π S. Indeed, for any x R p, πsx) = e 1, Lx R q,..., e q, Lx R q) = Lx. Fix any Borel set B R q. Using also the Fubini theorem, we get I B x)l γ)dx) = I B πsz))hz)dz = R q R p = 1 I B πy))hs 1 y))dy dets) R p 1 dy 1... dy q h S dets) B R 1 y 1,..., y p )dy q+1... dy p. p q 8

9 It follows that L γ has the density y 1,..., y q ) 1 h S 1 y 1,..., y p )dy q+1... dy p. dets) R p q Proof of Theorem 1.1. We will use Lemma 2.2 and adapt the method of the proof of [23, Theorem 27.7], based on [25] and [9]. Let T > be as in Lemma 2.2. Using Proposition 2.1 and the semigroup property of P t ), in order to prove the assertion it is enough to show that the law of Y t see 2.1)) is absolutely continuous for any t, T ). Recall that for an arbitrary Borel measure γ on R n, we have the unique measure decomposition γ = γ ac + γ s 2.1) where γ ac has a density and γ s is singular with respect to the Lebesgue measure. Define, for N N sufficiently large, say N N with 1/N < r, the measure ν N having density I {1/N x r} with respect to ν, i.e. ν N = ν I {1/N x r} and Zt N = Z s, t, <s t, 1 N Z s r the measure ν N has density I {1/N x r} with respect to the measure ν defined in 2.3)) and Z s = Z s Z s Z s = lim h Z s+h ). The process Zt N ) is a compound Poisson process and its Lévy measure is just ν N. By the hypotheses, for any N N, ν N has a density. Moreover since ν is infinite, we have that c N = ν N R d ) as N. It is well known that Z N t ) and Z t Z N t ) are independent Lévy processes see, for instance, [1] or [21, Chapter 1]). It follows, in particular, that the random variables Y N t = e t s)a BdZ N s and Y t Y N t = e t s)a BdZ Z N ) s are independent, 2.11) for any N N, t >. Fix t, T ) and denote by µ the law of the random variable Y t and by µ N the one of Y N t. 9

10 Since µ = µ N β N where β N is the law of Y t Y N t ), we have by 2.1) µ = µ N ) ac β N ) s + µ N ) s β N ) s + µ N ) ac β N ) ac + µ N ) s β N ) ac. By [23, Lemma 27.1]) we deduce that µ N ) ac β N ) s + µ N ) ac β N ) ac + µ N ) s β N ) ac is absolutely continuous and so µ s = µ N ) s β N ) s )s and µ s R n ) µ N ) s β N ) s R n ) µ N ) s R n ), for any N N. 2.12) Now we compute µ N which coincides with the law of esa BdZ N s. First, note that the law of Z N t is given by e c N t δ + e c N t k 1 c N t) k k! ν N ) k, where c N = ν N R d ), ν N = ν N, c N ν N ) k = ν N... ν N k-times). Then consider a sequence ξ i ) of independent random variables having the same exponential law of intensity c N. Introduce another sequence U i ) of independent random variables independent also of ξ i )) having the same law ν N. It is not difficult to check that the probability measure µ N coincides with the law of the following random variable: 1 {ξ1 >t} + k 1 1 {ξ ξ k t<ξ ξ k+1 } e ξ 1A BU e ξ ξ k )A BU k ). Note that the events H = {ξ 1 > t}, H k = {ξ ξ k t < ξ ξ k+1 } are all disjoint, k 1. Since, for any f B b R n ), f X k 1 Hk ) = fx k )1 Hk, k k 1

11 where X = and X k = e ξ 1A BU e ξ ξ k )A BU k, k 1, we get EfYt N ) = e c N t f) + R N, where R N = Ef = k 1 Ef k 1 1 {ξ ξ k t<ξ ξ k+1 } e ξ 1 A BU e ξ ξ k )A BU k ) ) 1 {ξ ξ k t<ξ ξ k+1 } e ξ 1 A BU e ξ ξ k )A ) ) BU k = c N ) k+1 e c N t t k+1) dt 1... dt k+1 k=1 t t k t t t k+1 fe t1a By e t t k )A By k ) νn dy 1 )... ν N dy k ) R dk = c N ) k+1 e c N t t k+1) dt 1... dt k+1 k 1 t t k t t t k+1 fy)µ t1,...,t k dy), f B b R n ), R n where µ t1,...,t k is the probability measure on R n which is the image of the product measure ν N... ν N independent of N) acting from R dk into R n, k-times) under the linear transformation J t1,...,t k J t1,...,t k y 1,..., y k ) = e t 1A By e t t k )A By k, where y i R d, i = 1,..., k. For any k n + 1, t 1, t i >, i = 2,..., k, we have t 1 <... < t t k T and J t1,...,t k = l t1,...,t t k see 2.5) and recall that t, T )). Applying Lemma 2.2, we obtain that, for any k n + 1, t i >, i = 1,..., k, the linear transformation J t1,...,t k is onto. Therefore, by Lemma 2.3, the measure µ t1,...,t k has a density g t1,...,t k L 1 R n ), for any k n + 1, t i >, i = 1,..., k. Using this fact, we write µ N = µ 1 N + µ 2 N, where µ 1 N = e c N t δ + n + c N ) k+1 e c N t t k+1) µ t1,...,t k dt 1... dt k+1, t t k <t<t t k+1 k=1 and µ 2 N has the following density on Rn : y c N ) k+1 e c N t t k+1) g t1,...,t k y)dt 1... dt k+1. k>n t t k <t<t t k+1 11

12 Therefore µ N ) s R n ) µ 1 NR n ) n = e c N t + c N ) k+1 e c N t t k+1) dt 1... dt k+1, t t k <t<t t k+1 k=1 as N, since c N by hypothesis. By 2.12), we immediately get that µ s =. This gives the assertion. The proof is complete. 3 Proof of the C -result We pass now to the proof of Theorem 1.3. To obtain C -regularity of the law at time t of the Ornstein-Uhlenbeck process 1.1) we will be estimating its characteristic function. We fix t >. It is enough to show that the law µ t of Y t see 2.1)) has a density p t L 1 R n ) C R n ) with all bounded derivatives. To this purpose, note that by 2.2) the characteristic function of µ t is ˆµ t y) = exp ) ψb e sa y)ds, y R n. We claim that there exist a t and c t > such that, for any y R n, y 1, exp This will imply in particular that ˆµ t L 1 R n ). ψb e sa y)ds) c t e at y α. 3.1) Then, by using the Fourier inversion formula see [23, Propositions 2.5]) we will get the assertion. It is not restrictive to assume that Q = and a = in 2.3), i.e., that Z t ) has no Gaussian component. For any y R n, we have exp ) ψb e sa y)ds = exp First, note that condition 1.4) is equivalent to the fact that {z R d : z,k 1} ds 1 cos B e sa y, z ) ) ) νdz). R d z, k 2 νdz) C k α, 3.2) 12

13 for sufficiently large k R d, say k c. To see this, it is enough to change in the condition 1.4), the vector h to the vector k/r. Fix y R n with y 1; using also the inequality 1 cosu) c 1 u 2, if u π, we find ds 1 cos B e sa y, z ) ) νdz) R d t c 1 ds B e sa y, z 2 νdz) {z R d : B e sa y,z 1} c 1 1 {s [,t] : B e sa y c } ds c 1 C {z R d : B e sa y,z 1} 1 {s [,t] : B e sa y c } B e sa y α ds. Set M t = sup{ B e sa h : s [, t], h 1, h R n }; since s [, t], we get c 1 C c 1 C y α M α t c 1 C y α M α t 1 {s [,t] : B e sa y c } B e sa y α ds 1 {s [,t] : B e sa y c } 1 {s [,t] : B e sa y c } B e sa y, z 2 νdz) B e sa y α ds y M t B e sa y 2 ds. y M t B e sa y y M t 1, Let us recall that the rank condition 1.2) is equivalent to the existence of C t > such that, for any u R n, B e sa u 2 ds C t u 2 see [27]). Moreover 1 {s : B e sa y c } This implies that, for any y R n, with y 1, c 1 C y α M α t c 1 CC t y α M α 2 t c 1 CC t y α M α 2 t 1 {s : B e sa y c } We get, for any y R n, y 1, c 1 C y α M α t c 1 C y α M α t B e sa y 2 c 2 ds t y M t y 2 Mt 2 B e sa y 2 ds y M t 1 {s : B e sa y c } c 2 t y 2 Mt 2. B e sa y 2 ds y M t ds 1 cos B e sa y, z ) ) νdz) c 1 CC t Mt α 2 y α c 1 Cc 2 tmt α 2. R d The assertion 3.1) is proved. 13

14 Finally, by the Fourier inversion formula, p t y) = 1 ) 2π) n e i y,h exp ψb e sa h)ds dh, y R n, 3.3) R n is the density of µ t. Differentiating under the integral sign, we get easily the assertion. The proof is complete. Remark 3.1. It follows from Theorem 1.3 that for any Borel function f with compact support one has P t f C b Rn ), for any t > i.e., P t f C R n ) with all bounded derivatives of any order) where P t fx) = R n fz)p t z e ta x)dz. We do not know if this regularizing effect holds for all f B b R n ) as we are unable to show that for a given multi-index β the partial derivative D β p t is integrable on R n. Acknowledgment The authors thank the referee for the careful reading of the original manuscript, and for giving useful comments. References [1] D. APPLEBAUM, Lévy processes and stochastic calculus, Cambridge Studies in Advanced Mathematics 93 Cambridge University Press, 24). [2] O. E. BARNDORFF-NIELSEN and N. SHEPARD, Non-Gaussian Ornstein- Uhlenbeck-based models and some of their uses in financial economics, J. R. Stat. Soc. Ser. B Stat. Methodol ) [3] G. BIRKHOFF, G. ROTA, Ordinary differential equations, fourth edition John Wiley & Sons, 1989). [4] J. M. BISMUT, Calcul des variations stochastique et processus de sauts French) [Stochastic calculus of variations and jump processes], Z. Wahrsch. Verw. Gebiete ) [5] P. J. BROCKWELL, Lévy-driven CARMA processes, Ann. Inst. Statist. Math ) [6] R. CONT and P. TANKOV, Financial modelling with jump processes Chapman & Hall/CRC, Boca Raton, FL, 24). [7] G. DA PRATO and J. ZABCZYK, Stochastic Equations in Infinite Dimensions Cambridge University Press, 1992). 14

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