aa r X i v : . [ m a t h . P R ] D ec Reflected BSDE with stochastic Lipschitzcoefficient
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 stochastic differential equations with stochastic Lipschitzcoefficient. We derive the existence and uniqueness of the solutionsfor those equations via Snell envelope and the fixed point theorem.
Keywords:
Reflected backward stochastic differential equation; stochas-tic Lipschitz coefficient; Snell envelope
AMS 2000 Subject Classification:
Nonlinear backward stochastic differential equations (BSDE in short) werefirstly introduced by Pardoux and Peng (1990). Since then, a lot of work havebeen devoted to the study of BSDEs as well as to their applications. Thisis due to the connections of BSDEs with mathematical finance ( see e.g. ElKaroui et al. (1997c)) as well as to stochastic optimal control (see e.g. Peng(1993)) and stochastic games ( see e.g. Hamad`ene and Lepeltier (1995)). ElKaroui et al. (1997a) introduced the notion of one barrier reflected BSDE(RBSDE in short), which is actually a backward equation but the solution ∗ 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]
1s forced to stay above a given barrier. This type of BSDEs is motivated bypricing American options (see El Karoui et al. (1997b)) and studying themixed game problems ( see e.g. Cvitanic and Karatzas (1996), Hamad`ene andLepeltier (2000)).The existence and uniqueness of solution of BSDE in Pardoux and Peng(1990) and of RBSDE in El Karoui et al. (1997a) are both proved under theLipschitz assumption on the coefficient. However, the Lipschitz condition istoo restrictive to be assumed in many applications. For instance, the pricingproblem of an American claim is equivalent to solving the linear BSDEd Y t = [ r ( t ) Y t + θ ( t ) Z t ]d t + Z t d B t , Y T = ξ, where r ( t ) is the interest rate and θ ( t ) is the risk premium vector. In general,both of them may be unbounded, therefore Pardoux and Peng’s result maybe invalid. And so is it in the case of RBSDE.Consequently, many papers have devoted to relax the Lipschitz conditionin both cases of BSDE and RBSDE (see e.g. Lepeltier and Martin (1997), ElKaroui and Huang (1997), Bender and Kohlmann (2000), Wang and Huang(2009), Matoussi (1997), Lepeltier et al. (2005) and the references therein).During them, El Karoui and Huang (1997) established a general result ofexistence and uniqueness for BSDEs driven by a general cadlag martingalewith stochastic Lipschitz coefficient. Later, Bender and Kohlmann (2000)showed the same result for BSDEs driven by a Brownian motion. Motivatedby the above works, the purpose of the present paper is to consider a classof one-dimensional RBSDEs with stochastic Lipschitz coefficient. We try toget the existence and uniqueness of solutions for those RBSDEs by means ofthe Snell envelope and the fixed point theorem.The rest of the paper is organized as follows. In Section 2, we introducesome notations including some spaces. Section 3 is devoted to prove theexistence and uniqueness of solutions to RBSDEs with stochastic Lipschitzcoefficient. Let ( B t ) t ≥ be a d -dimensional standard Brownian motion defined on a prob-ability space (Ω , F , P ). We denote ( F t ) t ≥ the natural filtration of ( B t ) t ≥ ,augmented by all P -null sets of F . The Euclidean norm of a vector y ∈ R n will be defined by | y | . 2et T > • L the space of F T -measurable random variables ξ such that E [ | ξ | ] < + ∞ . • S the space of predictable processes { ψ t : t ∈ [0 , T ] } such that E [ sup ≤ t ≤ T | ψ t | ] < + ∞ . • H the space of predictable processes { ψ t : t ∈ [0 , T ] } such that E Z T | ψ t | d t < + ∞ . Let β > a t ) t ≥ be a nonnegative F t -adapted process. Define A ( t ) = Z t a ( s )d s, ≤ t ≤ T. We further introduce the following spaces: • L ( β, a ) the space of F T -measurable random variables ξ such that E [ e βA ( T ) | ξ | ] < + ∞ . • S ( β, a ) the space of predictable processes { ψ t : t ∈ [0 , T ] } such that E [ e βA ( T ) sup ≤ t ≤ T | ψ t | ] < + ∞ . • H ( β, a )the space of predictable processes { ψ t : t ∈ [0 , T ] } such that E Z T e βA ( t ) | ψ t | d t < + ∞ . In this paper, we consider the following RBSDE: ( Y t = ξ + R Tt f ( s, Y s , Z s )d s + K T − K t − R Tt Z s d B s ,Y t ≥ S t , ≤ t ≤ T a.s. and R T ( Y t − S t )d K t = 0 , a.s. (1)3here the coefficient f : Ω × [0 , T ] × R × R d × R → R satisfies the followingassumptions:( H1 ) ∀ t ∈ [0 , T ] , ( y i , z i ) ∈ R × R d , i = 1 ,
2, there are two nonnegative F t -adapted processes µ ( t ) and γ ( t ) such that | f ( t, y , z ) − f ( t, y , z ) | ≤ µ ( t ) | y − y | + γ ( t ) | z − z | ; (2)( H2 ) ∃ ǫ > a ( t ) := µ ( t ) + γ ( t ) ≥ ǫ ;( H3 ) For all ( y, z ) ∈ R × R d , the process f ( · , · , y, z ) is progressively measur-able and such that ∀ t ∈ [0 , T ] , f ( t, , a ∈ H ( β, a ).Furthermore, we make the following assumptions:( H4 ) The terminal value ξ ∈ L ( β, a );( H5 ) The ”obstacle” { S t , ≤ t ≤ T } is a continuous progressively measur-able real-valued process satisfying E [sup ≤ t ≤ T e βA ( t ) ( S + t ) ] < ∞ and S T ≤ ξ a.s.We now give the definition of solution to RBSDE (1). Definition 2.1
Let β > and a a nonnegative F t -adapted process. A so-lution to RBSDE (1) is a triple ( Y, Z, K ) satisfying (1) such that ( Y, Z ) ∈ S ( β, a ) × H ( β, a ) and K ∈ S is continuous and increasing with K = 0 . We first give a priori estimate of the solution of RBSDE (1).
Lemma 3.1
Let ( Y t , Z t , K t ) ≤ t ≤ T be a solution of RBSDE (1) with data ( ξ, f, T ) . Then there exists a constant C β depending only on β such that E (cid:20) sup ≤ t ≤ T | Y t | e βA ( t ) + Z T e βA ( s ) | Z s | d s + Z T e βA ( s ) a ( s ) | Y s | d s + K T (cid:21) ≤ C β E (cid:20) | ξ | e βA ( T ) + Z T e βA ( s ) | f ( s, , | a ( s ) d s + sup ≤ t ≤ T e βA ( t ) ( S + t ) (cid:21) . roof. Applying Itˆo’s formula to e βA ( t ) | Y t | , we have e βA ( t ) | Y t | + Z Tt e βA ( s ) | Z s | d s + β Z Tt a ( s ) e βA ( s ) | Y s | d s = e βA ( T ) | ξ | + 2 Z Tt e βA ( s ) Y s f ( s, Y s , Z s )d s + 2 Z Tt e βA ( s ) Y s d K s − Z Tt e βA ( s ) Y s Z s d B s ≤ e βA ( T ) | ξ | + β Z Tt a ( s ) e A ( s ) | Y s | d s + 2 Z Tt e βA ( s ) βa ( s ) | f ( s, Y s , Z s ) | d s +2 Z Tt e βA ( s ) Y s d K s − Z Tt e βA ( s ) Y s Z s d B s ≤ e βA ( T ) | ξ | + β Z Tt a ( s ) e A ( s ) | Y s | d s + 6 β [ Z Tt e βA ( s ) a ( s ) | Y s | d s + Z Tt e βA ( s ) | Z s | d s ]+ 6 β Z Tt e βA ( s ) | f ( s, , | a ( s ) d s + 2 Z Tt e βA ( s ) Y s d K s − Z Tt e βA ( s ) Y s Z s d B s . Consequently, e βA ( t ) | Y t | + (1 − β ) Z Tt e βA ( s ) | Z s | d s + ( β − β ) Z Tt a ( s ) e βA ( s ) | Y s | d s ≤ e βA ( T ) | ξ | + 6 β Z Tt e βA ( s ) | f ( s, , | a ( s ) d s +2 Z Tt e βA ( s ) S s d K s − Z Tt e βA ( s ) Y s Z s d B s . (3)where we have used the fact that d K s = I [ Y s = S s ] d K s and the stochasticLipschitz property of f . For a sufficient large β >
0, taking expectation onboth sides above, we get E [ Z Tt a ( s ) e βA ( s ) | Y s | d s + Z Tt e βA ( s ) | Z s | d s ] ≤ c β E (cid:20) e βA ( T ) | ξ | + Z Tt e βA ( s ) | f ( s, , | a ( s ) d s + 2 Z Tt e βA ( s ) S + s d K s (cid:21) . (4)5oreover, by the Burkholder-Davis-Gundy’s inequality, one can derive that E [ sup ≤ t ≤ T | Z Tt e βA ( s ) Y s Z s d B s | ] ≤ E [ | Z T e βA ( s ) Y s Z s d B s | ] + E [ sup ≤ t ≤ T | Z t e βA ( s ) Y s Z s d B s | ] ≤ c E ((cid:20)Z T e βA ( s ) | Y s | | Z s | d s (cid:21) ) ≤ c E (cid:20) ( sup ≤ t ≤ T e βA ( t ) | Y t | ) ( Z T e βA ( s ) | Z s | d s ) (cid:21) ≤ E [( sup ≤ t ≤ T e βA ( t ) | Y t | )] + 2 c E [ Z T e βA ( s ) | Z s | d s ] . Combining this with (3) and (4), we have E (cid:20) sup ≤ t ≤ T | Y t | e βA ( t ) + Z T a ( s ) e βA ( s ) | Y s | d s + Z T e βA ( s ) | Z s | d s (cid:21) ≤ k β E (cid:20) e βA ( T ) | ξ | + Z T e βA ( s ) | f ( s, , | a ( s ) d s + 2 Z T e βA ( s ) S + s d K s (cid:21) . (5)We now give an estimate of K T . From the equation K T = Y − ξ − Z T f ( s, Y s , Z s )d s + Z T Z s d B s E [ K T ] ≤ d β E (cid:20) sup ≤ t ≤ T | Y t | e βA ( t ) + | ξ | + Z T | Z s | d s + Z T a ( s ) e − βA ( s ) d s Z T e βA ( s ) | f ( s, Y s , Z s ) | a ( s ) d s (cid:21) ≤ d β E (cid:20) e βA ( T ) | ξ | + Z T a ( s ) e βA ( s ) | Y s | d s + Z T e βA ( s ) | Z s | d s + Z T e βA ( s ) | f ( s, , | a ( s ) d s (cid:21) ≤ d β E (cid:20) e βA ( T ) | ξ | + Z T e βA ( s ) | f ( s, , | a ( s ) d s + 2 Z T e βA ( s ) S + s d K s (cid:21) ≤ d β E (cid:20) e βA ( T ) | ξ | + Z T e βA ( s ) | f ( s, , | a ( s ) d s + sup ≤ t ≤ T e βA ( t ) ( S + t ) (cid:21) + 12 E [ K T ] . Hence, E [ K T ] ≤ d β E (cid:20) e βA ( T ) | ξ | + Z T e βA ( s ) | f ( s, , | a ( s ) d s + sup ≤ t ≤ T e βA ( t ) ( S + t ) (cid:21) , (6)where we use the notation d β for a constant depending only on β and whosevalue could be changing from line to line. We get the desired result byestimates (6) and (5). (cid:3) We first consider the special case that is the coefficient does not depend on(
Y, Z ), i.e. f ( ω, t, y, z ) ≡ g ( ω, t ). We have the following result. Theorem 3.1
Let β > large enough and a a nonnegative F t -adapted pro-cess. Assume ga ∈ H ( β, a ) and ( H4 )-( H5 ) hold. Then RBSDE (1) withdata ( ξ, g, S ) has a solution. Proof.
For 0 ≤ t ≤ T , we define e Y t = ess sup ν ≥ t E [ Z ν g ( s ) ds + S ν I { ν
0, by H¨olderinequality, we have E "(cid:18)Z T | g ( s ) | d s (cid:19) = E "(cid:18)Z T | g ( s ) a ( s ) || a ( s ) | d s (cid:19) ≤ E (cid:20)(cid:18)Z T | g ( s ) a ( s ) | e βA ( s ) d s (cid:19) (cid:18)Z T a ( s ) e − βA ( s ) d s (cid:19)(cid:21) ≤ β E (cid:20)(cid:18)Z T | g ( s ) a ( s ) | e βA ( s ) d s (cid:19)(cid:21) < + ∞ . Consequently, by Doob-Meyer decomposition theorem in Dellacherie andMeyer (1980), there exists an increasing continuous process ( K t ) ≤ t ≤ T whichbelongs to S ( K = 0) and a martingale M t ∈ S such that e Y t = M t − K t . ∀ t ∈ [0 , T ]Since M t ∈ S , there exists Z t ∈ H such that M t = M + Z t Z s d B s . ∀ t ∈ [0 , T ]Let Y t = e Y t − R t f ( s )d s , by Proposition 5.1 of El Karoui et al. (1997a), wederive that the triple ( Y, Z, K ) verities Y t = ξ + Z Tt g ( s )d s + K T − K t − Z Tt Z s d B s . Moreover, Y t ≥ S t and R T ( Y t − S t )d K t = 0. By Lemma 3.1, ( Y t , Z t , K t ) ≤ t ≤ T is a solution of RBSDE (1). (cid:3) Furthermore, we have the following uniqueness result.
Proposition 3.1
With the same assumptions of Theorem 3.1, the RBSDE(1) with data ( ξ, g, S ) has at most one solution. Proof.
Let (
Y, Z, K ) and ( Y ′ , Z ′ , K ′ ) be two solutions of RBSDE (1). Let∆ Y = Y − Y ′ , ∆ Z = Z − Z ′ , ∆ K = K − K ′ . ≤ t ≤ T , we have∆ Y t = ∆ K T − ∆ K t − Z Tt ∆ Z s d B s . Applying Itˆo’s formula to e βA ( t ) | ∆ Y t | , we obtain − E [ e βA ( t ) | ∆ Y t | ] = − E [ Z Tt e βA ( s ) ∆ Y s d(∆ K s )] + E [ Z Tt e βA ( s ) | ∆ Z s | d s ] . Noting that R Tt e βA ( s ) ∆ Y s d(∆ K s ) ≤
0, it follows that ∆ Y t = ∆ Z t = 0 andthen ∆ K t = 0, 0 ≤ t ≤ T a.s. (cid:3) We can now state and prove our main result.
Theorem 3.2
Assume (H1)-(H5) hold for a sufficient large β . Then RB-SDE (1) with data ( ξ, f, S ) has a unique solution. Proof.
Let H ( β, a ) = S ( β, a ) × H ( β, a ). Given ( U, V ) ∈ H ( β, a ), considerthe following RBSDE: Y t = ξ + Z Tt f ( s, U s , V s )d s + K T − K t − Z Tt Z s d B s . (7)By Young’s inequality, we have | f ( t, U t , V t ) | a ( t ) ≤ a ( t ) | U t | + | V t | + | f ( t, , | a ( t ) ] , it follows from (H3) and Theorem 3.1 that the RBSDE (7) has a uniquesolution.Define a mapping Φ from H ( β, a ) to itself. Let ( U ′ , V ′ ) be another elementin H ( β, a ), set ( Y, Z ) = Φ(
U, V ) , ( Y ′ , Z ′ ) = Φ( U ′ , V ′ ) , where ( Y, Z, K ) (resp. ( Y ′ , Z ′ , K ′ )) is the unique solution of the RBSDEassociated with data ( ξ, f ( t, U t , V t ) , S ) (resp.( ξ, f ( t, U ′ t , V ′ t ) , S )).Let ∆ Y = Y − Y ′ , ∆ Z = Z − Z ′ , ∆ U = U − U ′ , ∆ V = V − V ′ , ∆ f s = f ( s, U s , V s ) − f ( s, U ′ s , V ′ s ) , ∆ K = K − K ′ . ≤ t ≤ T , we have∆ Y t = Z Tt ∆ f s d s + ∆ K T − ∆ K t − Z Tt ∆ Z s d B s Applying Itˆo’s formula to e βA ( t ) | ∆ Y t | , using ( H1 ) and the fact that d K s = I [ Y s = S s ] d K s and d K ′ s = I [ Y ′ s = S s ] d K ′ s , we get e βA ( t ) | ∆ Y t | + β Z Tt a ( s ) e βA ( s ) | ∆ Y s | d s + Z Tt e βA ( s ) | ∆ Z s | d s ≤ Z Tt e βA ( s ) ∆ Y s ∆ f s d s + 2 Z Tt e βA ( s ) ∆ Y s d(∆ K s ) − Z Tt e βA ( s ) ∆ Y s ∆ Z s d B s ≤ Z Tt e βA ( s ) ∆ Y s ∆ f s d s − Z Tt e βA ( s ) ∆ Y s ∆ Z s d B s ≤ β Z Tt a ( s ) e βA ( s ) | ∆ Y s | d s + 6 β Z Tt e βA ( s ) | ( a ( s ) | ∆ U s | + | ∆ V | )d s − Z Tt e βA ( s ) ∆ Y s ∆ Z s d B s , it follows that E [ Z Tt a ( s ) e βA ( s ) | ∆ Y s | d s ] + E [ Z Tt e βA ( s ) | ∆ Z s | d s ] ≤ ( 12 β + 6 β ) (cid:26) E [ Z Tt a ( s ) e βA ( s ) | ∆ U s | ] + E [ Z Tt e βA ( s ) | ∆ V | d s ] (cid:27) . For β > k ( Y, Z ) k β = E (cid:20)Z T e βA ( s ) ( a ( s ) | Y s | + | Z s | ) ds (cid:21) . Thus, φ has a unique fixed point and the theorem is proved. (cid:3) References [1] Bender, C., Kohlmann, M., 2000. BSDEs with stochastic Lipschitz condition.http://cofe.uni-konstanz.de/Papers/dp00 08.pdf
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