L p -solutions of Reflected Backward Doubly Stochastic Differential Equations
aa r X i v : . [ m a t h . P R ] J a n L p -Solutions of Reflected Backward DoublyStochastic Differential Equations Wen L¨u ∗† School of Mathematics, Shandong University, Jinan,
School of Mathematics, Yantai University, Yantai 264005, China
Abstract
In this paper, we deal with a class of one-dimensional reflectedbackward doubly stochastic differential equations with one continuouslower barrier. We derive the existence and uniqueness of L p -solutionsfor those equations with Lipschitz coefficients. Keywords:
Reflected backward doubly stochastic differential equation;Lipschitz coefficient; L p -solution AMS 2000 Subject Classification:
The general nonlinear case backward stochastic differential equation (BSDEin short) was first introduced in Pardoux and Peng (1990), who proved theexistence and uniqueness result when the coefficient is Lipschitz. El Karouiet al. (1997a) introduced the notion of one barrier reflected BSDE , which isactually a backward equation but the solution is forced to stay above a givenbarrier. This type of BSDEs is motivated by pricing American options (seeEl Karoui et al. (1997b)) and studying the mixed game problems (see e.g.Cvitani´c and Karatzas (1996), Hamad`ene and Lepeltier (2000)). In order ∗ Support by the National Basic Research Program of China (973 Program) grant No.2007CB814900 and The Youth Fund of Yantai University (SX08Z9). † Email address: [email protected]
1o give a probabilistic representation for a class of quasilinear stochasticpartial differential equations, Pardoux and Peng (1992) first considered aclass of backward doubly stochastic differential equations (BDSDEs) withtwo different directions of stochastic integrals.However in most of the previous works, solutions are taken in L spaceor in L p , p >
2. This limits the scope for several applications. To correct thisshortcoming, El Karoui et al. (1997c) obtained the first result on the exis-tence and uniqueness of solution in L p , p ∈ (1 ,
2) with a Lipschitz coefficient.Briand et al. (2003) generalized this result to the BSDEs with monotonecoefficients. Following this way, Aman (2009) considered the L p -solutionsof BDSDEs with a monotone coefficient. Moreover, Hamad`ene and Popier(2008) established the existence and uniqueness of the L p -solutions of BSDEswith reflection having a Lipschitz coefficient.More recently, Bahalai et al. (2009) obtained the existence and unique-ness of solution for BDSDEs with one continuous lower barrier, having acontinuous coefficient. Motivated by above works, the purpose of this paperis to prove the existence and uniqueness of L p -solutions for reflected BDSDEswith Lipschitz coefficients.The rest of the paper is organized as follows. In Section 2, we introducesome preliminaries including some spaces. With the help of some a prioriestimates, Section 3 is devoted to the existence and uniqueness of L p -solutionsfor those equations. Let
T > { W t } t ≥ , { B t } t ≥ be two mutually inde-pendent standard Brownian motions defined on a complete probability space(Ω , F , P ) with values in R d and R , respectively. For t ∈ [0 , T ], we define F t = F Wt ∨ F Bt,T , where F Wt = σ { W s , ≤ s ≤ t } , F Bt,T = σ { B s − B t , t ≤ s ≤ T } completedwith the P -null sets. We note that the collection {F t ; t ∈ [0 , T ] } is neitherincreasing nor decreasing, so it does not constitute a classical filtration. TheEuclidean norm of a vector y ∈ R n will be defined by | y | .Throughout the paper, we always assume that p ∈ (1 , M pd = { ψ : [0 , T ] × Ω → R d , predictable, such that E [( R T | ψ s | ds ) p ] < ∞} ;2 p = { ψ : [0 , T ] × Ω → R , progressively measurable, s.t. E (sup t ∈ [0 ,T ] | ψ t | p ) < ∞} ; S pci = { A : [0 , T ] × Ω → R + , continuous, increasing, s.t. A = 0 and E | A T | p < ∞} .The object in this paper is the following reflected BDSDE: Y t = ξ + R Tt f ( s, Y s , Z s ) ds + R Tt g ( s, Y s , Z s ) dB s + K T − K t − R Tt Z s dW s ,Y t ≥ L t , ≤ t ≤ T a.s. and R T ( Y t − L t ) dK t = 0 , a.s. (1)where the dW is a standard forward Itˆo integral and the dB is a backwardItˆo integral.On the items ξ, f, g and L , we make the following assumptions:( H1 ) The terminal condition ξ : Ω → R , F T -measurable such that E | ξ | p < ∞ ;( H2 ) the functions f, g : [0 , T ] × Ω × R × R d → R are jointly measurableand satisfy:(i) E [( R T | f s | ds ) p ] < ∞ , E [( R T | g s | ds ) p ] < ∞ ,where f s =: f ( s, , , g s =: g ( s, , ∀ t ∈ [0 , T ] , ( y , z ) , ( y , z ) ∈ R × R d , there exist constants C > < α < | f ( t, y , z ) − f ( t, y , z ) | ≤ C ( | y − y | + | z − z | ) , | g ( t, y , z ) − g ( t, y , z ) | ≤ C | y − y | + α | z − z | ;( H3 ) The barrier { L t , t ∈ [0 , T ] } is a real valued progressively measurableprocess such that E (sup ≤ t ≤ T ( L + t ) p ) < ∞ and L T ≤ ξ a.s..Let’s give the notion of L p -solution of reflected BDSDE (1). Definition 2.1 An L p -solution of the reflected BDSDE (1) is a triple of pro-gressively measurable processes ( Y, Z, K ) satisfying (1) such that ( Y, Z, K ) ∈S p × M pd × S pci . The following lemma is a slight generalization of Corollary 2.3 in Briandet al. (2003).
Lemma 2.1
Let ( Y, Z ) ∈ S p × M pd is a solution of the following BDSDE : | Y t | = ξ + Z Tt e f ( s, Y s , Z s ) ds + Z Tt e g ( s, Y s , Z s ) dB s + A T − A t − Z Tt Z s dW s , here:(i) e f and e g are functions which satisfy the assumptions as f and g ,(ii) P -a.s. the process ( A t ) t ∈ [0 ,T ] is of bounded variation type.Then for any ≤ t ≤ u ≤ T , we have | Y t | p + c ( p ) Z Tt | Y s | p − { Y s =0 } | Z s | ds ≤ | Y u | p + p Z Tt | Y s | p − b Y s dA s + p Z Tt | Y s | p − e f ( s, Y s , Z s ) ds + c ( p ) Z Tt | Y s | p − { Y s =0 } | e g ( s, Y s , Z s ) | ds + p Z Tt | Y s | p − b Y s e g ( s, Y s , Z s ) dB s − p Z Tt | Y s | p − b Y s Z s dW s , where c ( p ) = p ( p − and b y = y | y | { y =0 } . In order to obtain the existence and uniqueness result for solution of thereflected BDSDE (1), we first provide some a priori estimates of solution of(1).In what follows, d, d , d , · · · will be denoted as a constant whose valuedepending only on C, α, p and possibly T . We also denote by θ , θ , · · · theconstants which taking value in (0 , ∞ ) arbitrarily. Lemma 3.1
Let the assumptions (H1)-(H3) hold and let ( Y, Z, K ) be a so-lution of the reflected BDSDE (1). If Y ∈ S p then Z ∈ M pd and there existsa constant d > such that E "(cid:18)Z T | Z s | ds (cid:19) p ≤ d E " sup t ∈ [0 ,T ] | Y t | p + (cid:18)Z T | f s | ds (cid:19) p + (cid:18)Z T | g s | ds (cid:19) p . Proof.
For each integer n ≥
0, let’s define the stopping time τ n = inf { t ∈ [0 , T ] , Z t | Z s | ds ≥ n } ∧ T. a ∈ R , using Itˆo’s formula and assumption (H2), we get | Y | + Z τ n e as | Z s | ds = e aτ n | Y τ n | − a Z τ n e as | Y s | ds + 2 Z τ n e as Y s f ( s, Y s , Z s ) ds + Z τ n e as | g ( s, Y s , Z s ) | ds + 2 Z τ n e as Y s dK s +2 Z τ n e as Y s g ( s, Y s , Z s ) dB s − Z τ n e as Y s Z s dW s ≤ e aτ n | Y τ n | − a Z τ n e as | Y s | ds + 1 θ Z τ n e as | Y s | ds + θ Z τ n e as [4 C ( | Y s | + | Z s | ) + 2 | f s | ] ds +(1 + θ ) Z τ n e as (cid:0) C | Y s | + α | Z s | (cid:1) ds + (1 + 1 θ ) Z τ n e as | g s | ds + 1 θ sup t ∈ [0 ,τ n ] e at | Y t | + θ | K τ n | +2 Z τ n e as Y s g ( s, Y s , Z s ) dB s − Z τ n e as Y s Z s dW s . On the other hand, from the equation K τ n = Y − Y τ n − Z τ n f ( s, Y s , Z s ) ds − Z τ n g ( s, Y s , Z s ) dB s + Z τ n Z s dW s , we have | K τ n | ≤ d (cid:20) | Y | + | Y τ n | + ( Z τ n | f s | ds ) + Z τ n ( | Y s | + | Z s | ) ds + | Z τ n g ( s, Y s , Z s ) dB s | + | Z τ n Z s dW s | (cid:21) . − d θ ) | Y | + (1 − C θ − α (1 + θ )) Z τ n e as | Z s | ds − d θ Z τ n | Z s | ds ≤ ( d θ + e aτ n ) | Y τ n | + (cid:18) θ + 4 C θ + C (1 + θ ) − a (cid:19) Z τ n e as | Y s | ds +2 θ Z τ n e as | f s | ds + (1 + 1 θ ) Z τ n e as | g s | ds + 1 θ sup t ∈ [0 ,τ n ] e at | Y t | + d θ (cid:20)Z τ n | f s | ds + Z τ n | Y s | ds + | Z τ n g ( s, Y s , Z s ) dB s | + | Z τ n Z s dW s | (cid:21) +2 | Z τ n e as Y s g ( s, Y s , Z s ) dB s | + 2 | Z τ n e as Y s Z s dW s | . Choosing now θ , θ small enough and a > θ + 4 C θ + C (1 + θ ) − a < , we obtain Z τ n | Z s | ds ≤ d sup t ∈ [0 ,τ n ] | Y t | + Z τ n e as | g s | ds + Z τ n e as | f s | ds + | Z τ n e as Y s g ( s, Y s , Z s ) dB s | + | Z τ n e as Y s Z s dW s | + θ | Z τ n g ( s, Y s , Z s ) dB s | + θ | Z τ n Z s dW s | (cid:19) , (2)it follows that E (cid:18)Z τ n | Z s | ds (cid:19) p ≤ d E " sup t ∈ [0 ,τ n ] | Y t | p + (cid:18)Z τ n | g s | ds (cid:19) p + (cid:18)Z τ n | f s | ds (cid:19) p + (cid:18) | Z τ n e as Y s g ( s, Y s , Z s ) dB s | (cid:19) p + (cid:18) | Z τ n e as Y s Z s dW s | (cid:19) p + θ p | Z τ n g ( s, Y s , Z s ) dB s | p + θ p | Z τ n Z s dW s | p (cid:21) .
6y the Burkh¨older-Davis-Gundy and Young’s inequalities, we have E "(cid:12)(cid:12)(cid:12)(cid:12)Z τ n e as Y s g ( s, Y s , Z s ) dB s (cid:12)(cid:12)(cid:12)(cid:12) p ≤ d E "(cid:18)Z τ n | Y s | | g ( s, Y s , Z s ) | ds (cid:19) p ≤ d E " ( sup t ∈ [0 ,τ n ] | Y t | ) p (cid:18)Z τ n | g ( s, Y s , Z s ) | ds (cid:19) p ≤ ( d θ + θ ) E [ sup t ∈ [0 ,τ n ] | Y t | p ]+ θ E "(cid:18)Z τ n | g s | ds (cid:19) p + (cid:18)Z τ n | Z s | ds (cid:19) p and E "(cid:12)(cid:12)(cid:12)(cid:12)Z τ n e as Y s Z s dW s (cid:12)(cid:12)(cid:12)(cid:12) p ≤ d θ E " sup t ∈ [0 ,τ n ] | Y t | p + θ E "(cid:18)Z τ n | Z s | ds (cid:19) p . Plugging the two last inequalities in the previous one and using theBurkh¨older-Davis-Gundy inequality once again, it follows after choosing θ , θ small enough (s.t. (2) holds too): E "(cid:18)Z τ n | Z s | ds (cid:19) p ≤ d E " sup t ∈ [0 ,τ n ] | Y t | p + (cid:18)Z τ n | f s | ds (cid:19) p + (cid:18)Z τ n | g s | ds (cid:19) p . Finally, we get the desired result by Fatou’s Lemma. (cid:3)
Lemma 3.2
Assume that (H1)-(H3) hold, let ( Y, Z, K ) be a solution of thereflected BDSDE (1) where Y ∈ S p . Then there exists a constant d > such hat E " sup t ∈ [0 ,T ] | Y t | p + (cid:18)Z T | Z s | ds (cid:19) p + | K T | p ≤ d E " | ξ | p + (cid:18)Z T | f s | ds (cid:19) p + (cid:18)Z T | g s | ds (cid:19) p + sup t ∈ [0 ,T ] ( L + t ) p + Z T | Y s | p − I { Y s =0 } | g s | ds . Proof.
From Lemma 2.1, for any a ∈ R and any 0 ≤ t ≤ T , we have e apt | Y t | p + c ( p ) Z Tt e aps | Y s | p − { Y s =0 } | Z s | ds ≤ e apT | ξ | p + p Z Tt e aps | Y s | p − b Y s f ( s, Y s , Z s ) ds + p Z Tt e aps | Y s | p − b Y s dK s + c ( p ) Z Tt e aps | Y s | p − { Y s =0 } | g ( s, Y s , Z s ) | ds − ap Z Tt e aps | Y s | p ds + p Z Tt e aps | Y s | p − b Y s g ( s, Y s , Z s ) dB s − p Z Tt e aps | Y s | p − b Y s Z s dW s . (3)By assumption (H2) and Young’s inequality, we obtain p E (cid:20)Z Tt e aps | Y s | p − b Y s f ( s, Y s , Z s ) ds (cid:21) ≤ E [ p Z Tt e aps | Y s | p − | f s | ds + Cp Z Tt e aps | Y s | p − ( | Y s | + | Z z | ) ds ] ≤ ( p − θ pp − E sup s ∈ [0 ,T ] | Y s | p ! + θ − p E (cid:18)Z Tt e aps | f s | ds (cid:19) p +( Cp + p C c ( p ) θ ) E (cid:20)Z Tt e aps | Y s | p ds (cid:21) + c ( p )2 θ E (cid:20)Z Tt e aps | Y s | p − { Y s =0 } | Z s | ds (cid:21) (4)8nd c ( p ) E [ Z Tt e aps | Y s | p − { Y s =0 } | g ( s, Y s , Z s ) | ds ] ≤ c ( p )(1 + 1 θ ) E [ Z Tt e aps | Y s | p − { Y s =0 } | g s | ds ]+ c ( p ) C (1 + θ ) E [ Z Tt e aps | Y s | p ds ]+ c ( p ) α (1 + θ ) E [ Z Tt e aps | Y s | p − I { Y s =0 } | Z s | ds ] . (5)Moreover, since dK s = { Y s ≤ L s } dK s , we get from Young’s inequality p E [ Z Tt e aps | Y s | p − b Y s dK s ] ≤ p E [ Z Tt e aps | Y s | p − b Y s I { Y s ≤ L s } dK s ] ≤ p E [ Z Tt e aps | L s | p − c L s dK s ] ≤ p E [( sup s ∈ [0 ,T ] L + s ) p − Z Tt e aps dK s ] ≤ p − θ p − p E [ sup s ∈ [0 ,T ] ( L + s ) p ] + θ p E ( Z Tt e aps dK s ) p ≤ d " θ − p − p E ( sup s ∈ [0 ,T ] ( L + s ) p ) + θ p E | K T | p . On the other hand, by assumption (H2), the Burkh¨older-Davis-Gundy in-equality and Lemma 3.1, we have E | K T | p ≤ d E " sup s ∈ [0 ,T ] | Y s | p + (cid:18)Z T | f s | ds (cid:19) p + (cid:18)Z T | g s | ds (cid:19) p , (6)it follows that p E [ Z Tt e aps | Y s | p − b Y s dK s ] ≤ d E " θ p sup s ∈ [0 ,T ] | Y s | p + θ − p − p sup s ∈ [0 ,T ] ( L + s ) p + θ p (cid:18)Z T | f s | ds (cid:19) p + θ p (cid:18)Z T | g s | ds (cid:19) p (cid:21) . (7)9ombining (4)-(6), taking expectation on both sides of (3) to obtain E (cid:20) e apt | Y t | p + c ( p ) Z Tt e aps | Y s | p − I { Y s =0 } | Z s | ds (cid:21) ≤ E ( e apT | ξ | p + d θ − p − p sup s ∈ [0 ,T ] ( L + s ) p + h ( p − θ pp − + d θ p i sup s ∈ [0 ,T ] | Y s | p + (cid:20) ( Cp + p C c ( p ) θ ) + c ( p ) C (1 + θ ) − ap (cid:21) Z Tt e aps | Y s | p ds + (cid:20) c ( p )2 θ + c ( p ) α (1 + θ ) (cid:21) Z Tt e aps | Y s | p − I { Y s =0 } | Z s | ds + θ − p (cid:18)Z Tt e aps | f s | ds (cid:19) p + d θ p (cid:18)Z Tt | f s | ds (cid:19) p + d θ p (cid:18)Z Tt e aps | g s | ds (cid:19) p + (cid:20) c ( p )(1 + 1 θ ) + d θ p (cid:21) Z Tt e aps | Y s | p − { Y s =0 } | g s | ds (cid:27) . Choosing θ small enough and a > Cp + p C c ( p ) θ ) + c ( p ) C (1 + θ ) − ap < , (8)we get E (cid:20) e apt | Y t | p + c ( p ) Z Tt e aps | Y s | p − I { Y s =0 } | Z s | ds (cid:21) ≤ d E " | ξ | p + sup s ∈ [0 ,T ] ( L + s ) p + (cid:18)Z Tt | f s | ds (cid:19) p + (cid:18)Z Tt | g s | ds (cid:19) p + Z Tt e aps | Y s | p − I { Y s =0 } | g s | ds (cid:21) + d θ p E [ sup s ∈ [0 ,T ] | Y s | p ] . (9)Next using the Burkh¨older-Davis-Gundy inequality we have E " sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12) p Z T e aps | Y s | p − b Y s Z s dW s (cid:12)(cid:12)(cid:12)(cid:12) ≤ E [ sup t ∈ [0 ,T ] e apt | Y t | p ] + d E (cid:18)Z T e aps | Y s | p − { Y s =0 } | Z s | ds (cid:19) (10)10nd E " sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)Z T e aps | Y s | p − b Y s g ( s, Y s , Z s ) dB s (cid:12)(cid:12)(cid:12)(cid:12) ≤ E " sup t ∈ [0 ,T ] e apt | Y t | p + d E Z T e aps | Y s | p − { Y s =0 } | g ( s, Y s , Z s ) | ds ≤ E [ sup t ∈ [0 ,T ] e apt | Y t | p ] + d E (cid:20)Z T e aps | Y s | p − { Y s =0 } | g s | ds + Z T e aps | Y s | p ds + Z T e aps | Y s | p − I { Y s =0 } | Z s | ds (cid:21) . (11)Next going back to (3), using the Burkh¨older-Davis-Gundy inequality to-gether with the inequalities (9)-(11), we get after choosing θ small enough(s.t. inequality (8) holds too) E " sup t ∈ [0 ,T ] e apt | Y t | p + Z Tt e aps | Y s | p − I { Y s =0 } | Z s | ds ≤ d E (cid:20) | ξ | p + ( Z T | f s | ds ) p + ( Z T | g s | ds ) p + sup t ∈ [0 ,T ] ( L + t ) p + Z T e aps | Y s | p − I { Y s =0 } | g s | ds . We then complete the proof by the inequality (6). (cid:3)
Lemma 3.3
Let ( Y ′ , Z ′ , K ′ ) and ( Y, Z, K ) be the solution of the reflectedBDSDE (1) associated with ( ξ ′ , f ′ , g ′ , L ) and ( ξ, f, g, L ) respectively, where ( ξ ′ , f ′ , g ′ , L ) and ( ξ, f, g, L ) satisfy assumptions (H1)-(H3). Then E " sup t ∈ [0 ,T ] | Y ′ t − Y t | p + (cid:18)Z T | Z ′ s − Z s | ds (cid:19) p ≤ d E (cid:20) | ξ ′ − ξ | p + ( Z T | f ′ ( s, Y s , Z s ) − f ( s, Y s , Z s ) | ds ) p +( Z T | g ′ ( s, Y s , Z s ) − g ( s, Y s , Z s )) | ds ) p (cid:21) . roof. The proof of the lemma is a combination of the proofs of Lemmas3.1 and 3.2 with a slight change. Indeed, let ξ =: ξ ′ − ξ, ( Y , Z, K ) =: ( Y ′ − Y, Z ′ − Z, K ′ − K ) . One can easily to check that (
Y , Z, K ) is a solution to the following BDSDE: Y t = ξ + Z Tt h ( s, Y s , Z s ) ds + Z Tt k ( s, Y s , Z s ) dB s + K T − K t − Z Tt Z s dW s , where h ( s, y, z ) =: f ′ ( s, y + Y s , z + Z s ) − f ( s, Y s , Z s ) ,k ( s, y, z ) =: g ′ ( s, y + Y s , z + Z s ) − g ( s, Y s , Z s ) . Obviously, the functions h and k are Lipschitz w.r.t ( y, z ).Let’s note that Z t e aps Y s dK s = − Z t e aps ( Y ′ s − L s ) dK s − Z t e aps ( Y s − L s ) dK ′ s ≤ Z t e aps | Y s | p − c Y s dK s = − Z t e aps | Y s | p − { Y s =0 } ( Y ′ s − L s ) dK s − Z t e aps | Y s | p − I { Y s =0 } ( Y s − L s ) dK ′ s ≤ . The rest of the proof follows Itˆo’s formula, Lemma 2.1 and the steps similarto those in the proofs of Lemmas 3.1 and 3.2. (cid:3)
In order to obtain the existence and uniqueness result, we make the followingsupplementary assumption:(H4) g ( · , , ≡ Lemma 3.4
Let p = 2 . Assume that (H1)-(H3) hold. Then the reflectedBDSDE (1) has a unique solution ( Y, Z, K ) ∈ S × M d × S ci .
12e now state and prove our main result.
Theorem 3.1
Assume (H1)-(H4), then the reflected BDSDE (1) has a uniquesolution ( Y, Z, K ) ∈ S p × M pd × S pci . Proof.
The uniqueness is an immediate consequence of Lemma 3.3. Wenext to prove the existence.For each n, m ∈ N ∗ , define ξ n = q n ( ξ ) , f n ( t, x, y ) = f ( t, x, y ) − f t + q n ( f t ) , L mt = q m ( L t ) , where q k ( x ) = x k | x |∨ k . One can easily to check that the items ξ n , f n and L m satisfy the assumptions (H1)-(H3), it follows from Lemma 3.4 that, for each n, m ∈ N ∗ , there exists a unique solution ( Y n , Z n , K n ) ∈ L for the reflectedBDSDE associated with ( ξ n , f n , g, L m ), but in fact also in L p , according as-sumption (H4) and the Lemmas 3.1 and 3.2.Next, from Lemma 3.3, for ( i, n ) ∈ N × N ∗ , we have E ( sup t ∈ [0 ,T ] | Y n + it − Y nt | p + (cid:18)Z T | Z n + is − Z ns | ds (cid:19) p ) ≤ d E ( | ξ n + i − ξ n | p + (cid:18)Z T | q n + i ( f s ) − q n ( f s ) | ds (cid:19) p ) . Clearly, the right side of above inequality tend to 0 as n → ∞ , uniformlyon i so that ( Y n , Z n ) is a Cauchy sequence in S p × M pd . Let’s denote by( Y, Z ) ∈ S p × M pd it limit. By the equation K nt = Y n − Y nt − Z t f n ( s, Y ns , Z ns ) ds − Z t g ( s, Y ns , Z ns ) dB s + Z t Z ns dW s , similar computation can derive that ( K nt ) n ≥ is also a Cauchy sequence in S pci , then there exists a non-decreasing process K t ∈ S pci ( K = 0) such that E ( | K nt − K t | p ) → , as n → ∞ and Z T ( Y s − L ms ) dK s = 0 .
13y the dominated convergence theorem, we then get Z T ( Y s − L ms ) dK s → Z T ( Y s − L s ) dK s , as m → ∞ . It follows that the limit (
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