Markov Chain Approximation of Pure Jump Processes 3

Home , Hunt process, Jump process

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The proof of tightness is based on methods developed in [BC08, Lemma 2.1] and only works if the approximating Markov chains Xn are symmetric. In order to prove n the convergence of the finite-dimensional distributions of X n N to those of X, Mosco convergence of the corresponding symmetric Dirichlet forms{ is} use∈ d. This type of convergence is equivalent to strong convergence of the corresponding semigroups. It was first obtained in [Mos94] in the case when all the forms are defined on the same Hilbert space, and then it was generalized in [Kim06] (see also [CKK13] and [KS03]) to the case where the forms are defined on different spaces. We are interested in the convergence and approximation problems for non-symmetric pure jump processes. We will use two approaches: (i) via Dirichlet forms, and (ii) via semimartingale convergence results. The first approach (Section 2.1–2.3) follows the n roadmap laid out in [CKK13]: To obtain tightness of X n N we use semimartingale convergence results developed in [JS03]. More precisely, we{ first} ∈ ensure that the processes Xn, n N, are regular Markov chains (in particular, they are semimartingales), then we compute∈ their semimartingale characteristics, and finally we provide conditions for the n tightness of X n N in terms of the corresponding conductances. This is based on a result from [JS03]{ which} ∈ states that a sequence of semimartingales is tight if the corresponding characteristics are C-tight (i.e. tight and all accumulation points are processes with con- n tinuous paths). To get the convergence of the finite-dimensional distributions of X n N in the non-symmetric case we can still use Mosco convergence, but for non-symmetric{ } ∈ Dirichlet forms. Just as in the symmetric case, this type of convergence is equivalent to the strong convergence of the corresponding semigroups. It was first obtained in [Hin98] for forms defined on the same Hilbert space, and and then it was generalized in [T06¨ ] to forms living on different spaces. Our second approach (Section 3.1), is based on a result from [JS03] which provides general conditions under which a sequence of semimartingales converges weakly to a semimartingale. If the processes Xn, n N, are regular Markov chains and if the limit- ing process X is a so-called (pure jump)∈ homogeneous diffusion with jumps, we obtain conditions (in terms of conductances and characteristics of X) which imply the desired convergence. As an application, we can now answer question (ii) and provide conditions for the approximation of a given Markov process, both in the Dirichlet form set-up (Section 2.4) and the semimartingale setting (Section 3.2).

Notation. Most of our notation is standard or self-explanatory. Throughout this paper, Zd 1 Zd 1 Zd 1 we write n := n = n m : m for the d-dimensional lattice with grid size n . For p { ∈ }p Zd p Zd p 1 we use Ln as a shorthand for L ( n), the standard L -space on n. If p = 2, the ≥ k d 2 R scalar product is given by f,g Ln := a Zd f(a)g(a). We write Cc ( ) for the k-times h i ∈ n continuously diﬀerentiable, compactly supported functions, and CLip(Rd) is the space of P c Lipschitz continuous functions with compact support. Br(x) is the open ball with radius r > 0 and centre x, diag = (x, x): x Rd denotes the diagonal in Rd. Finally, the sum A + B of subsets A, B {Rd is deﬁned∈ as} A + B = a + b : a A, b B . ⊆ { ∈ ∈ } MARKOV CHAIN APPROXIMATION OF PURE JUMP PROCESSES 3

2. Convergence of Markov chains using Dirichlet forms n Our starting point is a sequence of continuous-time Markov chains Xt t 0 with state Zd { } ≥ space n and inﬁnitesimal generator n n f(a)= (f(b) f(a))C (a, b), f n , A − ∈DA b Zd X∈ n where the domain is given by

Zd R n Zd n := f : n : f(b) C (a, b) < for all a n . DA → | | ∞ ∈ b Zd n X∈ n o n n Zd Zd A sufficient condition for the existence of X is that the kernel C : n n [0, ) satisfies the following two properties × → ∞ (T1) n N, a Zd : Cn(a, a) = 0; ∀ ∈ ∀ ∈ n (T2) n N : sup Cn(a, b) < , ∀ ∈ a Zd ∞ ∈ n b Zd X∈ n n see e.g. [Nor98]; in this case, the chain Xt t 0 is regular, i.e. it has only finitely many jumps on finite time-intervals. If the chain{ } is≥ in state a Zd, it jumps to state b Zd n n ∈ ∈ with probability C (a, b)/ Zd C (a, c) after an exponential waiting time with param- c n n ∈ n eter Zd C (a, c). Moreover, Xt t 0 is conservative and defines a semimartingale. c n ≥ ∈ P { 1 } 2 Observe that (T1) implies Ln∞ Ln Ln n . Indeed, we have P ∪ ∪ ⊆DA f(b) Cn(a, b) f sup Cn(a, b), | | ≤k k∞a Zd b Zd ∈ n b Zd X∈ n X∈ n n n n 1 f(b) C (a, b) sup C (a, b) f(b) = f Ln sup C (a, b) | | ≤ a Zd | | k k a Zd b Zd ∈ n b Zd b Zd ∈ n b Zd X∈ n X∈ n X∈ n X∈ n and, using the Cauchy–Schwarz inequality,

f(b) Cn(a, b)= f(b) Cn(a, b) Cn(a, b) | | | | b Zd b Zd X∈ n X∈ n p p 1/2 1/2 f(b) 2Cn(a, b) Cn(a, b) ≤ | | Zd Zd b n b n X∈ X∈ n 2 f Ln sup C (a, b). ≤k k a Zd ∈ n b Zd X∈ n n We are interested in conditions which ensure the convergence of the family X n N as n . { } ∈ →∞ 2.1. Tightness. The proof of convergence relies on convergence criteria for semimartingales; our standard reference will be the monograph [JS03]. Let St t 0 bea d-dimensional { } ≥ d d semimartingale on the stochastic basis (Ω, , t t 0, P), and denote by h : R R a truncation function, i.e. a bounded and continuousF {F } ≥ function which such that h(→x) = x in a neighbourhood of the origin. Since a semimartingale has càdlàg (right-continuous, finite left limits) paths, we can write ∆St := St St , t > 0, and ∆S0 := S0, for the jumps of S and set − − S¯(h) := (∆S h(∆S )) and S(h) := S S¯(h) . t s − s t t − t s t X≤ 4 A.MIMICA,N.SANDRIC,´ AND R.L. SCHILLING

The process S(h)t t 0 is a special semimartingale, i.e. it admits a unique decomposition { } ≥ (2.1) S(h)t = S0 + M(h)t + B(h)t, where M(h)t t 0 is a local martingale and B(h)t t 0 is a predictable process of bounded variation{ on compact} ≥ time-intervals. { } ≥

d d Deﬁnition 2.1. Let St t 0 be a semimartingale and h : R R be a truncation function. The characteristics{ } ≥ of the semimartingale (relative to→ the truncation h) is a triplet (B, A, N) consisting of the bounded variation process B = B(h)t t 0 appearing in (2.1), the compensator N = N(ω,ds,dy) of the jump measure { } ≥

µ(ω,ds,dy) := δ(s,∆Ss(ω))(ds,dy)

s:∆Ss(ω)=0 X 6 of the semimartingale St t 0 and the quadratic co-variation process { } ≥ Aik = Si,c,Sk,c , i,k =1,...,d, t 0, t h t t i ≥ c of the continuous part St t 0 of the semimartingale. { } ≥ ik i k The modified characteristics is the triplet (B, A,˜ N) where A˜(h) := M(h) , M(h) 2 , t h t t iL i, k =1,...,d, with M(h)t t 0 being the local martingale appearing in (2.1). { } ≥ Using [JS03, Proposition II.2.17 and Theorem II.2.42] we can easily obtain the (mod- n ified) characteristics of Xt t 0 from the infinitesimal generator; as before, we write h : Rd Rd for the truncation{ } ≥ function: → t n n n n B (h)t = h(b)C (Xs ,Xs + b) ds, 0 Zd Z b n X∈ t ñ ik n n n A (h)t = hi(b)hk(b)C (Xs ,Xs + b) ds, 0 Zd Z b n X∈ n n n n N (ds,b)= C (Xs ,Xs + b) ds n n and, since Xt t 0 is purely discontinuous, A 0. { } ≥ ≡n N In order to show the tightness of the family Xt t 0, n , we need further conditions: { } ≥ ∈ (T3) ρ> 0 : lim sup sup Cn(a, a + b) < ; ∀ n a Zd ∞ →∞ n b >ρ ∈ |X| (T4) lim lim sup sup Cn(a, a + b) = 0; r n a Zd →∞ →∞ n b >r ∈ |X| n (T5) ρ> 0 i =1,...,d : lim sup sup biC (a, a + b) < ; ∃ ∀ n a Zd ∞ →∞ n b <ρ ∈ |X|

n (T6) ρ> 0 i, k =1,...,d : lim sup sup bibkC (a, a + b) < . ∃ ∀ n a Zd ∞ →∞ n b <ρ ∈ |X|

Theorem 2.2. Assume that (T1)–(T6) are satisﬁed. If the family of initial distributions Pn n n N (X0 ) is tight, then the family of Markov chains Xt t 0, n , is tight. ∈• { } ≥ ∈ Pn n Pn n Proof. We denote by the law of Xt t 0 such that (X0 ) n 1 is tight. Accord- { } ≥ n { N ∈• } ≥ ing to [JS03, Theorem VI.4.18], the family Xt t 0, n , will be tight, if for all T > 0 and all ε> 0, { } ≥ ∈ Pn n Zd (2.2) lim lim sup N ([0, T ], a n : a >r ) >ε =0, r n { ∈ | | } →∞ →∞ MARKOV CHAIN APPROXIMATION OF PURE JUMP PROCESSES 5

n ˜n t n and the families of processes B (h)t t 0, A (h)t t 0 and 0 a Zd g(a)N (ds,a) , { } ≥ { } ≥ ∈ n t 0 d ≥ n N, are tight for every bounded function g : R R whichnR P vanishes in a neighbour-o hood∈ of the origin. → Clearly, (2.2) is a direct consequence of (T4), and it is enough to show the tightness n ˜n t n N of the families B (h)t t 0, A (h)t t 0 and 0 a Zd g(a)N (ds,a) , n . { } ≥ { } ≥ ∈ n t 0 ∈ n ≥N According to [JS03, Theorem VI.3.21] tightnessnR P of B (h)t t 0, n o , follows if we can show that { } ≥ ∈ (i) for every T > 0 there exists some r > 0 such that

n n lim P sup B (h)t >r = 0; n t [0,T ] | | →∞ ∈ (ii) for every T > 0 and r> 0 there exists some τ > 0 such that

n n n lim P sup B (h)u B (h)v >r =0. n u,v [0,T ], u v τ | − | →∞ ∈ | − |≤ d Fix T > 0; without loss of generality we may assume that h(x)= x for all x Bρ(0) R where ρ> 0 is given in (T5). For i =1,...,d, we have ∈ ⊂ t sup Bn(h)i = sup h (b)Cn(Xn,Xn + b) | t| i s s t [0,T ] t [0,T ] 0 Zd ∈ ∈ Z b n X∈ t t n n n n n n sup biC (Xs ,Xs + b) ds + sup hi(b) C (Xs ,Xs + b) ds ≤ t [0,T ] 0 t [0,T ] 0 | | ∈ Z b <ρ ∈ Z b ρ |X| |X|≥

n n T sup biC (a, a + b) + T h sup C (a, a + b) ≤ Zd k k∞ Zd a n b <ρ a n b ρ ∈ |X| ∈ |X|≥

and for all 0 u v T such that u v τ we get ≤ ≤ ≤ | − | ≤ v Bn(h)i Bn(h)i = h (b)Cn(Xn,Xn + b) ds u − v i s s u Zd Z b n X∈

n n τ sup biC (a, a + b) + τ h sup C (a, a + b). ≤ Zd k k∞ Zd a n b <ρ a n b ρ ∈ |X| ∈ |X|≥

The assertion now follows from ( T3) and (T5). Since the proof of tightness of the other two families is very similar, we omit the details; note that these proofs require the (not yet used) conditions (T4) and (T6).

2.2. On a Class of Jump Processes and their Dirichlet Forms. In order to iden- n N tify the (weak) limit of the family Xt t 0, n , we will use Dirichlet forms. We restrict ourselves to a class of pure jump{ } processes≥ ∈ whose inﬁnitesimal generators have the following form

f(x) := lim (f(y) f(x))k(x, y) dy, f L2(Rd, dx), A ε 0 c − ∈DA ⊆ ↓ ZBε(x) where k : Rd Rd diag [0, ) is a Borel measurable function defined off the diagonal diag := (x, x×): x\ Rd→. Observe∞ that many interesting processes fall into this class. For instance,{ (non-)symmetric∈ } Lévy processes generated by a Lévy measure of the form ν(dy)= ν(y)dy (here, k(x, y)= ν(y x)). But this class goes beyond Lévy processes; for − 6 A.MIMICA,N.SANDRIC,´ AND R.L. SCHILLING

α(x) d d example it contains a process generated by k(x, y)= x y − − , where α : R (0, 2), the so-called stable-like processes (in the sense of R.F.| − Bass| [Bas88]), cf. Example→ 2.18. 1 1 Denote by ks(x, y) := 2 (k(x, y) + k(y, x)) and ka(x, y) := 2 (k(x, y) k(y, x)) the symmetric and antisymmetric parts of k(x, y), respectively. It is well known,− cf. [FOT11, Example 1.2.4], that the assumption

2 1 Rd x (1 y x )ks(x, y) dy Lloc( , dx) 7→ Rd ∧| − | ∈ Z ensures that k(x, y) deﬁnes a regular symmetric Dirichlet form ( , ) on L2(Rd, dx), where E F

(f,g) := (f(y) f(x))(g(y) g(x))k(x, y) dxdy, f,g ¯, E Rd Rd diag − − ∈ F Z × \ ¯ := f L2(Rd, dx): (f, f) < . F { ∈ E ∞} 1/2 The form domain is the 1 -closure of the Lipschitz continuous functions with compact Lip Rd F E 2 ¯ support Cc ( ) and α(f, f) := (f, f)+ α f L2 , α > 0, induces a norm on . The fact that (f,g)= (g,E f) holds, allowsE us to replacek k k(x, y) in the deﬁnition of theF form E E with its symmetric part ks(x, y). This explains why there is no condition on ka(x, y). In order to deal with the non-symmetric setting we need a slightly stronger assumption

2 1 Rd x (1 y x )ks(x, y) dy Lloc( , dx) 7→ Rd ∧| − | ∈ (C1) Z 2 ka(x, y) α0 := sup dy < . d d x R y R : ks(x,y)=0 ks(x, y) ∞ ∈ Z{ ∈ 6 } Under this assumption, see [FU12] and [SW15], the non-symmetric bilinear form

Lip Rd H(f,g) := lim (f(y) f(x))k(x, y) dyg(x) dx, f,g Cc ( ), − ε 0 Rd c − ∈ ↓ Z ZBε(x) is well deﬁned and it has the representation

1 Lip Rd H(f,g)= (f,g) (f(y) f(x))g(y)ka(x, y) dxdy, f,g Cc ( ). 2E − Rd Rd diag − ∈ Z × \ Moreover, H has an extension onto such that (H, ) deﬁnes a regular lower bounded semi-Dirichlet form on L2(RdF, dx ×F) in the sense of [MR92F ]; in particular, there is x a properly associated Hunt process ( Xt t 0, P x) which is deﬁned up to an exceptional set. The proofs of [FU12, Theorem{ 2.1]} and≥ { [SW15} , Proposition 2.1] reveal that 1 2+ √2 (2.3) (1 α ) (f, f) H (f, f) (1 α ) (f, f), f , 4 ∧ 0 E1 ≤ α0 ≤ 2 ∨ 0 E1 ∈F and

(2.4) (1 α )H (f, f) H (f) (1 α ) H (f), f , ∧ 0 1 ≤ α0 ≤ ∨ 0 1 ∈F 2 1/2 where Hα(f) := H(f, f)+ α f L2 , α > 0. In particular, is also the H1 -closure of Lip Rd ¯ k k F Cc ( ) in . Note that H = if α0 = 0. A weakerF version of the followingE result already appears in [SU12, Theorem 2.4], the main diﬀerence is that in [SU12] the shift-boundedness of ks(x, y) was needed.

d 1/2 Proposition 2.3. Under (C1) the family C∞(R ) is dense in with respect to H . c F 1 MARKOV CHAIN APPROXIMATION OF PURE JUMP PROCESSES 7

Rd 1/2 Lip Rd Proof. Clearly, it suffices to prove that Cc∞( ) is 1 -dense in Cc ( ). d E d Let χ C∞(R ) satisfying 0 χ(x) 1 for all x R , supp χ B¯ (0) and ∈ c ≤ ≤ ∈ ⊆ 1 Rd χ(x) dx = 1. For ε> 0 set d d R χε(x) := ε− χ(x/ε), x R . ∈ d By definition, χ C∞(R ), supp χ B¯ (0) and χ (x) dx = 1. The Friedrichs ε ∈ c ε ⊆ ε Rd ε mollifier J of g CLip(Rd) is defined as 1/n ∈ c R d J1/ng(x)= χ1/n g(x)= n χ1/n(x y)g(y) dy = χ(y)g(x y/n) dy, ∗ Rd − ¯ − Z ZB1(0) d x R ; for brevity, we use gn := J1/ng. It is easy to see that gn n 1 converges uniformly ∈ { } ≥ to g(x), gn g and supp gn supp g +B¯1(0) for all n N; in particular, gn n 1 convergesk tok∞g(x≤k) inkL∞2(Rd, dx). Hence,⊆ it remains to prove that∈ { } ≥

lim (gn g,gn g)=0. n →∞ E − − First, observe that for all n N and x, y Rd ∈ ∈

gn(x) gn(y) χ(z) g(x z/n) g(y z/n) dz Lg x y , | − | ≤ ¯ | − − − | ≤ | − | ZB1(0) where Lg > 0 is the Lipschitz constant of g(x). Pick R > 0 such that supp gn B¯R(0) for all n N. Then, we have that ⊆ ∈ 2 (gn(y) gn(x)) 1Rd Rd diag(x, y) − × \ 2 =(gn(y) gn(x)) 1B (0) Bc (0) + 1Bc (0) B (0) + 1B (0) B (0) diag (x, y) − 2R × 2R 2R × 2R 2R × 2R \ 2 2 2 2 g (x)1 c (x, y)+ g (y)1 c (x, y)+ L x y 1 (x, y) n B2R(0) B2R(0) n B2R(0) B2R(0) g B2R(0) B2R(0) ≤ × × | − | × 2 2 2 2 1 c 1 c 1 g B2R(0) B (0)(x, y)+ g B (0) B2R(0)(x, y)+ Lg x y B2R(0) B2R(0)(x, y). ≤k k∞ × 2R k k∞ 2R × | − | × Since

2 (gn g,gn g)= (gn(y) g(y) gn(x)+ g(x)) ks(x, y) dxdy E − − Rd Rd diag − − Z × \ 2 2 2(gn(y) gn(x)) + 2(g(y) g(x)) ks(x, y) dx dy, ≤ Rd Rd diag − − Z × \ the assertion follows directly from (C1) and the dominated convergence theorem. n N Let us now describe the Dirichlet form related to the Markov chains Xt t 0, n , N { } ≥ 2∈ introduced in the previous Section 2.1. ﬁrst, recall that for n , we denote by Ln the Zd ∈ standard Hilbert space on n with scalar product d 2 f,g 2 := n− f(a)g(a), f,g L . h iLn ∈ n a Zd X∈ n The following result is a direct consequence of [FU12] and [SW15]. Proposition 2.4. Assume that Cn, n N, satisfy (T1), (T2) and (C1)1. For every n N we deﬁne the bilinear forms ∈ ∈ n d n (f,g) := n− (f(b) f(a))(g(b) g(a))C (a, b) E − − s a Zd b Zd X∈ n X∈ n n 1 n d n H (f,g) := (f,g) n− (f(b) f(a))g(b)C (a, b). 2E − − a a Zd b Zd X∈ n X∈ n 1 n n (C1) is assumed to hold for each chain Xt t≥0 by replacing the kernel k(x, y) by C (a,b) { } 8 A.MIMICA,N.SANDRIC,´ AND R.L. SCHILLING

n 1 n n n 1 n n where Cs (a, b) := 2 (C (a, b)+ C (b, a)), resp., Ca (a, b) := 2 (C (a, b) C (b, a)) are the symmetric and antisymmetric parts of Cn(a, b). − n n 2 (i) H (f,g) is a well deﬁned non-symmetric bilinear form on := f Ln : n(f, f) < . F { ∈ (ii) (EHn, n) is∞} a regular lower bounded semi-Dirichlet form (in the sense of [MR92]); n F n 2 n 2 for all and . In particular, the associated (iii) H (f,g)= f,g Ln f Ln g h−A n i ∈ ∈F Hunt process is Xt t 0. Recall that { } ≥ n n 2 f(a)= (f(b) f(a))C (a, b) and Ln n . A − ⊆DA b Zd X∈ n (iv) The estimates (2.3) and (2.4) hold for = n, H = Hn and E E n 2 n Ca (a, b) α0 = α0 := sup n . Zd C (a, b) a n Zd s ∈ b n CnX(∈a,b)=0 s 6 Corollary 2.5. Assume that Cn, n N, satisfy (T1), (T2) as well as ∈ n (T2∗) sup C (a, b) < . b Zd ∞ ∈ n a Zd X∈ n Then n = L2 for every n N. F n ∈ Proof. Clearly, (T2)and (T2∗) imply (C1). The claim follows from Jensen’s and H¨older’s inequalities. In order to study the convergence of the forms Hn as n we need a few further → ∞ notions. Denote bya ¯ := [a1 1/2n, a1 +1/2n) [ad 1/2n, ad +1/2n) the half-open − Zd ×···× −1 Rd cube with centre a =(a1,...,ad) n and side-length n− , and for x =(x1,...,xd) we set ∈ ∈ [x]n := ([nx1 +1/2] /n, . . . , [nxd +1/2] /n) , where [u] is the integer part of u R. Note that for a Zd and x a¯ we have [x] = a. ∈ ∈ n ∈ n

a¯ Z(1d , 1) a n n ∈ 2 d/ √ 1/2n

Figure 1. Deﬁnition of the discretization

2 Rd 2 2 2 Rd By rn : L ( , dx) Ln and en : Ln L ( , dx) we denote the restriction and extension operators which→ are deﬁned by →

r f(a)= nd f(x)dx, a Zd n ∈ n Za¯ e f(x)= f(a), x a.¯ n ∈ 2 Rd 2 N These operators have the following properties: for f L ( , dx) and fn Ln, n : ∈ 2 2 ∈ ∈2 2 2 2 2 1 (i) sup rn Ln f L , lim rnf Ln = f L and enfn L2 = enfn L = fn L2 ; n N k k ≤k k n k k k k k k k k k k n ∈ →∞ MARKOV CHAIN APPROXIMATION OF PURE JUMP PROCESSES 9

(ii) r e f = f and r f, f 2 = f, e f 2 ; n n n n h n niLn h n niL (iii) lim enrnf f L2 = 0; n →∞ k − k see [CKK13, Lemma 4.1]. Let us recall from [KS03] the notions of strong and weak 2 Rd 2 Rd 2 convergence. Let L ( , dx) be dense in (L ( , dx), L2 ). A sequence fn Ln, C ⊆ 2 d k·k ∈ n N, converges strongly to f L (R , dx) if for every gm m 1 satisfying ∈ ∈ { } ≥ ⊆ C lim gm f L2 =0, m →∞ k − k we have that

2 lim lim sup rngm fn Ln =0. m n k − k →∞ →∞ The sequence f L2 , n N, converges weakly to f L2(Rd, dx), if n ∈ n ∈ ∈ lim fn,gn L2 = f,g L2 n n →∞h i h i 2 2 Rd for every sequence gn n 1, gn Ln, converging strongly to g L ( , dx). In the following lemma{ } ≥ we give∈ an equivalent characterization∈ of the strong and weak convergence, which simpliﬁes the use of these types of convergence. 2 2 Rd Lemma 2.6. (i) A sequence fn n 1, fn Ln, converges strongly to f L ( , dx) if, and only if, { } ≥ ∈ ∈

lim enfn f L2 =0. n →∞ k − k 2 2 Rd (ii) A sequence fn n 1, fn Ln, converges weakly to f L ( , dx) if, and only if, { } ≥ ∈ ∈ 2 d lim enfn,g = f,g , g L (R , dx). n →∞h i h i ∈ 2 N 2 Rd Proof. (i) Let fn Ln, n , and f L ( , dx). Assume that fn n 1 converges ∈ ∈ ∈ { } ≥ strongly to f, and let gm m 1 be an approximating sequence of f satisfying the conditions from the definition{ } ≥ of⊆ strong C convergence. Then, by the properties of the operators r and e , n N, we find n n ∈ e f f 2 e f e r f 2 + e r f f 2 k n n − kL ≤k n n − n n kL k n n − kL = f r f 2 + e r f f 2 k n − n kLn k n n − kL f r g 2 + r g r f 2 + e r f f 2 ≤k n − n mkLn k n m − n kLn k n n − kL f r g 2 + g f 2 + e r f f 2 . ≤k n − n mkLn k m − kL k n n − kL Letting first n and then m , the necessity of the claim follows. For the ‘if’ → ∞ → ∞ part, we proceed as follows. Let gm m 1 be any approximating sequence of f. Using the strong convergence e f f{ and} the≥ ⊆ fact C that r e f = f , we see n n → n n n n r g f 2 = r g r e f 2 g e f 2 k n m − nkLn k n m − n n nkLn ≤k m − n nkL g f 2 + f e f 2 . ≤k m − kL k − n nkL Letting first n and then m proves the assertion. →∞ →∞ 2 N 2 Rd (ii) Let fn Ln, n , and f L ( , dx). Assume that fn n 1 converges weakly to ∈ ∈ 2 d ∈ { } ≥ f. Since for every g L (R ) the sequence rng n 1 converges strongly to g, it follows immediately that ∈ { } ≥ 2 d lim enfn,g L2 = f,g L2 , g L (R , dx). n →∞h i h i ∈ 2 Rd 2 For the sufficiency part we pick g L ( , dx) and any sequence gn n 1, gn Ln ∈ { } ≥ ∈ such that gn n 1 converges strongly to g. By our assumption, { } ≥ lim fn,rng L2 = lim enfn,g L2 = f,g L2 ; n n n n →∞h i →∞h i h i 10 A.MIMICA,N.SANDRIC,´ AND R.L. SCHILLING this shows, in particular, that e f ,g 2 f 2 for all g with g 2 = 1, and so h n n iL ≤ k kL k kL supn N enfn L2 < . In order to prove the claim we have to show that ∈ k k ∞ lim fn,gn rng L2 =0. n n →∞h − i Using the properties of the operators rn and en along with the Cauchy-Schwarz inequality yields

For further details on strong and weak convergence we refer to [KS03] and [T06¨ ].

2.3. Convergence of the Finite-Dimensional Distributions. We can now combine the relative compactness from Section 2.1 and the convergence results from Section 2.2 to n N show the convergence of the ﬁnite-dimensional distributions of the chains Xt t 0, n , { } ≥ ∈ to those of a non-symmetric pure jump process Xt t 0. The latter will be determined by a kernel k : Rd Rd diag R satisfying (C1{ ). } ≥ × \ → n N Theorem 2.7. Assume that the chains Xt t 0, n , satisfy (T1), (T2) and (C1). { } ≥ ∈ d d Let Xt t 0 be a non-symmetric process determined by a kernel k : R R diag R { } ≥ n N × \ → satisfying (C1). Denote by Pt t 0, n , and Pt t 0 the transition semigroups of n N { } ≥ ∈ { } ≥ Xt t 0, n , and Xt t 0, respectively, { } ≥ n ∈ { } ≥ 2 Rd If Pt rnf n 1 converges strongly to Ptf for all t 0 and f L ( , dx), then there { } ≥ ≥ ∈ n exists a Lebesgue null set B such that the ﬁnite-dimensional distributions of Xt t 0, c { } ≥ n N, converge along Q on B to those of Xt t 0. ∈ { } ≥ Proof. Choose an arbitrary countable family CLip(Rd) which is dense in CLip(Rd) C ⊆ c c with respect to . By the Markov property, the properties of the operators rn and k·k∞ en, and a standard diagonal argument, we can extract a subsequence ni i 1 N such { } ≥ ⊆ that for all m 1, all t1,...,tm Q, 0 t1 tm < , and all f1,...fm , E n ≥ n ∈ ≤ ≤ ···≤E ∞ ∈ C n· [rnf1(Xt ) rnfm(Xt )] n 1 converges strongly to ·[f1(Xt1 ) fm(Xtm )]. { 1 ··· m } ≥ ··· Again by a diagonal argument, we conclude that there is a further subsequence ni′ i 1 d { } ≥ ⊆ ni i 1 and a Lebesgue null set B R , such that the above convergence holds pointwise {on B} ≥c. The assertion now follows⊆ from [EK86, Proposition 3.4.4].

Denote by D(Rd) the space of all c`adl`ag functions f : [0, ) Rd. The space D(Rd) ∞ → equipped with Skorokhod’s J1 topology becomes a Polish space, the so-called Skorokhod space, cf. [EK86]. Corollary 2.8. Assume that the conditions of Theorems 2.2 and 2.7 hold, and let B be the Lebesgue null set from Theorem 2.7. Denote by µn and µ the initial distributions of n N Xt t 0 and Xt t 0, n , respectively. If µ(B)=0 and if µn µ weakly, then the {following} ≥ convergence{ } ≥ holds∈ in Skorokhod space: →

n d (2.5) Xt Xt t 0. t 0 n ≥ { } ≥ −−−→→∞ { } Proof. The assertion follows by combining Theorems 2.2, 2.7,[JS03, Lemma VI.3.19] and the remark following that lemma.

n Theorem 2.7 states that (2.5) follows if we can prove “strong convergence” of Pt Pt, t> 0. A suﬃcient condition for this convergence is given in the following theorem.→ MARKOV CHAIN APPROXIMATION OF PURE JUMP PROCESSES 11

n N Theorem 2.9. Assume that (C1) holds for both Xt t 0, n , and Xt t 0. The n { } ≥ ∈ 2 {Rd } ≥ semigroups Pt rnf n 1 converge strongly to Ptf for all t 0 and f L ( , dx) if the following conditions{ } are≥ satisﬁed: ≥ ∈

n n (C2) 0 < lim inf α0 lim sup α0 < ; n ≤ n ∞ →∞ →∞ 2 (C3) ρ> 0 : sup (1 y )ks(x, x + y) dy < ; ∀ x Bρ(0) Rd ∧| | ∞ ∈ Z 2 n (C4) ρ> 0 : lim sup sup (1 b )Cs (a, a + b) < ; ∀ n a Bρ(0) ∧| | ∞ →∞ ∈ b Zd X∈ n (C5) ε> 0 n N n m n, f L2 : ∀ ∃ 0 ∈ ∀ 0 ≤ ≤ ∈ m 1/2 n(r e f,r e f) m(f, f)1/2 + ε; E n m n m ≤E (C6) for all suﬃciently small ε> 0 and large m N ∈ ¯n Lip Rd lim m,ε(f, f)= m,ε(f, f), f Cc ( ) n →∞ E E ∈

where for all f CLip(Rd) ∈ c 1 2 m,ε(f, f) := (f(y) f(x)) ks(x, y) dx dy, E 2 (x,y) Bm(0) Bm(0): x y >ε − dZZ{ ∈ × | − | } ¯n n 2 n 1 m,ε(f, f) := (f(y) f(x)) Cs (a, b) a¯ ¯b(x, y) dx dy, × E 2 (x,y) Bm(0) Bm(0): x y >ε − ZZ{ ∈ × | − | }

(C7) (i) x y 2k (x, x + y) dy L2 (Rd, dx), 7→ | | s ∈ loc ZB1(0) 2 d d (ii) x ks(x, x + y) dy L (R , dx) L∞(R , dx), 7→ Bc(0) ∈ ∪ Z 1 (iii) x y k (x, x + y) k (x, x y) dy L2 (Rd, dx), 7→ | || s − s − | ∈ loc ZB1(0) 2 Rd (iv) x (1 y ) ka(x, x + y) dy Lloc( , dx); 7→ Rd ∧| | | | ∈ Z 2 Rd 2 d n Lloc( ,dx) (C8) (1 y ) ks(x, x + y) n Cs ([x]n, [x]n + [y]n) dy 0; Rd ∧| | | − | −−−−−−→n Z →∞ (C9) for all suﬃciently large R> 1 2

ks(x, x + y) dy dx < ; Bc (0) B ( x) ∞ Z 2R Z R − (C10) for all suﬃciently large R> 1 2 n d n →∞ ks(x, x + y) n Cs ([x]n, [x]n + [y]n) dy dx 0; Bc (0) B ( x) | − | −−−→ Z 2R Z R − (C11) y k (x, x + y) k (x, x y) | || s − s − ZB1(0) L2 (Rd,dx) d n d n loc n Cs ([x]n, [x]n + [y]n)+ n Cs ([x]n, [x]n [y]n) dy 0; n − − | −−−−−−→→∞ 12 A.MIMICA,N.SANDRIC,´ AND R.L. SCHILLING

2 Rd d n Lloc( ,dx) (C12) (1 y ) ka(x, x + y) n Ca ([x]n, [x]n + [y]n) dy 0; Rd ∧| | | − | −−−−−−→n Z →∞ (C13) for all suﬃciently large R> 1 2 n d n →∞ ka(x, x + y) n Ca ([x]n, [x]n + [y]n) dy dx 0. Bc (0) B ( x) | − | −−−→ Z 2R Z R −

Proof. According to the properties of the operators rn and en, n N, Propositions 2.3 and 2.4, and [T06¨ , Theorem 2.41 and Remark 2.44] the assertion follows∈ if n 2 d (i) for every sequence fn n 1, fn , converging weakly to some f L (R , dx) { }n ≥ ∈ F ∈ and satisfying lim inf H1 (fn, fn) < , we have that f ; n ∞ ∈F 2 →∞Rd n (ii) for every g Cc ( ) and every sequence fn n 1, fn , converging weakly to f , ∈ { } ≥ ∈F ∈F n lim H (rng, fn)= H(g, f). n →∞ 2 Indeed, (i) and (ii) imply that for any sequence fn n 1, fn Ln, converging strongly 2 Rd n { } ≥ ∈ to f L ( , dx), the sequence Pt fn n 1 converges strongly to Ptf for every t 0, cf. ∈ { } ≥ 2 d ≥ [T06¨ , Theorem 2.41 and Remark 2.44]. For any ﬁxed f L (R , dx) we set fn = rnf. n ∈ Since rnf f strongly we conclude that Pt rnf Ptf strongly as claimed. Let us now→ prove that (C1)–(C13) imply (i) and→ (ii). We begin with (i). According to Proposition 2.4, we have 4(1 αn) n(f) ∨ 0 Hn(f), f n. E1 ≤ 1 αn 1 ∈F ∧ 0 n 2 d Let fn n 1, fn , be an arbitrary sequence converging weakly to some f L (R , dx) { } ≥ ∈F n ∈ n such that lim infn H1 (fn) < ; the condition (C2) ensures that lim infn 1 (fn) < . Finally, [CKK13→∞, Theorem 4.6]∞ states that under (C3)–(C6) →∞ E ∞ n (f, f) lim inf (fn, fn), n E ≤ →∞ E which proves (i). In order to prove (ii) we proceed as follows. According to Proposition 2.4 we have for g C2(Rd) and f n ∈ c n ∈F n n H (r g, f )= r g, f 2 , n N. n n h−A n niLn ∈ Using (C7) it is shown in [SW15, Theorem 2.2] that the generator ( , ) of Pt t 0 (or, equivalently, of (H, )) has the following properties: A DA { } ≥ F 2 Rd (i) Cc ( ) ; (ii) for every⊆Dg AC2(Rd), ∈ c 1 g(x)= (g(x + y) g(x) g(x),y B1(0)(y))ks(x, x + y) dy A Rd − − h∇ i Z 1 + g(x),y (k (x, x + y) k (x, x y)) dy 2 h∇ i s − s − ZB1(0)

+ (g(x + y) g(x))ka(x, x + y) dy; Rd − Z (iii) for all g C2(Rd) and all f , ∈ c ∈F H(g, f)= g, f 2 . h−A iL MARKOV CHAIN APPROXIMATION OF PURE JUMP PROCESSES 13

n Therefore, it suﬃces to prove that rng n 1 converges strongly to g for every g C2(Rd). Observe that for any g C{A2(Rd) and} ≥n N, A ∈ c ∈ c ∈ nr g(a)= (r g(a + b) r g(a))Cn(a, a + b) A n n − n b Zd X∈ n 1 n = (rng(a + b) rng(a) rn g(a), b b 1 (b))Cs (a, a + b) − −h ∇ i {| |≤ } b Zd X∈ n 1 + r g(a), b (Cn(a, a + b) Cn(a, a b)) 2 h n∇ i s − s − b 1 |X|≤ + (r g(a + b) r g(a))Cn(a, a + b). n − n a b Zd X∈ n Using the triangle inequality we get 5 n n e r g g 2 A A 2 , k nA n −A kL ≤ k i − ikL i=1 X 1 n where for ρ := 1 + (2n)− √d the Ai and Ai are given by

A := g(x + y) g(x) g(x),y 1 (y) k (x, x + y) dy 1 − − h∇ i B1(0) s ZBρ(0) An := r g([x] + [y] ) r g([x] ) r g([x] ), [y] 1 ([y] ) 1 n n n − n n −h n∇ n ni B1(0) n × ZBρ(0) nd Cn([x] , [x] + [y] ) dy × s n n n

A2 := g(x + y) g(x) ks(x, x + y) dy c − ZBρ(0) n d n A2 := rng([x]n + [y]n) rng([x]n) n Cs ([x]n, [x]n + [y]n) dy c − ZBρ(0) 1 A := g(x),y 1 (y)(k (x, x + y) k (x, x y)) dy 3 2 h∇ i B1(0) s − s − ZBρ(0) 1 An := r g([x] ), [y] 1 ([y] ) 3 2 h n∇ n ni B1(0) n × ZBρ(0) nd (Cn([x] , [x] + [y] ) Cn([x] , [x] [y] )) dy × s n n n − s n n − n A := g(x + y) g(x) k (x, x + y) dy 4 − a ZBρ(0) An := r g([x] + [y] ) r g([x] ) nd Cn([x] , [x] + [y] ) dy 4 n n n − n n a n n n ZBρ(0) A5 := g(x + y) g(x) ka(x, x + y) dy c − ZBρ(0) n d n A5 := rng([x]n + [y]n) rng([x]n) n Ca ([x]n, [x]n + [y]n) dy. c − ZBρ(0) In the remaining part of the proof we assume R > 1+ √d/2 such that supp g BR(0) 1 1 ⊆ and we write ρ := 1 + (2n)− √d and σ := 1 (2n)− √d. − n A A 2 k 1 − 1kL 14 A.MIMICA,N.SANDRIC,´ AND R.L. SCHILLING

g(x + y) g(x) g(x),y 1 (y) ≤ − − h∇ i B1(0) ZB2R(0) ZBρ(0) 2 1 2 r g([ x] + [y] )+ r g([x] )+ r g([x] ), [y] 1 ([y] ) k (x, x + y) dy dx − n n n n n h n∇ n ni B1(0) n s + r g([x] + [y] ) r g([x] ) r g([x] ), [y] 1 ([y] ) n n n − n n −h n∇ n ni B1(0) n × ZB2R(0) ZBρ(0) 2 1 2 k (x, x + y) ndCn([x] , [x] + [y] ) dy dx × s − s n n n g(x + y) g(x) g(x),y ≤ − − h∇ i ZB2R(0) ZBσ(0) 2 1 2 r g([x] + [y] )+ r g([x] )+ r g([x] ), [y] k (x, x + y) dy dx − n n n n n h n∇ n ni s 2 1 2 + 4 g +2ρ g ks(x, x + y) dy dx ∞ ∞ k k k∇ k B (0) Bρ(0) Bσ (0) Z 2R Z \ 1 2 2 2 2 d n + g y ks(x, x + y) n Cs ([x]n, [x]n + [y]n) dy dx k∇ k∞ | | − ZB2R(0) ZBσ(0)

+ 2 g + ρ g ∞ ∞ k k k∇ k × 1 2 2 d n ks(x, x + y) n Cs ([x]n, [x]n + [y]n) dy dx . × B (0) Bρ(0) Bσ (0) − Z 2R Z \

By monotone and dominated convergence theorem, Taylor’s theorem, (C7) (i) and (ii), n and (C8), we conclude that A A 2 0. Next, k 1 − 1kL →

n A2 A2 L2 k − k 1 2 2 g(x + y) g(x) rng([x]n + [y]n)+ rng([x]n) ks(x, x + y) dy dx ≤ c − − ZB2R(0) ZBρ(0) 2 1 2 + g(x + y) rng([x]n + [y]n) ks(x, x + y) dy dx Bc (0) Bc(0) − Z 2R Z ρ

+ rng([x]n + [y]n) rng([x]n) c − × ZB2R(0) ZBρ(0) 2 1 2 k (x, x + y) ndCn([x] , [x] + [y] ) dy dx × s − s n n n

1 2 2 d n + rng([x]n + [y]n) ks(x, x + y) n C s ([x]n, [x]n + [y]n) dy dx Bc (0) Bc(0) − Z 2R Z ρ 2 1 2 g(x + y) g(x) rng([x]n + [y]n)+ rng([x]n) ks(x, x + y) dy dx ≤ c − − ZB2R(0) ZBρ(0) 2 1 2 + g(x + y) rng([x]n + [y]n) ks(x, x + y) dy dx Bc (0) B ( x) − Z 2R Z R −

MARKOV CHAIN APPROXIMATION OF PURE JUMP PROCESSES 15

1 2 2 d n +2 g ks(x, x + y) n Cs ([x]n, [x]n + [y]n) dy dx k k∞ c − ZB2R(0) ZBρ(0) 2 1 2 d n + g ks(x, x + y) n Cs ([x]n, [x]n + [y]n) dy dx . k k∞ Bc (0) B ( x) − Z 2R Z R −

Again, by monotone and dominated convergence theorem, Taylor’s theorem, (C7) (ii), n (C8), (C9) and (C10), we have that A A 2 0. Further, k 2 − 2kL → n A A 2 k 3 − 3kL 1 g(x),y 1 (y) r g([x] ), [y] 1 ([y] ) ≤ 2 h∇ i B1 (0) − h n∇ n ni B1(0) n × ZB2R(0) ZBρ(0) 2 1 2 k (x,x + y) k (x,x y) dy dx × s − s − 1 + r g([x] ), [y] 1 ([y] ) 2 h n∇ n ni B1(0) n × ZB2R(0) ZBρ(0) 2 1 2 k (x,x + y ) k (x,x y) ndCn([x] , [x] + [y] )+ ndCn([x] , [x] [y] ) dy dx × s − s − − s n n n s n n − n 2 1 1 2 g(x),y r g([x] ), [y] k (x,x + y) k (x,x y) dy dx ≤ 2 h∇ i − h n∇ n ni s − s − ZB2R(0) ZBσ(0) 2 1 2 + ρ g ks(x,x + y) ks(x,x y) dy dx k∇ k∞ − − ZB2R(0) ZBρ(0) Bσ (0) \ + g y k∇ k∞ | |× ZB2R(0) ZBσ(0) 1 2 2 k (x,x + y) k (x,x y) ndCn([x] , [x] + [y] )+ ndCn([x] , [x] [y] ) dy dx × s − s − − s n n n s n n − n 1 + ρ g ∞ 2 k∇ k B (0) Bρ(0) Bσ (0) Z 2R Z \ 1 2 2 k (x,x + y) k (x,x y) ndCn([x] , [x] + [y] )+ ndCn([x] , [x] [y] ) dy dx , × s − s − − s n n n s n n − n

which, by monotone and dominated convergence, Taylor’s theorem, (C7) (iii), (C8) and n (C11), implies that A A 2 0. Next, k 3 − 3kL → n A A 2 k 4 − 4kL 1 2 2 g(x + y) g(x) r g([x] + [y] )+ r g([x] ) k (x, x + y) dy dx ≤ − − n n n n n a ZB2R(0) ZBρ(0)

+ rng([x]n + [y]n) rng([x]n) − × ZB2R(0) ZBρ(0) 2 1 2 k (x, x + y) ndCn([x] , [x] + [y] ) dy dx × a − a n n n 2 1 2 g(x + y) g(x) r g([x] + [y] )+ r g([x] ) k (x, x + y) dy dx ≤ − − n n n n n a ZB2R(0) ZBρ(0)

16 A.MIMICA,N.SANDRIC,´ AND R.L. SCHILLING

1 2 2 d n +2 g y ka(x, x + y) n Ca ([x]n, [x]n + [y]n) dy dx . k∇ k∞ | | − ZB2R(0) ZBρ(0)

Now, by monotone and dominated convergence, Taylor’s theorem, (C7) (iv) and (C12), n we see A A 2 0. Finally, k 4 − 4kL → n A A 2 k 5 − 5kL 1 2 2 g(x + y) g(x) rng([x]n + [y]n)+ rng([x]n) ka(x, x + y) dy dx ≤ c − − ZB2R(0) ZBρ)(0) 2 1 2 + g(x + y) rng([x]n + [y]n) ka(x, x + y) dy dx Bc (0) Bc(0) − Z 2R Z ρ

+ rng([x]n + [y]n) rng([x]n) c − × ZB2R(0) ZBρ(0) 2 1 2 k (x, x + y) ndCn([x] , [x] + [y] ) dy dx × a − a n n n

1 2 2 d n + rng([x]n + [y]n) ka(x, x + y) n Ca ([x]n, [x ]n + [y]n) dy dx Bc (0) Bc(0) − Z 2R Z ρ 2 1 2 g(x + y) g(x) rng([x]n + [y]n)+ rng([x]n) ka(x, x + y) dy dx ≤ c − − ZB2R(0) ZBρ(0) 2 1 2 + g(x + y) rng([x]n + [y]n) ka(x, x + y) dy dx Bc (0) B ( x) | − || | Z 2R Z R − 1 2 2 d n +2 g ka(x, x + y) n Ca ([x]n, [x]n + [y]n) dy dx k k∞ c − ZB2R(0) ZBρ(0) 2 1 2 d n + g ka(x, x + y) n Ca ([x]n, [x]n + [y]n) dy dx . k k∞ Bc (0) B ( x) − Z 2R Z R −

By monotone and dominated convergence, Taylor’s theorem, (C7) (iv), (C9), (C12) and n (C13), we get that A A 2 0, which concludes the proof. k 5 − 5kL → The conditions of Theorem 2.9 can be slightly changed to give a further set of suﬃcient n conditions of the convergence of Pt rnf n 1; the advantage is that we can state these { n } ≥ n conditions only using ks and k resp. Cs and C , which makes them sometimes easier to check. Corollary 2.10. Assume that (C1)–(C6) and (C7)(i), (ii) hold, that

(2.6) x y k(x, x + y) k(x, x y) dy L2 (Rd, dx) 7→ | | − − ∈ loc ZB1(0)

n n and that (C8)–(C11) hold with ks and Cs replaced by k and C , respectively. Then n 2 Rd Pt rnf n 1 converges strongly to Ptf for all t 0 and f L ( , dx). { } ≥ ≥ ∈ Proof. According to [SW15, Theorem 3.1], the above assumptions imply that the generator ( , ) of (H, ) satisﬁes A DA F Rd (i) Cc∞( ) ; ⊆DA MARKOV CHAIN APPROXIMATION OF PURE JUMP PROCESSES 17

d (ii) for every g C∞(R ), ∈ c 1 g(x)= g(x + y) g(x) g(x),y B1(0)(y) k(x, x + y) dy A Rd − − h∇ i Z 1 + g(x),y k(x, x + y) k(x, x y) dy. 2 h∇ i − − ZB1(0) Note that in [SW15, Theorem 3.1] slightly stronger conditions are assumed (namely (H3) which is a symmetrized version of (2.6) and the tightness assumption (H5)), but they are exclusively used to deal with the formal adjoint A∗; this follows easily from an inspection of the proofs of [SW15, Theorems 2.2 and 3.1]. From this point onwards we can follow the proof of Theorem 2.9.

Recall that a set is an operator core for ( , ) if = . If we happen 2 RdC⊆DA A DA A|C A to know that Cc ( ) is an operator core for ( , ), then there is an alternative proof A DA n of Theorem 2.9 and its Corollary 2.10 based on [EK86, Theorem 1.6.1]: Pt rnf n 1 2 d { n } ≥ converges strongly to Ptf for all t 0 and all f L (R , dx) if (and only if) rng n 1 converges strongly to g for every≥g C2(Rd). ∈ {A } ≥ A ∈ c 2.4. Approximation of a Given Process. We will now show how we can use the results of Sections 2.1–2.3 to approximate a given non-symmetric pure-jump process by a sequence of Markov chains. We assume that Xt t 0 is of the type described at the beginning of Section 2.2; in particular the kernel k{: R}d ≥ Rd diag R satisﬁes (C1). We are going to construct a sequence of approximating (in× the\ weak→ sense) Markov chains. Let 0

np (2.7) C (a, b) := a¯ ¯b | − |  Z Z √ 0, a b 2 d .  | − | ≤ np Remark 2.11. Th family of kernels deﬁned in (2.7) has the following properties: (i) The kernels Cn,p, n N, automatically satisfy (T1). ∈ (ii) For any increasing sequence ni i N N such that the lattices are nested, i.e. Zd Zd C5{ } ∈ ⊂C6 ni ni+1 , the conditions ( ) and ( ) hold true, cf. [CKK13, Theorem 5.4]. ⊆ i This is, in particular, the case for ni =2 , i N. (iii) Due to (C1) and Lebesgue’s diﬀerentiation theorem∈ (see [Fol84, Theorem 3.21]), we have for (Lebesgue) almost all (x, y) Rd Rd diag, ∈ × \ lim n2d k(u, v) dv du = k(x, y) n →∞ Z[x]n Z[y]n and

lim n2d k(u, v) k(x, y) dvdu =0. n [x] [y] | − | →∞ Z n Z n Let us check the conditions (T2)–(T6). Proposition 2.12. The conditions (T2) and (T3) hold true if

(T1.D) ρ> 0 : sup k(x, y)dy < . ∀ x Rd Bc(x) ∞ ∈ Z ρ 18 A.MIMICA,N.SANDRIC,´ AND R.L. SCHILLING

Proof. We will only discuss (T2) since (T3) follows in a similar way. Observe that for every d N and 0

2√d/np | − | [ This shows that

sup Cn,p(a, b)= nd sup k(x, y) dy dx Zd Zd ¯ a n Zd a n p a¯ b ∈ b n ∈ a b >2√d/n Z Z X∈ | − |X nd sup k(x, y) dy dx ≤ a Zd a¯ Bc (x) ∈ n Z Z √d/2np sup k(x, y) dy, ≤ x Rd Bc (x) ∈ Z √d/2np which concludes the proof. This means that under (T1.D), the kernels Cn,p, n N, deﬁne a family of regular n N ∈ Markov chains Xt t 0, n . Using the same arguments as above, it is easy to see that (T4) holds{ if } ≥ ∈

(T3.D) lim sup k(x, y)dy =0. r x Rd Bc(x) →∞ ∈ Z r Proposition 2.13. Assume that (T1.D) holds. Then the following statements are true. (i) (T5) will be satisﬁed if (T4.D.1) there is some ρ> 0 such that for i =1,...,d

lim sup sup (yi xi)k(x, y) dy < , d ε 0 x R Bρ(x) Bε(x) − ∞ ↓ ∈ Z \

lim sup sup yi xi k(x, y) dx < , ε 0 Rd | − | ∞ x B√dεp (x) B√dεp (√d/2)ε(x) ↓ ∈ Z \ − lim sup ε sup k(x, y) dy < . d ε 0 x R Bρ(x) B p (x) ∞ ↓ ∈ Z \ ε (ii) (T6) will be satisﬁed if (T5.D.1) there is some ρ> 0 such that for i, k =1,...,d

lim sup sup (yi xi)(yk xk)k(x, y) dy < , d ε 0 x R Bρ(x) Bε(x) − − ∞ ↓ ∈ Z \

lim sup sup yi xi yk xk k(x, y) dx < , ε 0 Rd | − || − | ∞ x B√dεp (x) B√dεp (√d/2)ε(x) ↓ ∈ Z \ −

lim sup ε sup yi xi k(x, y) dy < . d ε 0 x R Bρ(x) B p (x) | − | ∞ ↓ ∈ Z \ ε Proof. We will only discuss (T5), since (T6) follows in an analogous way. Assume (T4.D.1). We have

n,p d sup bi C (a, a + b) = n sup bi k(x, y) dy dx a Zd a Zd a¯ a+b ∈ n b <ρ ∈ n 2√d/np< b <ρ Z Z |X| X| |

MARKOV CHAIN APPROXIMATION OF PURE JUMP PROCESSES 19

d n sup (yi xi)k(x, y) dy dx ≤ a Zd a¯ a+b − ∈ n 2√d/np< b <ρ Z Z X| |

d 1 + √dn − sup k(x, y) dy dx. a Zd a¯ a+b ∈ n 2√d/np< b <ρ Z Z X| | Next, for every d N, 0

a + b Bρ+√d/2n(a) B√d/np (a) Bρ+√d/n(x) B√d/np √d/2n(x). ⊆ \ ⊆ \ − 2√d/np< b <ρ [ | | Thus,

n,p sup bi C (a, a + b) Zd a n b <ρ ∈ |X|

d n sup (yi xi)k(x, y) dy dx ≤ a Zd a¯ B (x) B (x) − ∈ n Z Z ρ+√d/n \ √d/np √d/2n − d + n sup (yi xi)k(x, y) dy a Zd a¯ B (x) B (x) − ∈ n Z Z ρ+√d/n \ √d/np √d/2n −

(yi xi)k(x, y) dy dx − a+b − 2√d/np< b <ρ Z X| |

d 1 + √dn − sup k(x, y) dy dx Zd a n a¯ Bρ+√d/n(x) B√d/np √d/2n(x) ∈ Z Z \ −

sup (yi xi)k(x, y) dy ≤ x Rd B (x) B (x) − ∈ Z ρ+√d/n \ √d/np √d/2n −

+ sup yi xi k(x, y) dy Rd | − | x Bρ+√d/n(x) Bρ √d/n(x) ∈ Z \ −

+ sup yi xi k(x, y) dy Rd | − | x B√d/np (x) B√d/np √d/2n(x) ∈ Z \ − √d + sup k(x, y) dy n Rd x Bρ+√d/n(x) B√d/np √d/2n(x) ∈ Z \ −

sup (yi xi)k(x, y) dy ≤ x Rd B (x) B (x) − ∈ Z ρ+√d/n \ √d/np √d/2n − √d + ρ + sup k(x, y) dy n Rd x Bρ+√d/n(x) Bρ √d/n(x) ∈ Z \ −

+ sup yi xi k(x, y) dx Rd | − | x B√d/np (x) B√d/np √d/2n(x) ∈ Z \ − 20 A.MIMICA,N.SANDRIC,´ AND R.L. SCHILLING

√d + sup k(x, y) dy. n Rd x Bρ+√d/n(x) B√d/np √d/2n(x) ∈ Z \ − Together with (T1.D) and (T4.D.1) this proves the claim. Let us now discuss the conditions (C2)–(C6). n Proposition 2.14. (i) The condition (C1) implies lim supn α0 α0. Assume that there exist open balls Rd such that →∞ ≤ (ii) B1, B2 sup(x,y) B1 B2 ks(x, y) < ⊆ ∈ × , ks(x, y) > 0 Lebesgue-a.e. on B1 B2, and inf(x,y) B1 B2 ka(x, y) > 0. Then, ∞ n × ∈ × | | lim infn α0 > 0. In particular,→∞ the assumptions in (i) and (ii) guarantee that (C2) holds true. N Zd n,p Proof. (i) For n , 0 < p 1 and a, b n such that Cs (a, b) = 0 the Cauchy– Schwarz inequality∈ gives ≤ ∈ 6 2 2 1 ka(x, y) ka(x, y) dydx = ks=0 (x, y) | | ks(x, y) dy dx a¯ ¯b | | a¯ ¯b { 6 } k (x, y) Z Z Z Z s 2 p 1 kap(x, y) ks=0 (x, y)| | dy dx ks(x, y) dy dx, ≤ ¯ { 6 } k (x, y) ¯ Za¯ Zb s Za¯ Zb i.e. 2 d n,p 2 n ¯ ka(x, y) dy dx 2 Ca (a, b) a¯ b | | d 1 ka(x, y) n,p = d n ks=0 (x, y)| | dy dx. Cs (a, b) n k (x, y) dy dx ≤ ¯ { 6 } k (x, y) R a¯R ¯b s Za¯ Zb s Consequently, R R n,p 2 2 n Ca (a, b) 1 ka(x, y) α0 = sup n,p sup sup ks=0 (x, y)| | dy d d d ¯ { 6 } a Zn d Cs (a, b) ≤ a Zn x R d b ks(x, y) ∈ b Zn ∈ ∈ b Zn Z Cn,pX∈(a,b)=0 Cn,pX∈(a,b)=0 s 6 s 6 2 ka(x, y) sup | | dy = α0. d d ≤ x R y R : ks(x,y)=0 ks(x, y) ∈ Z{ ∈ 6 }

(ii) Let m := inf ka(x, y) sup ks(x, y) =: M. For all n N large (x,y) B1 B2 | | ≤ ∈ ∈ × (x,y) B1 B2 enough, we have ∈ × n,p 2 n,p 2 2 n Ca (a, b) Ca (a, b) m d α0 = sup n,p sup n,p n− . Zd Cs (a, b) ≥ Zd Cs (a, b) ≥ M a n Zd a n Zd Zd ∈ b n ∈ b n b n n,pX∈ a¯ B1 X∈ X∈ Cs (a,b)=0 ⊆ ¯b B2 ¯b B2 6 ⊆ ⊆ Proposition 2.15. (C3) implies (C4) for Cn,p given by (2.7). Proof. For any ρ> 0, n N and 0

√d, we have that ∈ n,p d sup Cs (a, b) sup n ks(x, y) dy dx c a Bρ(0) ≤ a Bρ(0) a¯ B (x) ∈ a b 1 ∈ Z Z 1 √d/n | X− |≥ − MARKOV CHAIN APPROXIMATION OF PURE JUMP PROCESSES 21

sup sup ks(x, y) dy c ≤ a Bρ(0) x a¯ B (x) ∈ 1 √d/n ∈ Z −

sup ks(x, x + y) dy. ≤ x B (0) Bc (0) ρ+√d/2n 1 √d/n ∈ Z − Next, it is easy to see that ¯b B (a) B (x), a Zd , x a,¯ ⊆ 1+√d/2n ⊆ 1+√d/n ∈ n ∈ a b <1 | −[| Zd Rd ¯ and a b 2 x y for all a, b n, a b > √d/n, and all x, y , x a¯, y b. Thus,| − | ≤ | − | ∈ | − | ∈ ∈ ∈ 2 n,p 2 n,p sup a b Cs (a, b) = sup a b Cs (a, b) a Bρ(0) | − | a Bρ(0) | − | ∈ a b <1 ∈ 2√d/np< a b <1 | X− | X| − | d 2 4 sup n x y ks(x, y) dy dx ≤ a Bρ(0) a¯ ¯b | − | ∈ 2√d/np< a b <1 Z Z X| − | d 2 4 sup n x y ks(x, y) dy dx ≤ a Bρ(0) a¯ B (x) | − | ∈ Z Z 1+√d/n 2 4 sup sup y ks(x, x + y) dy ≤ a Bρ(0) x a¯ B (0) | | ∈ ∈ Z 1+√d/n 2 4 sup y ks(x, x + y) dy, ≤ a B (0) B (0) | | ∈ ρ+√d/2n Z 1+√d/n which proves the assertion. We will now discuss some examples where the conditions (C1)–(C13) are satisfied. Example 2.16 (Symmetric jump processes). Assume that k(x, y) = k(y, x) Lebesgue a.e. on Rd Rd. For 0 < p 1 we define the corresponding family of conductances Cn,p, n N×, by (2.7). If (T1)–(≤ T6) hold, then the family of underlying Markov chains n ∈ N Xt t 0, n , is tight. Due to symmetry, the second condition in (C1), (C7) (iii), (iv), ({C12})≥ and∈ (C13) are trivially satisfied, and (C2) is not needed. The condition (C4) follows from (C3), while (C5) and (C6) automatically hold true (take, for example, a i subsequence ni = 2 , i N). For an alternative approach to the problem of discrete approximation of symmetric∈ jump processes we refer the readers to [CKK13]. Example 2.17 (Non-symmetric Lévy processes). A class of non-symmetric Lévy processes which satisfy conditions (C1)–(C13) can be constructed in the following way. Let d ν1(dy) = n1(y) dy and ν2(dy) = n2(y) dy be Lévy measures and let B R be a Borel set. Define a new Lévy measure ν(dy) by ⊆

ν(dy) := 1B(y) ν1(dy)+ 1Bc (y) νd(dy).

In general, ν(dy) is not symmetric and a suitable choice of the densities n1(y) and α d C1 C13 1 c n2(y) ensures ( )–( ). For example, take n1(y) = y − − B1(0)(y) and n2(y) := β d | | y − − 1Bc(0)(y), where α, β (0, 2), and B is any Borel set. Obviously, the kernel | | 1 ∈ α d β d k(x, y) := y x − − 1 (x y)+ y x − − 1 c (x y), x = y, | − | B − | − | B − 6 satisfies (C1) and the Dirichlet form (H, ) corresponds to a pure jump Lévy process with Lévy measure ν(dy) defined as above.F Finally, it is not hard to check that k(x, y) satisfies the conditions in (C2)–(C13) (for a subsequence n =2i, i N). i ∈ 22 A.MIMICA,N.SANDRIC,´ AND R.L. SCHILLING

Example 2.18 (Stable-like processes). Let α : Rd (0, 2) be a Borel measurable function. Consider the following integro-diﬀerential operator→ 1 dy (2.8) Lf(x) := γ(x) f(y + x) f(x) f(x),y B1(0)(y) α(x)+d Rd − − h∇ i y Z | | d where f C∞(R ) and ∈ c α(x) 1 Γ(α(x)/2+ d/2) γ(x) := α(x)2 − πd/2Γ(1 α(x)/2) − (Γ(x) is Euler’s Gamma function). It is well known that

i x,ξ α(x) ˆ Rd Lf(x)= e h i ξ f(ξ) dξ, f Cc∞( ), Rd | | ∈ Z ˆ d i ξ,x where f(ξ) := (2π)− Rd e− h if(x) dx denotes the Fourier transform of the function f. This shows that L = ( ∆)α(x) is a stable-like operator. If α satisfies a Hölder R condition, [Bas88] shows that− −L generates a unique “stable-like” Markov process (in dimension d = 1), the multivariate case is discussed by [Hoh00, Neg94] if α(x) is smooth and, recently, in [Küh16a] for α(x) satisfying a Hölder condition. Note that a stable- like process is, in general, non-symmetric. If α(x) α is constant, then L generates a rotationally invariant (hence, symmetric) α-stable Lévy≡ process. Assume that 0 <α α(x) α(x) α< 2 for all x Rd, and ≤ ≤ ≤ ∈ 1 (β(u) log u )2 | | du < , u1+α ∞ Z0 where β(u) := sup x y u α(x) α(y) . The (non-symmetric) kernel | − |≤ | − | α(x) d d k(x, y) := γ(x) y x − − , x,y R , | − | ∈ satisfies (C1) and defines a regular lower bounded semi-Dirichlet form on L2(Rd, dx); we call the corresponding Hunt process a stable-like process. For 0 < p 1, define the corresponding family of conductances Cn,p, n N, by (2.7). Clearly, for≤ any p (0, 1] such that 1/p α, the conductances Cn,p, n ∈N, satisfy (T1)–(T6). Thus, the∈ family ≥ n ∈ N of corresponding Markov chains Xt t 0, n , is tight. If there exist open balls B , B Rd such that { } ≥ ∈ 1 2 ⊆ inf α(x) α(y) > 0 (x,y) B1 B2 | − | ∈ × (this assumption implies (C2)), then it is easy to see with the above assumptions that i (C2)–(C6) hold true (for the subsequence ni = 2 , i N). It is also very easy to verify the conditions (C7) (i), (ii), (2.6) and (C9) (in the context∈ of Corollary 2.10). On the other hand, conditions (C8) and (C10) (again in the context of Corollary 2.10) follow directly from the dominated convergence theorem and Lebesgue’s differentiation theorem d n,p α(x) d in order to show limn n C ([x]n, [x]n + [y]n) = γ(x) y − − . Indeed, for n N, →∞ | | ∈ p (0, 1], 1/p α, x Rd, y B (0), [y] > 2√d/np, we have ∈ ≥ ∈ ∈ 1 | n| d n,p n C ([x]n, [x]n + [y]n)

2d γ(u) dv du = n α(u)+d [x¯] [x] +[y¯] v u Z n Z n n | − | 2d γ(u) dv du = n α(u)+d [x¯] [x]n u+[y¯] v Z n Z − n | | 2d dv du 2d dv du n γ α(u)+d + n γ α(u)+d ¯ ¯ c ≤ [x] B ([y]n) B1(0) v [x] B ([y]n) B (0) v Z n Z √d/n ∩ | | Z n Z √d/n ∩ 1 | | MARKOV CHAIN APPROXIMATION OF PURE JUMP PROCESSES 23

d dv d dv n γ α+d + n γ α+d ≤ B ([y] ) v B ([y] ) v Z √d/n n | | Z √d/n n | | α+d d/2 α d α+d d/2 α d = γ 2 d V y − − + γ 2 d V y − − d| | d| | α+d+1 d/2 α d γ 2 d V y − − , ≤ d| | where γ := supx Rd γ(x) and Vd denotes the volume of the d-dimensional unit ball. Sim- ∈ ilarly, for n N, p (0, 1], 1/p α, x Rd, y Bc(0), [y] > 2√d/np, we have ∈ ∈ ≥ ∈ ∈ 1 | n| d n,p α+d+1 d/2 α d n C ([x] , [x] + [y] ) γ 2 d V y − − . n n n ≤ d| | The assertion now follows by an application of the dominated convergence and Lebesgue’s diﬀerentiation theorems. Finally, let us verify (C11) (in the context of Corollary 2.10). We proceed as follows

y k(x, x + y) k(x, x y) ndCn,p([x] , [x] + [y] )+ ndCn,p([x] , [x] [y] ) dy | | − − − n n n n n − n ZB1(0)

= nd y k(u, v) dvdu k(u, v) dv du dy ¯ ¯ ¯ ¯ B1(0) B (0) | | [x] [y] +[x]n − [x] [y] +[x]n Z \ √d/np Z n Z n Z n Z− n

= nd y k(u,u + v) dv du ¯ ¯ B1(0) B p (0) | | [x] [y] +[x]n u Z \ √d/n Z n Z n −

k(u,u + v) dv du dy − [x¯] [y¯] +[x]n u Z n Z− n −

d d α(u) = n y γ(u) v + [x]n u − − ¯ ¯ B1(0) B (0) | | [x] [y] − Z \ √d/np Z n Z n d α(u) γ(u) v [x] + u − − dvdudy − − n d α d 1 2(d + α) γ n y [y]n √d/2n − − − dy [x]n u du ¯ ≤ B1(0) B (0) | | | | − [x] | − | Z \ 1 √d/2n Z n − d α d 1 + 2(d + α) γ n y [y]n √d/2n − − − dy [x]n u du B (0) B (0) | | | | − [x¯] | − | Z 1 √d/2n \ √d/np Z n − α+d+2 d α d 2 (d + α) γ n y − − dy u du ≤ | | | | B1(0) B1 √d/2n(0) B√d/2n(0) Z \ − Z α+d+2 d α d +2 (d + α) γ n y − − dy u du | | | | B1 √d/2n(0) B√d/np (0) B√d/2n(0) Z − \ Z α α α 2α+1 d(d+5)/2 (d + α) γV 2 √d − √d − √d − = d 1 1 + 1 , α (d + 1) n − 2n − np − − 2n " # where we use Taylor’s theorem in the fourth step. Thus, if 1/p > α, (C11) follows directly from the dominated convergence theorem.

3. Semimartingale approach We can improve the convergence results of the previous section if we know that the lim- iting process is a semimartingale. Let Xt t 0 be the canonical process on the Skorokhod d d { } ≥ d d space D(R ) and set (R ) := σ Xt : t 0 ; it is known that (R ) = (D(R )), see [JS03, Chapter VI]. TheD canonical{ ﬁltration≥ } on the measurable spaceD (D(RBd), (Rd)) is d d d D given by D(R )= t(R ) t 0, t(R ) := σ Xu : u s . {D } ≥ D s>t { ≤ } T 24 A.MIMICA,N.SANDRIC,´ AND R.L. SCHILLING

Let b : Rd Rd be a Borel function and ν : Rd (Rd 0 ) [0, ) a Borel kernel → 2 × B R\{d } → ∞ satisfying Rd 0 (1 y )ν(x,dy) < for every x . Fix a truncation function \{ } ∧| | ∞ ∈ h : Rd Rd (see Section 2.1 for the definition) and define → R t Bt := b(Xs) ds, 0 Z t ñ,ik At := hi(y)hk(y) ν(Xs,dy) ds, i,k =1, . . . , d, d Z0 ZR N(ds,dy) := ν(Xs,dy) ds. Finally, assume

d d d (C1.S) for each x R there is a unique probability Px( ) on (D(R ), (R )) such that ∈ • D (i) x P (B) is Borel measurable for all B (Rd); 7→ x ∈D (ii) Px(X0 = x) = 1; (iii) Xt t 0 is a semimartingale on the stochastic basis { } ≥ (D(Rd), (Rd), P , D(Rd)) with modiﬁed characteristics (B, A,˜ N). D x Note that we assume that X is a pure jump semimartingale, i.e. the continuous part – the characteristic A – vanishes.

d d d Remark 3.1. Condition (C1.S) implies that (D(R ), (R ), Px x Rd , D(R ), Xt t 0) is a Markov process with an extended generator, definedD for all{ f } ∈C2(Rd) by { } ≥ ∈ b d ∂f(x) d ∂f(x) f(x)= bi(x) + f(y + x) f(x) hi(x) ν(x,dy), A ∂x Rd − − ∂x i=1 i i=1 i X Z X see [JS03, Remark IX.4.5]. For Lévy processes (i.e. for the situation when b(x) and ν(x,dy) do not depend on x), (C1.S) is trivially satisfied. If

c 2 lim sup ν(x, Br (0)) = 0, sup b(x) < , sup (1 y )ν(x,dy) < r x Rd x Rd | | ∞ x Rd Rd ∧| | ∞ →∞ ∈ ∈ ∈ Z and (C2.S) (see below) hold, then there is at least one semimartingale Xt t 0 with the given characteristics (B, 0, N), cf. [JS03, Theorem IX.2.31]. { } ≥ d Conditions ensuring uniqueness of the family Px( ), x R , are given in [JS03, Theorem III.2.32]. In the context of Feller (or Lévy-type) processes· ∈ a different approach to existence and uniqueness of Xt t 0 can be found in [Jac05, Theorem 4.6.7] and [BSW13, Chapter 3]. { } ≥ n Zd Zd N 3.1. Convergence. Let C : n n [0, ), n , be a family of kernels satisfying (T1) and (T2). As we have already× seen→ in∞ Section∈ 2, these kernels define a family of n N Zd regular Markov semimartingales Xt t 0, n , on n. If we apply the results of [JS03, Theorem IX.4.8] in our setting, we{ arrive} ≥ at∈ the following theorem. n Zd Zd Theorem 3.2. Let C : n n [0, ) be as before, and assume that (C1.S) holds as well as × → ∞

(C2.S) the functions b(x), x Rd hi(y)hk(y)ν(x,dy) and x Rd g(y)ν(x,dy) are; continuous for all i, k7→= 1,...,d and all bounded and7→ continuous functions g : Rd R vanishing in aR neighbourhood of the origin R → c (C3.S) R> 0 : lim sup ν(x, Br (0)) = 0; ∀ r x B (0) →∞ ∈ R MARKOV CHAIN APPROXIMATION OF PURE JUMP PROCESSES 25

(C4.S) R> 0, i =1,...,d : ∀ n lim sup hi(b)C ([x]n, [x]n + b) bi(x) = 0; n x B (0) − →∞ ∈ R b Zd X∈ n

(C5.S) R> 0, i =1,...,d : ∀ n lim sup hi(b)hk(b)C ([x]n, [x]n + b) hi(y)hk(y) ν(x,dy) = 0; n − Rd →∞ x BR(0) Zd ∈ b n Z X∈ (C6.S) R> 0, f C (Rd, R), vanishing in a neighbourhood of 0: ∀ ∀ ∈ b n lim sup g(b)C ([x]n, [x]n + b) g(y) ν(x,dy) =0. n − Rd →∞ x BR(0) Zd ∈ b n Z X∈

n N If the initial distributions of Xt t 0, n , converge weakly to that of Xt t 0, then ≥ ≥ n d { } ∈ { } Xt Xt t 0 in Skorokhod space. t 0 n ≥ { } ≥ −−−→→∞ { } 3.2. Approximation. Using the same notation as in Section 2.2, for 0 < p 1, we deﬁne a family of kernels Cn,p : Zd Zd [0, ), n N, by ≤ n × n → ∞ ∈

ν(a, ¯b a), a b > √d (3.1) Cn,p(a, b) := − | − | np 0, a b √d . ( | − | ≤ np

The kernles Cn,p, n N, automatically satisfy (T1), the conditions (T2) and (T3) are ensured by ∈

c (T1.S) ρ> 0 : sup ν(x, Bρ(0)) < . ∀ x Rd ∞ ∈

n Hence, under (T1.S), there is a family of regular Markov semimartingales Xt t 0, n N, with (modiﬁed) characteristics of the form { } ≥ ∈

t n n B (h)t = h(a) ν(Xs , a¯) ds, 0 Zd Z a n X∈ n At =0, t ˜n ik n A (h)t = hi(a)hk(a) ν(Xs , a¯) ds, 0 Zd Z a n X∈ n n N (ds,b)= ν(Xs , a¯) ds

with some ﬁxed truncation function h : Rd Rd. Let us give suﬃcient conditions for (C4.S→)–(C6.S). 26 A.MIMICA,N.SANDRIC,´ AND R.L. SCHILLING

Proposition 3.3. (i) The condition (C4.S) will be satisﬁed if one of the following two conditions holds:

d (C4.S.1) there exists ρ > 0 such that ν(x,dy) is symmetric on Bρ(0) for all x R and ∈ c sup ν(x, Br(0)) < , r> 0, x B (0) ∞ ∈ R c lim sup ν(x, Br (0)) = 0, r x B (0) →∞ ∈ R c lim sup ν([x]n, ) ν(x, ) TV(Bε (0)) =0, ε> 0, n x B (0)k • − • k →∞ ∈ R

bi(x) = lim hi(y) ν(x,dy) ε 0 c ↓ ZBε(0) hold for all R > 0 and i = 1,...,d, where µ( ) TV(A) denotes the total variation of the signed measure µ( A); k • k • ∩ C4.S.2 ( ) c sup ν(x, Br(0)) < , for all r > 0, x B (0) ∞ ∈ R c lim sup ν(x, Br(0)) = 0, r x B (0) →∞ ∈ R lim ε sup ν(x, B1(0) Bεp (0)) = 0, ε 0 x B (0) \ ↓ ∈ R p lim ε sup ν x, B√dεp+(√d/2)ε(0) B√dεp (√d/2)ε(0) =0, ε 0 − ↓ x BR(0) \ ∈ lim sup y ν([x]n,dy) ν(x,dy) TV =0, n | |k − k →∞ x BR(0) B1(0) B√d/np √d/2n(0) ∈ Z \ − c lim sup ν([x]n, ) ν(x, ) TV(Bε (0)) =0, ε> 0, n x B (0)k • − • k →∞ ∈ R

lim sup hi(y) ν(x,dy) bi(x) =0 ε 0 x B (0) Bc(0) − ↓ ∈ R Z ε hold for all R> 0 and i =1,...,d.

(ii) The condition (C5.S) will be satisﬁed if

(C5.S.1) c sup ν(x, Br(0)) < , r> 0, x B (0) ∞ ∈ R c lim sup ν(x, Br(0)) = 0, r x B (0) →∞ ∈ R lim ε sup y ν(x,dy)=0, ε 0 x BR(0) B1(0) B p (0) | | ↓ ∈ Z \ ε lim sup y 2 ν(x,dy)=0, ε 0 x B (0) B (0) | | ↓ ∈ R Z ε 2 lim sup y ν([x]n,dy) ν(x,dy) TV =0, n x B (0) B (0) | | k − k →∞ ∈ R Z 1 lim sup ν([x]n, ) ν(x, ) TV(Bc(0)) =0, ε> 0, n ε →∞ x BR(0)k • − • k hold for∈ all R> 0. MARKOV CHAIN APPROXIMATION OF PURE JUMP PROCESSES 27

(iii) The condition (C6.S) will be satisﬁed if (C6.S.1) for any R> 0, c sup ν(x, Br (0)) < , r> 0, x B (0) ∞ ∈ R c lim sup ν(x, Br(0)) = 0, r x B (0) →∞ ∈ R c lim sup ν([x]n, ) ν(x, ) TV(Bε (0)) =0, ε> 0. n x B (0)k • − • k →∞ ∈ R Proof. Let us ﬁrst check (C4.S). For every i 1,...,d we have ∈{ } h (b)Cn,p([x] , [x] + b) b (x) = h (b)ν([x] , ¯b) b (x) i n n − i i n − i Zd p b n b >√d/n X∈ | | X

h (b) ν([x] , ¯b) ν(x, ¯b ) + h (b)ν(x, ¯b) b (x) ≤ i n − i − i b >√d/np b >√d/np | | X | | X

h (b) h (y) ν([x] ,dy ) ν(x,dy) ≤ i − i n − b >√d/np Z¯ | | X b

+ hi(y) ν([x]n,dy) ν(x,dy) hi(y) ν([x]n,dy) ν(x,dy) ¯b − − − b >√d/np Z Bc Z (0) | | X √d/np √d/2n −

+ h (y) ν([x] ,dy) ν(x,dy) + h (b) h (y) ν(x,dy) i n − i − i √ p Bc Z (0) b > d/n Z¯b √d/np √d/2n | | X −

+ hi(y) ν(x,dy) hi(y) ν(x,dy) ¯b − Bc (0) b >√d/np Z Z √d/np √d/2n | | X −

+ hi(y) ν(x,dy) bi(x) . Bc (0) − Z √d/np √d/2n − Pick 0 <η<ρ 1 such that h(y) = y for all y B (0) (recall that ρ > 0 appears in ∧ ∈ η (C4.S.1)). For all n N, np > 3√d/2η, we have that ∈ h (b)Cn,p([x] , [x] + b) b (x) i n n − i b Zd X∈ n

(bi yi) ν([x]n,dy) ν(x,dy) ≤ ¯b − − √d/np< b <η √d/2n Z |X| −

+ hi(b) hi(y) ν([x]n,dy) ν(x,dy) ¯b − − b η √d/2n Z | |≥ X−

+ yi ν([x]n,dy) ν(x,dy) B (0) B (0) − Z √d/np+√d/2n \ √d/np √d/2n −

+ yi ν([x]n,dy) ν(x,dy) + h ν([x]n, ) ν(x, ) TV(Bc (0)) ∞ η Bη(0) B (0) − k k k • − • k Z \ √d/np √d/2n −

+ (bi yi) ν(x,dy) + hi(b) hi(y) ν(x,dy) ¯b − ¯b − √d/np< b <η √d/2n Z b η √d/2n Z |X| − | |≥ X−

28 A.MIMICA,N.SANDRIC,´ AND R.L. SCHILLING

+ yi ν(x,dy) + hi(y) ν(x,dy) bi(x) . c B p (0) B p (0) B (0) − Z √d/n +√d/2n \ √d/n √d/2n Z √d/np √d/2n − −

Fix R> 0 and ε> 0. According to (C4.S.1) there is some r0 > 0 such that c sup ν(x, Br(0)) <ε, r r0. x B (0) ≥ ∈ R ¯ Since hi(y) is uniformly continuous on the compact set B2r0 (0)), there is some δ > 0 such that h (y ) h (y ) < ε for all y ,y B¯ (0), y y < δ. Now, for n N, | i 1 − i 2 | 1 2 ∈ 2r0 | 1 − 2| ∈ np > 3√d/2η √d/2δ, we have that ∨ h (b)Cn,p([x] , [x] + b) b (x) i n n − i b Zd X∈ n

(bi yi) ν([x]n,dy) ν(x,dy) ≤ ¯b − − √d/np< b <η √d/2n Z |X| −

+ ε ν([x]n, ) ν(x, ) TV(B2r (0) B (0)) +4ε h k • − • k 0 \ η/2 k k∞

+ yi ν([x]n,dy) ν(x,dy) B (0) B (0) − Z √d/np+√d/2n \ √d/np √d/2n −

+ yi ν([x]n,dy) ν(x,dy) + h ν([x]n, ) ν(x, ) TV(Bc (0)) ∞ η Bη(0) B p (0) − k k k • − • k Z \ √d/n √d/2n −

+ (bi yi) ν(x,dy) + εν x, B 2r0 (0) Bη/2(0) +2ε h ¯b − \ k k∞ √d/np< b <η √d/2n Z |X| −

+ yi ν(x,dy) + hi(y) ν(x,dy) bi(x) . c B p (0) B p (0) B (0) − Z √d/n +√d/2n \ √d/n √d/2n Z √d/np √d/2n − −

This shows that (C4.S.1) implies (C4.S). Further, we have that

h (b)Cn,p([x] , [x] + b) b (x) i n n − i b Zd X∈ n √d ν([x] , ) ν(x, ) n TV(Bη (0) B√d/np √d/2n(0)) ≤ 2n k • − • k \ −

+ ε ν([x]n, ) ν(x, ) TV(B2r (0) B (0)) +4ε h k • − • k 0 \ η/2 k k∞ + y ν([x] ,dy) ν(x,dy) | |k n − kTV B√d/np+√d/2n(0) B√d/np √d/2n(0) Z \ − + y ν([x] ,dy) ν(x,dy) | |k n − kTV Bη(0) B√d/np √d/2n(0) Z \ − + h ν([x]n, ) ν(x, ) TV(Bc (0)) k k∞k • − • k η √d + ν x, Bη(0) B√d/np √d/2n(0) + εν x, B2r0 (0) Bη/2(0) +2ε h 2n \ − \ k k∞ 3√d + p ν x, B√d/np+√d/2n(0) B√d/np √d/2n(0) + hi(y) ν(x,dy) bi(x) . 2n \ − Bc (0) − Z √d/np √d/2n −

Thus, (C4.S.2) implies (C4.S), too. MARKOV CHAIN APPROXIMATION OF PURE JUMP PROCESSES 29

Now we turn to (C5.S). Fix i, k 1,...,d . We have ∈{ }

n hi(b)hk(b)C ([x]n, [x]n + b) hi(y)hk(y) ν(x,dy) − Rd Zd b n Z X∈

= hi(b)hk(b)ν([x]n, ¯b) hi(y)hk(y) ν(x,dy) − Rd b >√d/np Z | | X

h (b)h (b)ν([x] , ¯b) h (b)h (b)ν(x, ¯b) ≤ i k n − i k b >√d/np b >√d/np | | X | | X

+ hi(b)hk(b) hi(y)hk(y) ν(x,dy) + hi(y)hk(y) ν(x,dy) . ¯b − ¯b b >√d/np Z b √d/np Z | | X | |≤X

Pick 0 <η< 1 be such that h(y)= y for all y B (0). Thus, for all n N, np > 3√d/2η, ∈ η ∈

n hi(b)hk(b)C ([x]n, [x]n + b) hi(y)hk(y) ν(x,dy) − Rd Zd b n Z X∈

2 bi bk ν([x]n, ¯b) ν(x, ¯b) +2 h ν([x]n, ¯b) ν(x, ¯b) ≤ | || | − k k∞ − √d/np< b <η √d/2n b η √d/2n |X| − | |≥ X−

+ (bibk yiyk) ν(x,dy) ¯b − √d/np< b <η √d/2n Z |X| −

+ hi(b)hk(b) hi(y)hk(y) ν(x,dy) ¯b − b η √d/2n Z | |≥ X−

+ yi yk ν(x,dy) B (0) | || | Z √d/np+√d/2n 2 2 4 y ν([x]n,dy) ν(x,dy) TV +2 h ν([x]n, ) ν(x, ) TV(Bc (0)) ≤ | | k − k k k∞k • − • k η/2 ZBη(0) 3 + y ν(x,dy) 2n | | Bη(0) B√d/np √d/2n(0) Z \ −

+ hi(b)hk(b) hi(y)hk(y) ν(x,dy) ¯b − b η √d/2n Z | |≥ X−

+ y 2 ν(x,dy). B (0) | | Z √d/np+√d/2n

In the ﬁnal step we use the elementary estimates bibk yiyk (3/2n) y and bi bk 4 y 2 for all y ¯b. | − | ≤ | | | || | ≤ | | ∈ Fix R> 0 and ε> 0. As before (C5.S.1) shows that there are r0 > 0 and δ > 0, such that

c sup ν(x, Br(0)) <ε, r r0, x B (0) ≥ ∈ R 30 A.MIMICA,N.SANDRIC,´ AND R.L. SCHILLING and h (y )h (y ) h (y )h (y ) <ε for all y ,y B¯ (0), y y < δ. Consequently, | i 1 k 1 − i 2 k 2 | 1 2 ∈ 2r0 | 1 − 2| for any n N, np > 3√d/2η √d/2δ, ∈ ∨ n hi(b)hk(b)C ([x]n, [x]n + b) hi(y)hk(y) ν(x,dy) − Rd Zd b n Z X∈ 2 2 4 y ν([x]n,dy) ν(x,dy) TV +2 h ν([x]n, ) ν(x, ) TV(Bc (0)) ≤ | | k − k k k∞k • − • k η/2 ZBη(0) 2√d 2 + y ν(x,dy)+ εν x, B2r0 Bη(0) +2ε h n Bη(0) B p (0) | | \ k k∞ Z \ √d/n √d/2n − + y 2 ν(x,dy). B (0) | | Z √d/np+√d/2n Therefore, (C5.S) is a direct consequence of (C5.S.1). Let us, ﬁnally, discuss (C6.S). Fix any bounded continuous function g : Rd R which → vanishes, for some η > 0, on B (0). For all n N, np > √d/2η, we have η ∈

n g(b)C ([x]n, [x]n + b) g(y) ν(x,dy) − Rd Zd b n Z X∈

¯ = g(b)ν([x]n, b) g(y) ν(x,dy) − Rd b >√d/nd Z | | X

g(b)ν([x] , ¯b) g(b)ν(x, ¯b) ≤ n − b >√d/np b >√d/np | | X | | X

+ g(b)ν(x, ¯b) g(y) ν(x,dy) − Rd b >√d/np Z | | X

g ν([x] , ) ν(x, ) c + g(b) g(y) ν(x,dy) . n TV(Bη/2(0)) ≤k k∞k • − • k ¯b − b >η √d/2n Z | | X−

Fix R> 0 and ε> 0, and pick, as before, r0 > 0 and δ > 0 such that

c sup ν(x, Br (0)) <ε, r r0 x B (0) ≥ ∈ R and g(y ) g(y ) < ε for all y ,y B¯ (0), y y < δ. Thus, for all n N, | 1 − 2 | 1 2 ∈ 2r0 | 1 − 2| ∈ np > √d/2η √d/2δ, ∨ n g(b)C ([x]n, [x]n + b) g(y) ν(x,dy) − Rd Zd b n Z X∈ c g ν([x]n, ) ν(x, ) TV(B (0)) + εν x, B2r0 (0) Bη(0) +2ε g , ≤k k∞k • − • k η/2 \ k k∞ which, in view of (C6.S.1), concludes the proof.

2 Rd In the following we discuss the situation when Rd y ν(x,dy) < for all x . The following result is a direct consequence of [JS03, Theorem| | IX.4.15].∞ ∈ R n Zd Zd N Theorem 3.4. Let C : n n [0, ), n , be a family of kernels satisfying n × → N ∞ ∈ (T1) and (T2), and let Xt t 0, n , be the corresponding family of regular Markov { } ≥ ∈ MARKOV CHAIN APPROXIMATION OF PURE JUMP PROCESSES 31

Zd 2 Rd semimartingales on n. Assume that Rd y ν(x,dy) < for all x and that the following conditions hold: (C1.S) and | | ∞ ∈ R (C2.S2) the functions x b (x)+ (y h (y))ν(x,dy), x y y ν(x,dy), and 7→ i Rd i − i 7→ Rd i k x g(y)ν(x,dy) are continuous for all i, k =1,...,d and all bounded and 7→ Rd R R continuous functions g : Rd R vanishing in a neighbourhood of the origin; R → (C3.S2) for all R> 0, lim sup y 2 ν(x,dy) = 0; r x B (0) Bc(0) | | →∞ ∈ R Z r (C4.S2) for all R> 0 and all i =1,...,d,

n lim sup biC ([x]n, [x]n + b) bi(x) (yi hi(y)) ν(x,dy) = 0; n − − Rd − →∞ x BR(0) Zd ∈ b n Z X∈ 2 (C5.S ) for all R> 0 and all i =1,...,d,

n lim sup bibkC ([x]n, [x]n + b) yiyk ν(x,dy) = 0; n − Rd →∞ x BR(0) Zd ∈ b n Z X∈ 2 R d R (C6.S ) for all R> 0 and all bounded and continuous functions g : vanishing in a neighbourhood of the origin, →

n lim sup g(b)C ([x]n, [x]n + b) g(y) ν(x,dy) =0. n − Rd →∞ x BR(0) Zd ∈ b n Z X∈ n N If the initial distributions of Xt t 0, n , converge weakly to that of Xt t 0, then { } ≥ ∈ { } ≥ n d Xt Xt t 0 t 0 n ≥ { } ≥ −−−→→∞ { } in Skorokhod space. If we use the discretisation (3.1) of k(x, y), the proof of Proposition 3.3 applies and yields Proposition 3.5. (i) The condition (C4.S2) will be satisﬁed if either (C4.S.1) or (C4.S.2) and (C2.S2.1) for all R> 0,

lim sup y ν([x]n,dy) ν(x,dy) TV =0 n x B (0) B (0) | |k − k →∞ ∈ R Z 1 hold true. (ii) The condition (C5.S2) will be satisﬁed if (C5.S2.1) for all R> 0

lim ε sup y ν(x,dy)=0, ε 0 x B (0) Bc (0) | | ↓ ∈ R Z εp lim sup y 2 ν(x,dy)=0, ε 0 x B (0) B (0) | | ↓ ∈ R Z ε 2 lim sup y ν([x]n,dy) ν(x,dy) TV =0. n x B (0) Rd | | k − k →∞ ∈ R Z In the following proposition we discuss tightness conditions (T4)-(T6). 32 A.MIMICA,N.SANDRIC,´ AND R.L. SCHILLING

Proposition 3.6. Assume that (T1.S) holds. The discretisation deﬁned in (3.1) satisﬁes the conditions (T4)–(T6) if the following conditions hold: c (T2.S) lim sup ν(x, Br(0)) = 0; r x Rd →∞ ∈ (T3.S) either there exists some ρ> 0 such that ν(x,dy) is symmetric on Bρ(0) for all x Rd, or there exists some ρ> 0 such that for all i =1,...,d ∈

lim sup sup yi ν(x,dy) < , d ε 0 x R Bρ(0) B p (0) ∞ ↓ ∈ Z \ ε

lim sup ε sup ν (x, Bρ(0) Bεp (0)) < ; ε 0 x Rd \ ∞ ↓ ∈ (T4.S) there exists some (alternatively: for all) ρ> 0 such that

lim sup y 2 ν(x,dy) < . ε 0 d x R Bρ(0) Bε(0) | | ∞ ↓ ∈ Z \ In the remaining part of this section we discuss some examples satisfying (C1.S)– (C6.S).

Example 3.7 (Lévy processes). Let Xt t 0 be a Lévy process with semimartingale characteristics (Lévy triplet) (b, 0, ν(dy{)) with} ≥ respect to some truncation function h(x). Then, the conditions in (C1.S)–(C3.S) and, for the discretization (3.1) with p < 1/2, and (C4.S.1)–(C6.S.1) are trivially satisfied. Example 3.8 (Stable-like processes). Let α : Rd (0, 2) be a continuously differen- tiable function with bounded derivatives, such that 0→<α = α(x)= α< 2 for all x Rd. Under these assumptions, it has been shown in [Bas88, Hoh00, Küh16a, Neg94], [Sch98∈ , Theorem 3.5] and [SW13, Theorem 3.3] that the operator (2.7) defines a stable-like process, i.e. a unique Feller process Xt t 0 which is also a semimartingale. Its (modified) characteristics (with respect to a{ symmetric} ≥ truncation function h(x)) are of the form

B(h)t =0,

At =0, t ˜i,k dy At = hi(y)hk(y) d+α(X ) ds, Rd y s Z0 Z | | dy ds N(ds,dy)= . y d+α(Xs) | | If α(x) α (0, 2) is constant, then we have a rotationally invariant α-stable L´evy ≡ ∈ process. Clearly, Xt t 0 satisﬁes (C1.S)–(C3.S). Using the discretisation (3.1) with 0 < p 1, the continuity{ } ≥ of α(x) and the dominated convergence theorem ensure that (C4.S.1≤)–(C6.S.1) hold, too.

Example 3.9 (Lévy-driven SDEs). Let Lt t 0 be an n-dimensional Lévy process and d d n { } ≥ Φ: R R × be bounded and locally Lipschitz continuous. The SDE → d dXt = Φ(Xt )dLt, X0 = x R , − ∈ admits a unique strong solution which is a Feller semimartingale, see [Küh16b] or [SS10, Theorem 3.5]. In particular, if

(i) Lt = (lt, t), t 0, where lt t 0 is a d-dimensional Lévy process determined by the Lévy triplet≥ (0, 0, ν(dy{))} such≥ that the Lévy measure ν(dy) is symmetric; (ii) Φ(x)=(φ(x)I, 0), x Rd, where φ : Rd R is locally Lipschitz continuous and ∈ → 0 < infx Rd φ(x) supx Rd φ(x) < , and I and 0 are the d d identity matrix and the ∈d-dimensional| | ≤ column∈ | vector| ∞ of zeros, × MARKOV CHAIN APPROXIMATION OF PURE JUMP PROCESSES 33

then, according to [Sch98, Theorem 3.5], Xt t 0 is a d-dimensional Feller semimartingale which is determined by the (modiﬁed){ characteristics} ≥ (with respect to a symmetric truncation function h(x)) of the form

B(h)t =0,

At =0, t ˜i,k At = hi(y)hk(y) ν (dy/ φ(Xs) ) ds, Rd | | Z0 Z N(ds,dy)= ν dy/ φ(X ) . | s | It is obvious, that the solution Xt t 0 satisfies (C1.S)–(C3.S). For the discretization (3.1), the boundedness (away from{ zero} ≥ and infinity) of φ(x), allows to prove the first two and the fourth condition in (C4.S.1) and (C5.S.1) as well as the first two conditions in (C6.S.1). Hence, it remains to prove that

c lim sup ν ( / φ([x]n) ) ν ( / φ(x) ) TV(Bε (0)) =0, ε> 0, n x B (0)k • | | − • | | k →∞ ∈ R 2 lim sup y ν (dy/ φ([x]n) ) ν (dy/ φ(x) ) TV =0, n x B (0) B (0) | | k | | − | | k →∞ ∈ R Z 1 and

lim ε sup y ν (dy/ φ(x) )=0, ε 0 x BR(0) B1(0) B p (0) | | | | ↓ ∈ Z \ ε hold for all R> 0. To prove the ﬁrst relation we proceed as follows. Fix R > 0 and ε > 0. Due to the continuity and boundedness (away from zero and inﬁnity) of φ(x) we can use dominated convergence theorem to see that the function x ν (Bc(0)/ φ(x) ) 7→ ε | | is (uniformly) continuous on any ball BR(0); this implies the desired relation. Essentially the same argument implies that for every R> 0, the function

x y 2 ν (dy/ φ(x) ) 7→ | | | | ZB1(0) is (uniformly) continuous on BR(0). Finally, we check the last relation. Fix R> 0 and 0

lim ε sup y ν (dy/ φ(x) ) ε 0 x BR(0) B1(0) B p (0) | | | | ↓ ∈ Z \ ε 1 p p = lim ε − sup ε y ν (dy/ φ(x) ) ε 0 x BR(0) B1(0) B p (0) | | | | ↓ ∈ Z \ ε 1 p 2 lim ε − sup y ν (dy/ φ(x) ) ε 0 ≤ x BR(0) B1(0) B p (0) | | | | ↓ ∈ Z \ ε 1 p 2 lim ε − sup y ν (dy/ φ(x) )=0. ≤ ε 0 x B (0) B (0) | | | | ↓ ∈ R Z 1

Therefore, Xt t 0 satisﬁes (C1.S)–(C6.S) if we use the discretisation (3.1) with 0 < p< 1. { } ≥ 34 A.MIMICA,N.SANDRIC,´ AND R.L. SCHILLING

Acknowledgement. Financial support through the Croatian Science Foundation (under Project 3526) (for Ante Mimica), the Croatian Science Foundation (under Project 3526) and the Dresden Fellowship Programme (for Nikola Sandri´c) is gratefully acknowledged.

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(Mimica) Ante Mimica, *20-Jan-1981 – 9-Jun-2016, https://web.math.pmf.unizg.hr/~amimica/ † (Sandri´c) Faculty of Civil Engineering, University of Zagreb, 10000 Zagreb, Croatia E-mail address: [email protected]

(Schilling) Institut fur¨ Mathematische Stochastik, Fachrichtung Mathematik, TU Dres- den, 01062 Dresden, Germany E-mail address: [email protected]