Densities for Ornstein-Uhlenbeck processes with jumps

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1 Densities for Ornstein-Uhlenbeck processes with jumps 8 august 27 arxiv:78.184v1 [math.pr] 8 Aug 27 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 condition and an 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 Introduction and notation We study absolute continuity of the law, at time t >, of a n-dimensional Ornstein- Uhlenbeck process X x t ), which solves the SDE dx t = AX t dt + BdZ t, X = x R n, 1.1) where Z t ) is a given Lévy process with values in R d and d might be different and also smaller than n. Let us recall that Z t ) is a stochastic process having independent increments, càdlàg trajectories, continuous in probability, starting from, and defined on some stochastic basis Ω, F,F t ) t, P). Moreover A is a real n n matrix and B a n d real matrix. 1 Partially 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 Partially supported by the Polish Ministry of Science and Education project 1PO 3A Stochastic evolution equations with Lévy noise. 1

2 Ornstein-Uhlenbeck processes appear in many areas of science, for instance in physics see [7] and the references therein) and in mathematical finance see [1], [3] and the references therein). Ornstein-Uhlenbeck processes with jumps have recently received much attention see [24], [22], [21] and [17]). When in 1.1) the process Z t ) is a d-dimensional Wiener process, it is known that X x t has a density if and only if the following Kalman controllability condition holds see, e.g., [4] and [5]) H1) Rank[B,AB,...,A n 1 B] = n. 1.2) Here [B,AB,..., A n 1 B] denotes the n nd matrix composed of B,...,A n 1 B. Moreover under H1) the random variable X x t has a C -density, for any t >, x R n. This result can be extended to the case when the Lévy process Z t ) has a non-degenerate Gaussian component. Indeed in such case it is enough to remark that the convolution of two Borel measures µ 1 and µ 2 has a density if either µ 1 or µ 2 has a density see [21, 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 theorem is the following one. Theorem 1.1. Assume H1). 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 the law of the Ornstein-Uhlenbeck process at time t i.e., the law of X x t ) is absolutely continuous, for any t >, x R n. It also turns out see Proposition 2.1) that under the assumptions of the theorem the Ornstein-Uhlenbeck process Xt x ) is strong Feller. An interesting example of an Ornstein-Uhlenbeck process with degenerate noise satisfying H1) is a solution of the equation Xt 1 = Z t, X 1 = x1, Xt 2 = x 1 t + t Z s ds + x 2, t, x = x 1,x 2 ) R 2. It is a generalization of a famous example due to Kolmogorov, in which Z t ) was a real Wiener process d = 1, n = 2). In [1] Kolmogorov showed that the law of the random variable Xt 1,X2 t ) is absolutely continuous with respect to the Lebesgue measure, for any t >, x R 2, and, in fact, its density is a C -function on R 2. This Gaussian example has been also considered by Hörmander in [8]. In Theorem 3.2 we generalize the Kolmogorov result to the case when Z t ) is a Lévy process of α-stable type, α,2). Although there are similar results in the literature, our proof is based on a new idea. There are a number of papers dealing with the absolute continuity of laws of degenerate diffusion processes with jumps see [2], [13], [11], and more recent: [15] and the internet article [12]). They apply appropriate extensions of Malliavin calculus for jump processes assuming also the well-known Hörmander condition on commutators which becomes H1) for Ornstein-Uhlenbeck processes). In [2] it is assumed that the Lévy measure of Z t ) has a sufficiently smooth density. In [13], [11], [15], and [12] only 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 well as control theoretic arguments. 2

3 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 + t e t s)a BdZ s = e ta x + Y t, t, x R n, 2.1) where the stochastic convolution Y t can be defined as a limit in probability of Riemann sums. The law µ x t of X x t has the characteristic function Fourier transform) ˆµ x t, ˆµ x t h) = x,h ˆµ ei eta 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 ), 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 any R k, k N, respectively. Moreover B denotes the adjoint 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 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 Zt ) is a Lévy pure jump process. The processes W t) and Zt ) are independent. Let P t ) be the transition semigroup determined by Xt x ), i.e., P t fx) = E[fX x t )], t, x Rn, f B b R n ), 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 ). Applying a result due to Hawkes see [9]) we show now that the strong Feller property for P t ) is equivalent to the existence of a density for the law of X x t, for any t >, x Rn. This result holds for any Ornstein-Uhlenbeck process defined in 1.1) without requiring H1)). For related results in infinite dimensions, see [2] and [18]. 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 X x t is absolutely continuous with respect to the Lebesgue measure. 3

4 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 [9, Lemma 2.1], we know that the Markov operator T t, T t gx) = R n gx + z)e ta 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. Remark 2.2. If the Ornstein-Uhlenbeck process X x t ) has a density for any x Rn, t >, i.e., P t fx) = R n fe ta x + y)g t y)dy), then P t f is uniformly continuous on R n, for any f L R n ) and t >. This follows easily by approximating in L 1 R n ) the function g t with a sequence of continuous functions having compact support. The proof of Theorem 1.1 requires the following lemma of independent interest. Lemma 2.3. Assume the controllability condition H1). 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, are onto. l s1,...,s m y 1,...,y m ) = Proof. The proof is divided into two parts. I Part. We define T >. m e sja By j, 2.5) Let λ j ) be the distinct complex eigenvalues of A, j = 1,...,k. 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 [14]) we know, in particular, that there exists T > depending on n and on the coefficients a 1,...,a n ) such that any non-trivial 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 j=1 k n 1 c rj e λjt t r 2.6) j=1 r= 4

5 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 = t 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, [25, 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) 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. Proof of Theorem 1.1. We will use Lemma 2.3 and adapt the method of the proof of [21, Theorem 27.7], based on [23] and [6]. Let T > be as in Lemma 2.3. 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 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, ν N = ν I {1/N x r} and Zt N = Z s, t, <s t, 1 N Z s r where ν is given in 2.3) and Z s = Z s Z s Z s = lim h Z s+h ). The process Z N t ) is a compound Poisson process and its Lévy measure is just ν N. By the hypotheses, ν N has a density, for any N N. Moreover since ν is infinite, we have that c N = ν N R d ) as N. 5

6 It is well known that Z N t ) and Z t Z N t ) are independent Lévy processes see for instance [19, Chapter 1]). It follows, in particular, that the random variables Y N t = t e t s)a BdZ N s and Y t Y N t = t e t s)a BdZ Z N ) s are independent, 2.1) 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 Yt N. Since µ = µ N β N where β N is the law of Y t Yt N ), using a standard property of convolution of measures see [21, Lemma 27.1]) we deduce that, for the singular parts of the measures, µ s R n ) µ N ) s R n ), for any N N. 2.11) Now we compute µ N which coincides with the law of t esa BdZs N. First, note that the law of Zt N is given by e c Nt δ + e c Nt 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} + ) 1 {ξ ξ k t<ξ ξ k+1 } e ξ1a BU e ξ ξ k )A BU k. k 1 Moreover, since the events {ξ ξ k t < ξ ξ k+1 } are all disjoint, k N, we get, for any f B b R n ), R N = Ef k 1 = k 1 Ef 1 {ξ ξ k t<ξ ξ k+1 } EfYt N ) = e cnt f) + R N, where 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 Nt 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 Nt t k+1) dt 1...dt k+1 fy)µ t1,...,t k dy), k 1 t t k t t t k+1 R n where µ t1,...,t k is the probability measure on R n image of the product measure ν N... ν N k-times) under the linear transformation J t1,...,t k independent of N) acting from R dk into R n, J t1,...,t k y 1,...,y k ) = e t 1A By e t t k )A By k, 6

7 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 2.3, we obtain that, for any k n+1, t i >, i = 1,...,k, the linear transformation J t1,...,t k is onto. Therefore it follows easily that 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 + µ2 N, where n µ 1 N = e cnt δ + c N ) k+1 e C Nt 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 Nt t k+1) g t1,...,t k y)dt 1...dt k+1. k>n t t k <t<t t k+1 Therefore = e c Nt + n k=1 µ N ) s R n ) µ 1 NR n ) t t k <t<t t k+1 c N ) k+1 e C Nt t k+1) dt 1... dt k+1, as N, since c N by hypothesis. Using 2.11), we immediately get that µ s =. This gives the assertion. The proof is complete. 3 Regularity of densities for α stable noise Here, we will obtain C -regularity of the law at time t of the Ornstein-Uhlenbeck process 1.1) by looking at its characteristic function. We will consider the following assumptions. Hypothesis 3.1. There exists C > and α, 2) such that, for sufficiently small r >, the following estimates hold: z 2 νdz) Cr 2 α, z r z r z,h 2 νdz) C h 2 z r z 2 νdz), h R d. 3.1) Such hypotheses are a generalization of those introduced in [16] when d = 1, in order to ensure that the law of Z t has a C -density for any t > see [12]). Clearly, if Z t ) is a d-dimensional α-stable process i.e., ψh) = h α, for h R d, α,2)) then 3.1) holds. 7

8 Theorem 3.2. Assume H1) and Hypothesis 3.1. Then, at any time t >, x R n, the Ornstein-Uhlenbeck process X x t ) has a C -density with all bounded derivatives. Moreover, and one has, for any t >, x R n, P t fx) = 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 t h,x exp R n R n t ) ) ψb e sa h)ds dh dy ) ) ψb e sa h)ds dh dz, f B b R n ). 3.2) Proof. 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 the characteristic function of µ t is given by exp t ) ψb e sa y)ds, y R n. We claim that there exists c t > such that, for any y R n with sufficiently large y, exp t ψb e sa y)ds) e c t y α. 3.3) This will prove the theorem. It is not restrictive to assume that Q = and a = in 2.3) i.e., Z t ) has no Gaussian component). For any y R n, we have t ) t exp ψb e sa y)ds = exp ds 1 cos B e sa y,z ) ) ) νdz). R d Set M t = sup B e sa h. Using the inequality 1 cosu) c 1 u 2, if u π, s [,t], h 1,h R n together with the assumptions, we find for sufficiently large y, t ds 1 cos B e sa y,z ) ) t νdz) c 1 ds B e sa y,z 2 νdz) R d z π y M t c 1 C t B e sa y 2 ds z 2 νdz) c 1 C 2 z π y M t π y M t ) 2 α t B e sa y 2 ds. The controllability condition is equivalent to the existence of C t > such that, for any y R n, t B e sa y 2 ds C t y 2 see [25]). This implies that t ds 1 cos B e sa y,z ) ) νdz) c t y α, R d for all y R n with sufficiently large y. The assertion 3.3) is proved. Finally, note that by the inversion theorem of the Fourier transform, p t y) = 1 t ) 2π) n e i y,h exp ψb e sa h)ds dh, y R n, 3.4) R n is the density of µ t. Differentiating under the integral sign, we get easily the assertion. The proof is complete. 8

9 Remark 3.3. It follows from Theorem 3.2 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. References [1] 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 ) [2] J. M. BISMUT, Calcul des variations stochastique et processus de sauts French) [Stochastic calculus of variations and jump processes], Z. Wahrsch. Verw. Gebiete ) [3] R. CONT and P. TANKOV, Financial modelling with jump processes Chapman & Hall/CRC, Boca Raton, FL, 24). [4] G. DA PRATO and J. ZABCZYK, Stochastic Equations in Infinite Dimensions Cambridge University Press, 1992). [5] G. DA PRATO and J. ZABCZYK, Second Order Partial Differential Equations in Hilbert Spaces, London Mathematical Society Lecture Note Series 293 Cambridge University Press, 22). [6] M. FISZ and V. S. VARADARAJAN, A condition for absolute continuity of infinitely divisible distribution functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete /1963) [7] P. GARBACZEWSKI and R. OLKIEWICZ, Ornstein-Uhlenbeck-Cauchy process, Journal of Mathematical Physics 41 2), [8] L. HÖRMANDER, Hypoelliptic second order differential operators, Acta Math ) [9] J. HAWKES, Potential theory of Lévy processes, Proc. London Math. Soc. 3) ) [1] A. N. KOLMOGOROV, Zufällige Bewegungen, Ann. of Math. 2) ) [11] T. KOMATSU and A. TAKEUCHI, On the smoothness of PDF of solutions to SDE of jump type, Int. J. Differ. Equ. Appl. 2 21) [12] H. KUNITA, Malliavin calculus of canonical stochastic differential equations with jumps, preprint 21, 1pdf/ pdf. [13] A. NEGORO and M. TSUCHIYA, Stochastic processes and semigroups associated with degenerate Lévy generating operators, Stochastics Stochastics Rep ) [14] Z. NEHARI, On the zeros of solutions of n-th order linear differential equations, J. London Math. Soc ) [15] J. OH, The density for jump processes in canonical stochastic differential equation, Kanmgweon- Kyungki Math. Jour. 8 2) [16] S. OREY, On continuity properties of infinitely divisible distribution functions, Ann. Math. Statist ) [17] E. PRIOLA and J. ZABCZYK, Liouvile theorems for non-local operators, J. Funct. Anal ), [18] E. PRIOLA and J. ZABCZYK, Harmonic functions for generalized Mehler semigroups, Stochastic partial differential equations and applications-vii eds G. DA PRATO and L. TUBARO), Lect. Notes Pure Appl. Math. 245 Chapman & Hall/CRC, 26),

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