Approximating multivariate tempered stable processes

APPROXIMATING MULTIVARIATE TEMPERED STABLE PROCESSES BORIS BAEUMER AND MIHÁLY KOVÁCS Abstract. We give a simple method to approximate multidimensional exponentially tempered stable processes and show that the approximating process converges in the Skorokhod topology to the tempered process. The approximation is based on the generation of a random angle and a random variable with a lower dimensional Lévy measure. We then show that if an arbitrarily small normal random variable is added, the marginal densities converge uniformly at an almost linear rate. 1. Introduction Being able to approximate stable and tempered stable processes is important for investigation and simulation purposes. A stable or tempered stable process is a process whose increments are independent stable or tempered stable random vectors. An exponentially tempered stable random vector Xρ is obtained from a stable random vector X by exponentially cooling its jump size (or Lévy measure); the general class of tempered stable vectors was introduced by [12]. Tempered stable laws are used in physics as a model for turbulent velocity fluctuations [8, 11], as well as in finance [3, 2] and hydrology [10] as a model of transient anomalous diffusion [1]. As the random variables are infinitely divisible, they can be approximated using LePage’s method in splitting up the integral in their Lévy Khintchine representation into a compound Poissonian part (with tempered Pareto jumps) and an approximately normal part. In case of a stable random vector this involves adding a random number of random vectors comprised of one-dimensional Pareto jumps multiplied by a random direction drawn according to the mixing measure M . In Zhang et al. [13] the authors compared this approach with just drawing a random direction multiplied by a one-dimensional skewed stable variable which can also easily be generated [4]. As they are in the same domain of attraction of an operator stable [9], both approaches work well. However, in the case of tempered stable vectors, they are in the domain of attraction of the multivariate normal and hence another argument is needed. In this paper we explore approximations to a stationary stochastic process with independent increments for which we are given a coordinate transform T : Ωθ × Ωr → Rd for some measure spaces Ωθ and Ωr which decomposes the Lévy measure of the stochastic process into lower dimensional Lévy measures; i.e., assume that we have a probability measure M on Ωθ and for each θ ∈ Ωθ there is a measure φθ Date: May 2, 2011. 2010 Mathematics Subject Classification. Primary: 60G51; 60G52. Key words and phrases. Tempered stable processes, tempered stable vectors, infinitely divisible law, stable distribution. 1 2 B. BAEUMER AND M. KOVÁCS on Ωr such that (1.1) Z IA (x)φ(dx) = Z IA (T (θ, r))φ(dT (θ, r)) Ωθ ×Ωr = Z Ωθ Z IA (T (θ, r))φθ (dr)M (dθ) Ωr for any measurable set A and the induced degenerate measure on Rd via Z IA (T (θ, r))φθ (dr) (1.2) φ̃θ (A) := Ωr is also Lévy. Note that for most processes appearing in applications there is a canonical decomposition. We show that the processes obtained by using increments stemming from infinitely divisible distributions with Lévy measure τ φ̃Θ converge to the original stochastic process in the Skorokhod topology, where Θ is a random vector distributed according to M . In particular, processes obtained by using random increments of the form ΘSρ (Θ), where Θ is a random angle and Sρ (Θ) is a properly scaled onedimensional tempered stable, converge to tempered stable process. This provides an alternative method in any number of dimensions to the one-dimensional method to obtain tempered stable laws as random walk limits developed by Chakrabarty and Meerschaert [5]. We then investigate the rate of convergence in the exponentially tempered stable case with a single α [12] and show that the densities at a fixed time t converge in the L2 -multiplier norm a rate of C(1 + log2 n)/n. We further show that given a small random initial perturbation, the marginal densities of the processes converge uniformly in x and t at an almost linear rate. Furthermore, we also show that if the Lévy measure can be decomposed into finitely many tempered stable measures (e.g. of different α’s) then for each step randomly choosing which tempered α-stable to simulate also converges at an almost linear rate. 2. The general result Let X(t) be a stationary stochastic process in Rd with independent increments. Then X(t) has a unique Lévy-Khintchine representation E[exp(−ihk, X(t)i)] = exp(tψ(k)) and log-characteristic function 1 ψ(k) =ihk, vi − hk, Qki + 2   ihk, xi −ihk,xi φ(dx) e −1+ 1 + kxk2 Rd Z for some drift vector v, covariance matrix Q and Lévy measure φ satisfying Z kxk2 φ(dx) < ∞. 1 + kxk2 Definition 2.1. Assume a Lévy measure φ decomposes as in (1.1); i.e. φ(dx) = φθ (dr)M (dθ), and assume that φ̃θ , defined as in (1.2) is Lévy. We then call (φθ , M ) a Lévy decomposition of φ and we call φ̃θ the projected Lévy measure in the direction of θ. APPROXIMATING TEMPERED STABLE PROCESSES 3 Let Θ be a Ωθ -valued random variable with Pr{Θ ∈ A} = M (A). Let Xθ (τ ) be a random variable with characteristic function E[exp(−ihk, Xθ (τ )i] = exp(τ ψθ (k)) and  Z  1 ihk, xi ψθ (k) = ihk, vi − hk, Qki + e−ihk,xi − 1 + φ̃θ (dx). 2 1 + kxk2 Rd R Clearly, ψ(k) = Ωθ ψθ (k) M (dθ). Let Xθj (τ ) and Θj , j ∈ N, τ > 0, be random variables on the same probability space distributed as Xθ (τ ) and Θ, respectively, all independent, and define the approximate process t Sτ (t) = ⌊τ ⌋ X j XΘ j (τ ), j=1 where ⌊ τt ⌋ t τ. denotes the integer part of We are ready to state the first theorem. Theorem 2.2. Let X be a stationary stochastic process in Rd with independent increments and a Lévy measure with Lévy decomposition (φθ , M ). If supθ∈Ωθ |ψθ (k)| < ∞ for all k ∈ Rd , then Sτ → X in the Skorokhod topology as τ → 0+. Proof. By design Sτ has independent increments. According to [6, Corollary VII.4.43] all we have to show is that the characteristic function of Sτ (t) converges to the characteristic function exp(tψ(k)) of X(t) uniformly on compact intervals in t for all k ∈ Rd . Conditioning on Θ, the characteristic function of Sτ (t) is given by ⌊ τt ⌋ Z −ihk,Sτ (t)i . exp(τ ψθ (k))M (dθ) E[e ]= Ωθ Using the fact that an − bn = (a − b) that Pn−1 j=0 aj bn−j−1 and | exp(τ ψ(k))| ≤ 1, we see t | exp(tψ(k)) − E[e−ihk,Sτ (t)i ]| = | exp(tψ(k)) − exp(τ ψ(k))⌊ τ ⌋ t (2.1) + exp(τ ψ(k))⌊ τ ⌋ − E[e−ihk,Sτ (t)i ]| t ≤| exp(tψ(k)) − exp(τ ⌊ ⌋ψ(k))| τZ t + ⌊ ⌋| exp(τ ψ(k)) − exp(τ ψθ (k))M (dθ)|. τ Ωθ Using Taylor expansion the last term may be bounded as Z t exp(τ ψθ (k))M (dθ)| ⌊ ⌋| exp(τ ψ(k)) − τ Ωθ Z X ∞ ∞ (τ ψθ (k))j t X (τ ψ(k))j =⌊ ⌋| − M (dθ)| τ j=2 j! j! Ωθ j=2 (2.2) t ≤τ 2 ⌊ ⌋|ψ(k)|2 exp(τ |ψ(k)|) τ Z t |ψθ (k)|2 exp(τ |ψθ (k)|)M (dθ). + τ 2⌊ ⌋ τ Ωθ 4 B. BAEUMER AND M. KOVÁCS It follows from our assumption on ψθ that the integral is bounded for fixed k ∈ Rd and hence (2.2) converges to zero uniformly in t on compact sets. Since the first term in (2.1) converges to zero uniformly on compacta as well, the proof is complete.  Example 2.3. Let X(t) be an operator stable process. Using the Jurek coordinate system [7] the log-characteristic function can be written as  Z Z ∞ E ihk, rE θi c e−ihk,r θi − 1 + dr M (dθ) ψ(k) = E θk2 2 1 + kr r kθk=1 0 for some scaling matrix E with eigenvalues whose real part is larger than 1/2. By Theorem 2.2 this process can be approximated by generating steps via randomly (according to M ) choosing a direction Θ and generating a random variable with log-characteristic function  Z ∞ ihk, rE Θi c −ihk,r E Θi τ ψΘ (k) = τ e −1+ dr. E Θk2 2 1 + kr r 0 Example 2.4. Let X(t) be a tempered operator stable process with uniform scaling; i.e. its log-characteristic function can be written as Z Z ∞  e−rρ(θ) e−ihk,θir − 1 + ihk, θir ψ(k) = dr M (dθ) rα+1 kθk=1 0 Z α =c (hik, θi + ρ(θ)) − ρ(θ)α − αhik, θiρ(θ)α−1 M (dθ) kθk=1 for some 1 < α < 2, c > 0 and exponential taper ρ; i.e. a bounded measurable function ρ : S d−1 → R+ , where S d−1 = {θ ∈ Rd : kθk = 1}. By Theorem 2.2 this process can also be approximated by generating steps via randomly choosing a direction Θ (according to M ) and generating a one-dimensional tempered stable random variable YΘ (see [1]) with tempering variable ρ(Θ), time scale τ , and letting XΘ = ΘYΘ . 3. Rate of convergence for tempered stables In the case of most tempered stable processes we can go further and actually provide a rate at which the densities of the processes converge. We are going to show that the characteristic functions converge uniformly at a rate of o(log2 n/n) which translates into L2 multiplier convergence of the densities or uniform convergence if an arbitrarily small normal random variable is added to the process. Definition 3.1. Let ρ(θ) ≥ 0 be a bounded measurable function on S d−1 , 1 < α < 2 and a > 0. Let α Aθ : k 7→ a (ihk, θi + ρ(θ)) − aρ(θ)α − αaρ(θ)α−1 ihk, θi be the tempered fractional derivative symbol in the direction θ. If ρ(θ) = 0, then we call Aθ : k 7→ ahik, θiα the fractional derivative symbol in the direction of θ. We call ρ the taper and the extended real valued function  2−α essupkθk=1 ρ(θ) + ǫ FC : θ 7→ lim ǫ→0+ ρ(θ) + ǫ the normalised fractional content of the taper at θ. APPROXIMATING TEMPERED STABLE PROCESSES 5 For a probability measure M on S d−1 , define the (tempered) fractional derivative symbol to be Z Aθ (k) M (dθ). A : k 7→ kθk=1 We say that a tempered fractional derivative symbol is full if Z λM = min hη, θi2 M (dθ) > 0. kηk=1 kθk=1 It is easy to show that λM is the smallest eigenvalue of the co-variance matrix of M viewed as a measure on Rd and is zero if and only if M is supported on a subspace. Our main theorem is the following: Theorem 3.2. Let A be a full, tempered fractional derivative symbol and assume FC ∈ L2 (S d−1 , M (dθ)). Then there exists C ≥ 0 such that !n Z t 1 + log2 n A (k) (3.1) − etA(k) ≤ C e n θ M (dθ) n kθk=1 for all k ∈ Rd , n ∈ N and t ≥ 0. Proof. See Section 4 below.  Let ψ ∈ L∞ (Rd ) and let S(Rd ) denote the space of Schwartz functions. We call ψ an Lp -Fourier multiplier, 1 ≤ p < ∞, if the map S(Rn ) ∋ f 7→ Tψ f := F −1 (ψF (f )) R extends to a bounded linear operator on Lp (Rd ), where F (f )(k) = Rd e−ikx f (x) dx denotes the Fourier transform of f . It is well-known that for a bounded Borel R measure µ on Rd , its Fourier transform µ̂(k) = Rd e−ikx µ(dx) is an Lp -Fouriermultiplier and Tµ̂ f = µ ∗ f. The Fourier multiplier p-norm of µ is defined as kµkMp (Rd ) := sup kf kLp (Rd ) =1 kµ ∗ f kLp(Rd ) = kTµ̂ kB(Lp (Rd )) , where ∗ denotes convolution and k·kB(Lp (Rd )) denotes the operator norm on Lp (Rd ). Let µt and νt be probability measures with Fourier transforms Z etAθ (k) M (dθ) µ̂t (k) = kθk=1 and and let µn∗ R ν̂t (k) = et kθk=1 Aθ (k) M(dθ) denote the n-th convolution power of a measure µ. Corollary 3.3. Let A be a full, tempered fractional derivative symbol and assume FC ∈ L2 (S d−1 ). Then for all ǫ > 0 there exists C ≥ 0 such that kµn∗ t/n − νt kM2 (Rd ) ≤ C 1 + log2 n n for all n ∈ N and t ≥ 0. Proof. Since for a bounded Borel measure µ we have kµkM2 (Rd ) = supk∈Rd |µ̂(k)|, the statement follows from Theorem 3.2.  6 B. BAEUMER AND M. KOVÁCS The next corollary translates L2 multiplier convergence into uniform convergence in the presence of a small perturbation δN (0, 1), where N (0, 1) is the multivariate standard normal random variable. Corollary 3.4. Let X be a tempered stable process with characteristic function E[e−ihk,X(t)i ] = exp (tA(k)) where A is a full, tempered fractional derivative symbol with FC ∈ L2 (S d−1 ). For j ∈ N, τ > 0 and kθk = 1, let Θj and Yj,θ (τ ) be random variables on the same probability space, all independent, with Θj distributed as Pr{Θj ∈ Ω} = M (Ω) and the distribution of Yj,θ satisfying E[e−ihk,Yj,θ (τ )i ] = exp (τ Aθ (k)) . Then, for all δ > 0, there exists C > 0 such that the marginal densities of the P⌊t/τ ⌋ approximate process δN (0, 1) + j=1 Θj Yj,Θj (τ ) converge uniformly in x ∈ Rd and t ≥ 0 at a rate of Cτ (1 + log2 (1/τ )) to the marginal densities of δN (0, 1)+ X(t) as τ → 0+. Proof. See Section 4  The next theorem is important when it is used in conjunction with Corollary 3.3, as it allows the mixing of operators with different α’s or tempered and untempered operators. Theorem 3.5. Let Aj ∈ C(Rd ), j = 1, . . . , m be sectorial; i.e., assume that there is c > 0 such that Re (Aj (k)) ≤ −c|Aj (k)| for k ∈ Rd and all j = 1, . . . , m and let P µt and νt be probability measures with Fourier transforms µ̂t (k) = λj etAj (k) and P P ν̂t (k) = et λj Aj (k) . Then, for each collection of 0 < λj < 1 with λj = 1, there exists C > 0 such that 1 + log2 n kµn∗ t/n − νt kM2 (Rd ) ≤ C n for all n ∈ N and t ≥ 0. Proof. As in the proof of Corollary 3.3 we use the fact that X n P t n Aj − et λj Aj − ν k d = (3.2) kµn∗ λ e t j M2 (R ) t/n L∞ (Rd ) . Without loss of generality assume 0 < λ1 ≤ λj , which implies that λ1 ≤ 1/m. We divide the proof into 2 cases. Case 1. Assume that k ∈ Rd is such that X t λj | Aj (k)| ≤ 2m log n/(cλ1 n). n Then nt |Aj (k)| ≤ 2m log n/(λ21 cn) and by the binomial formula n  X X P P t t t λj e n Aj (k) − e λj n Aj (k) λj e n Aj (k) − et λj Aj (k) ≤n 2 ≤n + X X λj t t Aj (k) e n |Aj (k)| n 2 P t t λj Aj (k) e λj n |Aj (k)| n ≤nC(log n/n)2 = C log2 n/n. ! APPROXIMATING TEMPERED STABLE PROCESSES 7 Case 2. Assume that k ∈ Rd is such that X t λj | Aj (k)| ≥ 2m log n/(cλ1 n). n Then there exists j1 such that λj1 | nt Aj1 (k)| ≥ 2 log n/(cλ1 n) and hence X t t λj e n Aj (k) ≤λj1 e−c| n Aj1 (k)| + (1 − λj1 ) ≤λj1 e−2c log n/cλ1 n + (1 − λj1 ) =λj1 e−2 log n/λ1 n + (1 − λj1 ). It can be seen by differentiating with respect to x that if 0 ≤ x ≤ ln 2, then λe−x + (1 − λ) ≤ e−λx/2 and hence, as λ1 ≤ λj1 , λj1 e−2 log n/λn + (1 − λj1 ) ≤ e− log n/n . Hence X t λj e n Aj (k) n − et P λj Aj (k) ≤e− log n + e−2m log n/λ1 ≤C/n. Thus, by cases 1 and 2 above, there is C such that X n P t λj e n Aj − et λj Aj ∞ d ≤ C(1 + log2 n)/n, n ∈ N, t ≥ 0, L (R ) which finishes the proof in view of (3.2).  t The next corollary allows us to approximate each e n Aj with its polar approximation. R Corollary 3.6. Let Aj = kθk=1 Aj,θ Mj (dθ), j = 1, . . . , m, be tempered fractional derivative operators each satisfying the conditions of Theorem P R 3.2 and let µt and νt be probability measures with Fourier transforms µ̂t (k) = λj kθk=1 etAj,θ (k) Mj (dθ) P P and ν̂t (k) = et λj Aj (k) . Then for each collection of 0 < λj < 1 with λj = 1 there exists C > 0 such that kµn∗ t/n − νt kM2 (Rd ) ≤ C 1 + log2 n n for all n ∈ N and t ≥ 0. Proof. Straightforward extension, combining the proofs of Theorems 3.2 and 3.5 using the cases for which X Z t 2m λj log n/c̃n Aj,θ (k) Mj (dθ) ≤ n λ1 kθk=1 or not, with c̃ = cS c3 where λ1 , cS and c3 are the smallest of the λi , the constants in Property (1) of Proposition 4.1 and the constants c from Proposition 4.5 respectively.  We finish this section with an example highlighting the intended usage of Corollaries 3.3, 3.4 and 3.6. 8 B. BAEUMER AND M. KOVÁCS Example 3.7. Consider a process on R2 with log-characteristic R 2π 1 2π 0 Aθ (k) dθ with  1.6  ~  − sin1.6 (θ) − (1.6) sin0.6 (θ)hik, θi θi + sin(θ)  hik, ~ Aθ (k) = hik, ~ θi1.2    ~ 1.8 (2π − θ)hik, θi function ψ(k) = 0≤θ≤π π < θ < 3π/2 3π/2 < θ < 2π, where ~ θ = (cos θ, sin θ). In order to apply 3.6, let λ1 = 1/2, λ2 = λ3 = 1/4, M1 (dx) = π1 I[0,π] (x) dx, M2 (dx) = π2 I[π,3π/2] (x) dx, M3 (dx) = π82 I[3π/2,2π] (x)(2π − x) dx. So we can rewrite ψ(k) as Z 2π Z 2π Z 2π π hik, ~θi1.8 M3 (dθ). ψ(k) = λ1 Aθ (k)M1 (dθ) + λ2 Aθ (k)M2 (dθ) + λ3 4 0 0 0 By the developed theory the process can be faithfully approximated by generating increments with step-size τ , where each increment is generated by first generating a uniformly distributed q random variable Θ̃ over [0, 2π] and letting Θ = Θ̃ for Θ̃ < 3π/2 and Θ = 2π − π 2 − Θ̃π/2 otherwise and then generating a one-dimensional random variable XΘ with characteristic function E[exp(−ikXΘ )] = exp(τ ÃΘ (k)), where  1.6 1.6 0.6  (ik + sin(θ)) − sin (θ) − (1.6) sin (θ)ik Ãθ (k) = (ik)1.2  π 1.8 4 (ik) 0≤θ≤π π < θ < 3π/2 3π/2 < θ < 2π. The one-dimensional variables can be generated (or approximated) using the meth~ In Figure 1 we plot a sample ods in [1, 4]. The increment is then given by XΘ Θ. path over the time intervals t < 1, t < 100 and t < 1000, generated with τ = 1/1000. 4. Proof of Theorem 3.2 and Corollary 3.4 Proposition 4.1. Let A be a tempered fractional derivative symbol. Let u = hk, θi, where k ∈ Rd and θ ∈ S d−1 . Then (1) There exists a constant cS > 0, such that Re(Aθ (k)) ≤ −cS |Aθ (k)| for all k and θ; i.e., Aθ and A are sectorial. (2) There exist constants cL , cU > 0 such that     u2 u2 α α ≤ |Aθ (k)| ≤ cU min , |u| , |u| cL min ρ(θ)2−α ρ(θ)2−α for all k and θ with ρ(θ) > 0; if ρ(θ) = 0, cL |u|α ≤ |Aθ (k)| ≤ cU |u|α . Proof. Note that without loss of generality we can set ρ(θ) = a = 1 as the general case follows by replacing u (or equivalently k) with u/ρ(θ) and multiplying Aθ by aρ(θ)α . As α (iu + 1) − 1 − αiu = (iu)α + o(|u|α ) APPROXIMATING TEMPERED STABLE PROCESSES t=1 t=1000 t=100 0.6 9 20 50 y 10 0.2 y y 0.4 0 0 0 −50 −0.2 −0.4 −100 −10 −0.4−0.2 x 0 0 5 10 15 x −40−20 0 20 40 x Figure 1. A simulated sample path of Example 3.7, generated with τ = 0.001. Note that relatively large upwards jumps are present in the small and medium scale but virtually disappear in the larger scale on the right as the probability of jumps larger than x is less than exp(−x sin(θ)) for 0 < θ < π. as u → ∞ and by the Taylor expansion, (iu + 1)α − 1 − αiu = − α(α − 1) 2 u + o(u2 ) 2 as u → 0, the inequalities follow once we establish that | Re(Aθ )| and | Im(Aθ )| with ρ(θ) = 1 are continuous, increasing functions of |u|. First we show that Re ((iu + 1)α − 1) is decreasing for u > 0; it clearly is continuous and differentiable. To that end let φ = arctan u. Then |iu + 1| = sec φ and (4.1) f (φ) = Re ((i tan φ + 1)α − 1) = cos αφ secα φ − 1. Its derivative is given by (4.2) d f (φ) = − α sin(αφ) secα (φ) + cos(αφ)α secα−1 (φ) sec(φ) tan(φ) dφ = − α secα+1 (φ) (sin(αφ) cos(φ) − cos(αφ) sin(φ)) = − α secα+1 (φ) sin(αφ − φ) < 0 for φ > 0 and positive for φ < 0. Hence f is decreasing for positive φ or u and increasing for negative u, and since f (0) = 0 we have that |f | is increasing for increasing |φ| or |u|. Similarly we show that Im ((iu + 1)α − αiu) is increasing. Again let φ = arctan u and f (φ) = Im ((i tan φ + 1)α − αi tan φ) = sin αφ secα φ − α tan φ. 10 B. BAEUMER AND M. KOVÁCS Then d f (φ) =α cos(αφ) secα (φ) + sin(αφ)α secα−1 (φ) sec(φ) tan(φ) − α sec2 (φ) dφ =α secα+1 (φ) (cos(αφ) cos(φ) + sin(αφ) sin(φ)) − α sec2 φ (4.3) =α secα+1 (φ) cos(αφ − φ) − α sec2 (φ)
= sec2 (φ) α secα−1 (φ) cos ((α − 1)φ) − α . d f (φ) = 0. The last factor in(4.3) is similar to (4.1) and At φ = 0 we have that dφ its derivative is computed similarly to (4.2) and is given by −α(α − 1) secα (φ) sin((α − 2)φ) > 0 d for φ > 0 and negative for φ < 0. Hence dφ f (φ) > 0 for all φ6= 0 and since f (0) = 0 we have that |f | is increasing for increasing |u|.  Lemma 4.2. Let FC ∈ L1 (S d−1 , M (dθ)) and ǫ > 0. Then there exists a constant c > 0 such that Z |Aθ (k)| M (dθ) ≥ ǫ kθk=1 implies that if ρmax := essupkθk=1 ρ2−α (θ) > 0,   kFC k1 , kkkα > cǫ, min kkk2 ρmax otherwise kkkα > cǫ. Proof. Assume ρmax > 0. As FC ∈ L1 (S d−1 ), the set for which ρ(θ) = 0 is a null set. By Proposition 4.1,   Z Z hk, θi2 α M (dθ) , |hk, θi| |Aθ (k)| M (dθ) ≤ cU min ρ(θ)2−α kθk=1 kθk=1 (Z ) Z hk, θi2 α ≤cU min |hk, θi| M (dθ) M (dθ), 2−α kθk=1 ρ(θ) kθk=1   α 2 kFC k1 , kkk . ≤cU min kkk ρmax n o C k1 α Hence min kkk2 kF ≥ ǫ/cU . ρmax , kkk In case of ρmax = 0, |Aθ (k)| = a|hk, θi|α . Similarly we then obtain kkkα ≥ ǫ/cU .  Lemma 4.3. There exists c > 0 such that if ρmax := essupkθk=1 ρ(θ)2−α > 0, then   hk, θi2 kkk2 , kkkα , |Aθ (k)| ≥ c min ρmax kkk2 otherwise |Aθ (k)| ≥ ckkkα−2 hk, θi2 . APPROXIMATING TEMPERED STABLE PROCESSES 11 2 Proof. If ρ(θ) = 0, then |Aθ (k)| = a|hk, θi|α ≥ akkkα hk,θi kkk2 . If ρ(θ) > 0, by Proposition 4.1 we have that   hk, θi2 α , |hk, θi| |Aθ (k)| ≥cL min ρ(θ)2−α ( ) α 2 |hk, θi| k α ≥cL min , kkk h , θi ρmax kkk   kkk2 k 2 α k 2 ≥cL min h , θi , kkk h , θi ρmax kkk kkk   hk, θi2 kkk2 , kkkα . =cL min ρmax kkk2  Lemma 4.4. Let M be a probability measure on [0, 1]. Then for µ ≥ 0, Z 1 min(µ,1) R 1 uM(du) 0 e−µu M (du) ≤ e− 2 . 0 Proof. Clearly, for u ∈ [0, 1], e−µu ≤ 1 − (1 − e−µ )u. Hence Z 1 Z 1 R1 −µ −µu −µ e M (du) ≤ 1 − (1 − e ) uM (du) ≤ e−(1−e ) 0 uM(du) . 0 0 The assertion follows from the fact that for 0 ≤ µ ≤ 1, 1 − e−µ ≥ µ/2 and for µ > 1, 1 − e−µ ≥ 1/2.  Proposition 4.5. Let A be a full, tempered fractional derivative symbol. Assume FC ∈ L1 (S d−1 , M (dθ)). Then there exist 0 < c ≤ 1 and d > 0 such that for all ǫ, t > 0 and n ∈ N, Z t |Aθ (k)| M (dθ) ≥ ǫ n kθk=1 implies that Z t e− n |Aθ (k)| M (dθ) ≤ e− min{cǫ,d} . kθk=1 Proof. Combining Lemma 4.2 and Lemma 4.3, there exists cL , cU > 0 such that   kkk2 hk, θi2 |Aθ (k)| ≥cL min , kkkα ρmax kkk2   n ǫcL 1 hk, θi2 (4.4) ≥ min ,1 t cU kFC k1 kkk2 n ǫcL hk, θi2 = . t cU kFC k1 kkk2 Define a measure Mk on [0, 1] via Mk (Ω) = M {θ : hk, θi2 /kkk2 ∈ Ω}
12 B. BAEUMER AND M. KOVÁCS for each measurable Ω ⊂ [0, 1]. Then by Lemma 4.4 and inequality (4.4) there exists 0 < c ≤ 1 such that, Z Z ǫcL hk,θi2 t −t n |Aθ (k)| −n M (dθ) ≤ e n t cU kFC k1 kkk2 M (dθ) e kθk=1 1 −c kθk=1 ≤ Z e ǫcL U kFC k1 u Mk (du) 0 ≤e −  min  ǫcL ,1 cU kFC k1 2 R1 0 u Mk (du) ≤e −  min  ǫcL ,1 cU kFC k1 λM 2 ≤e− min{cǫ,d} .  Lemma 4.6. Let A be a full, tempered fractional derivative symbol. Then there exists a constant c > 0 such that for all ǫ > 0, Z |Aθ (k)| M (dθ) ≤ ǫ kθk=1 implies that |Aθ (k)| < cǫFC (θ). Proof. By Lemma 4.3, if ρmax := essupkθk=1 ρ(θ)2−α > 0, then   hk, θi2 kkk2 , kkkα |Aθ (k)| ≥ cL min ρmax kkk2 and hence ǫ≥ Z |Aθ (k)| M (dθ) ≥ cL min kθk=1   kkk2 , kkkα λM . ρmax Therefore, either kkk2 ≤ ǫρmax /λM cL or kkkα ≤ ǫ/λM cL . If ρmax = 0 we clearly also have kkkα ≤ ǫ/λM cL . In case of kkk2 ≤ ǫρmax /λM cL , this implies that   cU hk, θi2 α , |hk, θi| ≤ FC (θ)ǫ; |Aθ (k)| ≤cU min 2−α ρ(θ) λM cL in case of kkkα ≤ ǫ/λM cL , this implies that   hk, θi2 cU α |Aθ (k)| ≤cU min , |hk, θi| ǫ. ≤ ρ(θ)2−α λM cL As FC (θ) ≥ 1, the lemma is proven.  Proposition 4.7. Let A be a full, tempered fractional derivative symbol and assume that FC ∈ L2 (S d−1 , M (dθ)). Then there exist a constant c > 0 such that for all 0 < ǫ ≤ 1 and all n, t > 0, Z t |Aθ (k)| M (dθ) ≤ ǫ n kθk=1 implies that Z kθk=1 t t e n Aθ (k) M (dθ) − e n A(k) ≤ cǫ2 . APPROXIMATING TEMPERED STABLE PROCESSES 13 Proof. Note that by Lemma 4.6 there exists a constant c = cU /cL λM such that t |Aθ (k)| ≤cǫFC (θ). n R Then, using the fact that A(k) = kθk=1 Aθ (k) M (dθ) and that M is a probability measure, (4.5) e t n A(k) − Z e n Aθ (k) M (dθ) = = Z Z = Z t kθk=1 kθk=1 1 0 s kθk=1    t t t t A(k) − Aθ (k) e(s n A(k)+(1−s) n Aθ (k)) ds M (dθ) n n  t t t A(k) − Aθ (k) e n (sA(k)+(1−s)Aθ (k)) n n s=1 s=0 2 t t t t A(k) − Aθ (k) e(s n A(k)+(1−s) n Aθ (k)) ds M (dθ) − s n n 0 2 Z Z 1  t t t t A(k) − Aθ (k) e(s n A(k)+(1−s) n Aθ (k)) ds M (dθ) = s n n kθk=1 0 Z 2 1 t t ≤ A(k) − Aθ (k) M (dθ) 2 kθk=1 n n Z 2 1 3 t ≤ Aθ (k) M (dθ) + ǫ2 2 kθk=1 n 2 2 2 Z 3 c ǫ 2 |FC (θ)| M (dθ) + ǫ2 . ≤ 2 kθk=1 2 Z 1   Proof of Theorem 3.2. We divide the proof into two cases. Let first Z t |Aθ (k)| M (dθ) > log n/c̃n, n kθk=1 where c̃ = cS c3 . Here cS denotes the constant in Property (1) of Proposition 4.1 and c3 ≤ 1 is the constant c from Proposition 4.5. Then, by Property (1) of Proposition 4.1, et R kθk=1 Aθ (k) M(dθ) ≤ e−ncS log n/c̃n = 1/n1/c3 . Furthermore, by Proposition 4.5, we also have that !n Z t A (k) ≤ e−cS n min{c3 log n/c̃n,d} . e n θ M (dθ) kθk=1 Hence, there is C > 0 such that Z kθk=1 e t n A(k) !n − etA(k) ≤ C/n. 14 B. BAEUMER AND M. KOVÁCS In case Z t |Aθ (k)| M (dθ) ≤ log n/c̃n, n kθk=1 first note that for |a|, |b| ≤ 1, |an − bn | = |a − b| n−1 X |aj bn−1−j | ≤ n|a − b|. j=0 By Proposition 4.7 there exist c > 0 such that (4.6) !n Z Z n  t t A (k) A(k) θ n n e M (dθ) ≤n − e kθk=1 t t e n Aθ (k) M (dθ) − e n A(k) kθk=1 log2 n . n Hence, combining the two cases, there exists a C > 0 such that (3.1) holds. ≤nc(log n/c̃n)2 = C  Proof of Corollary 3.4. Uniform convergence of the densities follows from the L1 1 ˆ ˆ convergence of the characteristic function since kf k∞ ≤ (2π) d kf k1 , where f : k 7→ R ihk,xi e f (x) dx. Hence we need to estimate  !⌊ τt ⌋  Z 2 δ  e− 2 k·k etA(·) − eτ Aθ (·) M (dθ) kθk=1 ≤ e − δ2 k·k2 + e  e  − δ2 k·k2 =I1 + I2 . tA(·) e −e τ ⌊ τt L1 (Rd ) τ ⌊ τt ⌋A(·) ⌋A(·) −  L1 (Rd ) Z kθk=1 !⌊ τt ⌋   eτ Aθ (·) M (dθ) L1 (Rd ) Using that Re A(k) ≤ 0, we obtain δ 2 I1 ≤ e− 2 k·k 1 − eτ A(·) L1 (Rd ) . Comparing real and imaginary parts we easily see that there exists c > 0 such that 1 − eτ A(k) ≤ cτ (1 + |A(k)|) and hence there exists C such that I1 ≤ Cτ . In order to estimate I2 , note that by Proposition 4.1 there exists cU such that max |Aθ (k)| ≤ cU kkkα kθk=1 for all k ∈ Rd . We divide the estimate into two parts. Firstly, consider t ≤ 1. Then using the same technique as in (4.5) and (4.6) we see that !⌊ τt ⌋ Z Z 1 t 2 τ ⌊ τt ⌋A(k) τ Aθ (k) e ≤ ⌊ ⌋τ 2 − |Aθ (k) − A(k)| M (dθ) e M (dθ) 2 τ kθk=1 kθk=1 ≤Cτ kkk2α , APPROXIMATING TEMPERED STABLE PROCESSES 15 where the last inequality follows from Proposition 4.1. Hence, for t ≤ 1, δ I2 ≤ Cτ k · k2α e− 2 (k·k 2 ) L1 (Rd ) For t ≥ 1, apply Theorem 3.2 with n = ⌊ τt ⌋ to obtain !⌊ τt ⌋ Z t eτ ⌊ τ ⌋A(k) − eτ Aθ (k) M (dθ) ≤C1 kθk=1 . 1 + log2 ⌊ τt ⌋ ⌊ τt ⌋ ≤Cτ (1 + log2 (1/τ )) and hence in this case δ I2 ≤ Cτ (1 + log2 (1/τ )) e− 2 k·k 2 L1 (Rd ) . Thus, the marginal densities converge independently of t at the prescribed rate.  References [1] B. Baeumer and M. M. Meerschaert, Tempered stable Lévy motion and transient superdiffusion, Journal of Computational and Applied Mathematics 233 (2010), no. 10, 2438 – 2448. [2] P. Carr, H. Geman, D. B. Madan, and M. Yor, Stochastic volatility for Lévy processes, Mathematical Finance 13 (2003), no. 3, 345–382. [3] P. Carr, H. Geman, D. B. Madan, and M. Yor, The fine structure of asset returns: An empirical investigation, Journal of Business, 2002, vol. 75, no. 2) 75 (2002), no. 2, 305–332. [4] J. M. Chambers, C. L. Mallows, and B. W. 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Baeumer, Tempered anomalous diffusion in heterogeneous systems, Geophysical Research Letters 35 (2008), L17403. [11] E. A. Novikov, Infinitely divisible distributions in turbulence, Phys. Rev. E 50 (1994), no. 5, R3303–R3305. [12] J. Rosiński, Tempering stable processes, Stochastic Processes and their Applications 117 (2007), 677–707. [13] Y. Zhang, D. A. Benson, M. M. Meerschaert, E. M. LaBolle, and H.-P. Scheffler, Random walk approximation of fractional-order multiscaling anomalous diffusion, Physical Review E (Statistical, Nonlinear, and Soft Matter Physics) 74 (2006), no. 2, 026706. Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin, New Zealand E-mail address: [email protected] Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin, New Zealand E-mail address: [email protected]