aa r X i v : . [ m a t h . P R ] M a r BSE’S, BSDE’S AND FIXED POINT PROBLEMS ∗ Patrick CheriditoETH Zurich8092 Zurich, Switzerland Kihun NamRutgers UniversityPiscataway, NJ 08854, USAAugust 2016
Abstract
In this paper, we introduce a class of backward stochastic equations (BSEs) that extend classicalBSDEs and include many interesting examples of generalized BSDEs as well as semimartingalebackward equations. We show that a BSE can be translated into a fixed point problem in a spaceof random vectors. This makes it possible to employ general fixed point arguments to establishthe existence of a solution. For instance, Banach’s contraction mapping theorem can be used toderive general existence and uniqueness results for equations with Lipschitz coefficients, whereasSchauder-type fixed point arguments can be applied to non-Lipschitz equations. The approachworks equally well for multidimensional as for one-dimensional equations and leads to results inseveral interesting cases such as equations with path-dependent coefficients, anticipating equa-tions, McKean–Vlasov type equations and equations with coefficients of superlinear growth.
MSC 2010:
Key words:
Backward stochastic equation, backward stochastic differential equation, path-dependentcoefficients, anticipating equations, McKean–Vlasov type equations, coefficients of superlinear growth.
In this paper we study backward stochastic equations (BSEs) of the form Y t + F t ( Y, M ) + M t = ξ + F T ( Y, M ) + M T . (1.1)For a given maturity T ∈ R + , a filtered probability space (Ω , F , ( F t ) ≤ t ≤ T , P ) , a generator F anda terminal condition ξ ∈ L p ( F T ) d , a solution to (1.1) consists of a d -dimensional adapted process Y together with a d -dimensional martingale M such that equation (1.1) holds for all t ∈ [0 , T ] . If F ( Y, M ) is a finite variation process, (1.1) is a semimartingale backward equation, which as a specialcase, contains the semimartingale Bellman equation introduced by Chitashvili (1983); see also Mania ∗ We thank Ramon van Handel, Ying Hu, Peter Imkeller, Shige Peng, and Frederi Viens for fruitful discussions andhelpful comments. F is of the form F t ( Y, M ) = R t f ( s, Y, M ) ds , BSE (1.1) becomes a generalized backward stochastic differential equation (BSDE), Y t = ξ + Z Tt f ( s, Y, M ) ds + M T − M t , (1.2)in the spirit of Liang et al. (2011). If in addition, the probability space carries an n -dimensionalBrownian motion W and a Poisson random measure N on [0 , T ] × ( R m \ { } ) such that every square-integrable martingale M has a unique representation of the form M t = Z t Z Ms dW s + Z t Z R m \{ } U Ms ( x ) ˜ N ( ds, dx ) + K Mt for the compensated Poisson random measure ˜ N , suitable integrands Z M and U M , and a square-integrable martingale K M strongly orthogonal to W and ˜ N , one can write equations of the form Y t = ξ + Z Tt f ( s, Y, Z M , U M ) ds + M T − M t . (1.3)This generalizes the jump-diffusion extension of Tang and Li (1994) of the classical BSDEs introducedby Pardoux and Peng (1990) in three directions. First, in Tang and Li (1994) the filtration is generatedby the Brownian motion and the Poisson random measure, whereas here it is general; secondly, atany given time, the driver f in (1.3) can depend on the whole paths of the processes Y , Z M , U M andnot only on their current values; and finally, f can be a function of Y , Z M , U M viewed as randomelements instead of just their realizations Y ( ω ) , Z M ( ω ) and U M ( ω ) . As special cases, (1.3) containsBSDEs with drivers that depend on the past or future of Y , Z M and U M , such as the time-delayedBSDEs of Delong and Imkeller (2010a, 2010b) or the anticipating BSDEs of Peng and Yang (2009).It also includes mean-field BSDEs as in Buckdahn et al. (2009), or more generally, McKean–Vlasovtype BSDEs with coefficients depending on the distributions of Y , Z M and U M .Our approach to proving that a BSE has a solution is to translate it into a fixed point problemfor a mapping G : L p ( F T ) d → L p ( F T ) d . This makes it possible to apply general fixed point results.For instance, Banach’s contraction mapping theorem can be used to derive general existence anduniqueness results for equations with Lipschitz coefficients. In the non-Lipschitz case one can employSchauder type fixed point arguments. This yields results for equations with coefficients of superlineargrowth, but it requires compactness assumptions. By reducing a BSE to a fixed point problem in L p ( F T ) d , one eliminates the time-dimension. But one still has to find compact subsets of L p ( F T ) d . Wedo that by making use of Sobolev spaces corresponding to infinite-dimensional Gaussian measures.Our method works equally well for multidimensional as for one-dimensional equations, and inaddition to general results for BSEs, it also yields interesting findings for BSDEs. For instance, inSection 3, we obtain existence and uniqueness results for BSDEs with functional drivers dependingon the whole processes Y and M . In general, such results require Lipschitz continuity with a smallenough Lipschitz constant or, alternatively, a sufficiently short maturity. But in several interestingspecial cases, it is possible to derive the existence of a unique solution for arbitrary Lipschitz constantand maturity. In Section 4, we use compactness and a theorem by Krasnoselskii (1964), which com-bines the fixed point results of Banach and Schauder, to derive existence results for multidimensionalBSDEs with functional drivers of superlinear growth. For instance, Corollary 4.7 establishes the exis-tence of solutions to BSDEs with general path-dependent drivers and Corollary 4.10 the existence of a2olution to a multidimensional mean-field BSDE with driver of quadratic growth. The latter comple-ments results by e.g., Tevzadze (2008) and Cheridito and Nam (2015) on multidimensional quadraticBSDEs, which are known to not always have solutions (see e.g., Peng, 1999, or Frei and dos Reis,2011).The structure of the paper is as follows. In Section 2, we formally introduce BSEs and relatethem to fixed point problems in L p ( F T ) d . In Section 3, we derive existence and uniqueness resultsfor various BSEs and BSDEs with general functional Lipschitz coefficients from Banach’s contractionmapping theorem. In Section 4, we provide existence results for different non-Lipschitz equationsusing compactness and Krasnoselskii’s fixed point theorem. L p In this section, we introduce BSEs and show how they can be translated into fixed point problemsin L p -spaces. We fix a finite time horizon T ∈ R + and let (Ω , F , F , P ) be a filtered probability spacewith a filtration F := ( F t ) t ∈ [0 ,T ] satisfying the usual conditions. Then all martingales admit a RCLLmodification (i.e., right-continuous with left limits). By | . | we denote the Euclidean norm on R d , andfor a d -dimensional random vector X , we define k X k p := ( E | X | p ) /p if p < ∞ and k X k ∞ := ess sup ω ∈ Ω | X | . For p ∈ (1 , ∞ ] , we set: • L p ( F t ) d : all d -dimensional F t -measurable random vectors X satisfying k X k p < ∞• E t X := E [ X |F t ] • S p : all R d -valued RCLL adapted processes ( Y t ) ≤ t ≤ T satisfying k Y k S p := (cid:13)(cid:13) sup ≤ t ≤ T | Y t | (cid:13)(cid:13) p < ∞• S p : all Y ∈ S p with Y = 0 • M p : all martingales in S p .A BSE is specified by a generator F : S p × M p → S p and a terminal condition ξ ∈ L p ( F T ) d . Definition 2.1.
A solution to the BSE Y t + F t ( Y, M ) + M t = ξ + F T ( Y, M ) + M T (2.1) consists of a pair ( Y, M ) ∈ S p × M p such that (2.1) holds for all t ∈ [0 , T ] . Definition 2.2.
We say F satisfies condition (S) if for all y ∈ L p ( F ) d and M ∈ M p , the equation Y t = y − F t ( Y, M ) − M t (2.2) has a unique solution Y ∈ S p . V ∈ L p ( F T ) d , one obtains from Jensen’s inequality that y V := E V belongs to L p ( F ) d and from Doob’s L p -maximal inequality that M Vt := E V − E t V is in M p . If F satisfies (S), we denoteby Y V the solution of the equation Y t = y V − F t ( Y, M V ) − M Vt .A BSE depends on the generator F and terminal condition ξ . Provided that F satisfies condition(S), then the pair ( F, ξ ) also defines a map G : L p ( F T ) d → L p ( F T ) d through V ξ + F T ( Y V , M V ) . To relate solutions of the BSE (2.1) to fixed points of G , we define the two mappings π : S p × M p → L p ( F T ) d and φ : L p ( F T ) d → S p × M p by π ( Y, M ) := Y − M T and φ ( V ) := ( Y V , M V ) . Theorem 2.3.
Assume F satisfies (S) . Then the following hold: a) V = ( π ◦ φ )( V ) for all V ∈ L p ( F T ) d . In particular, φ is injective. b) If V ∈ L p ( F T ) d is a fixed point of G , then φ ( V ) is a solution of the BSE (2.1) . c) If ( Y, M ) ∈ S p × M p solves the BSE (2.1) , then π ( Y, M ) is a fixed point of G and ( Y, M ) = ( φ ◦ π )( Y, M ) . d) V is a unique fixed point of G in L p ( F T ) d if and only if φ ( V ) is a unique solution of the BSE (2.1) in S p × M p .Proof. a) is straight-forward to check.b) If V ∈ L p ( F T ) d is a fixed point of G , then y V − M VT = ( π ◦ φ )( V ) = V = G ( V ) = ξ + F T ( Y V , M V ) . (2.3)Since Y V satisfies Y Vt = y V − F t ( Y V , M V ) − M Vt for all t , (2.3) is equivalent to Y Vt + F t ( Y V , M V ) + M Vt = ξ + F T ( Y V , M V ) + M VT for all t, which shows that φ ( V ) = ( Y V , M V ) solves the BSE (2.1).c) Let ( Y, M ) ∈ S p × M p be a solution of the BSE (2.1). Set V := π ( Y, M ) = Y − M T . Then, y V = Y and M Vt = M t . In particular, Y t = Y − F t ( Y, M ) − M t = y V − F t ( Y, M V ) − M Vt for all t . It follows that ( Y, M ) = ( Y V , M V ) = φ ( V ) = ( φ ◦ π )( Y, M ) and y V = Y V = ξ + F T ( Y V , M V ) + M VT = G ( V ) + M VT . Since y V − M VT = V , this shows that V = G ( V ) .d) follows from a)–c). 4n the special case, where F does not depend on Y , condition (S) holds trivially, and it is enough tofind a fixed point of the mapping G ( V ) := G ( V ) − E G ( V ) in the subspace L p ( F T ) d := n V ∈ L p ( F T ) d : E V = 0 o . Corollary 2.4. If F does not depend on Y , the following hold: a) If V ∈ L p ( F T ) d is a fixed point of G , then the processes Y t := E ξ + E F T ( M ) − F t ( M ) − M t and M t := − E t V form a solution of the BSE (2.1) in S p × M p . b) If ( Y, M ) ∈ S p × M p solves the BSE (2.1) , then − M T is a fixed point of G . c) V is a unique fixed point of G in L p ( F T ) d if and only if the pair ( Y, M ) given by Y t := E ξ + E F T ( M ) − F t ( M ) − M t and M t := − E t V is a unique solution of the BSE (2.1) in S p × M p .Proof. a) If V = G ( V ) , then for ˜ V = V + E G ( V ) , one has M ˜ V = M V , and therefore, ˜ V = V + E G ( V ) = G ( V ) = ξ + F T ( M V ) = ξ + F T ( M ˜ V ) = G ( ˜ V ) . So it follows from Theorem 2.3 that the pair ( Y, M ) given by Y t := E ξ + E F T ( M ) − F t ( M ) − M t and M t := − E t V solves the BSE (2.1).b) If ( Y, M ) ∈ S p × M p solves the BSE (2.1), it follows from Theorem 2.3 that V := Y − M T is afixed point of G . So G ( − M T ) = G ( Y − M T ) = G ( V ) − E G ( V ) = V − E V = − M VT = − M T . c) V is a fixed point of G if and only if V + E G ( V ) is a fixed point of G . Therefore, the resultfollows from part d) of Theorem 2.3.The following lemma provides a sufficient condition for F to satisfy condition (S). For ( Y, M ) ∈ S p × M p and k ∈ N , define F ( k ) t ( Y, M ) := F t ( Y ( k,M ) , M ) , where Y ( k,M ) is recursively given by Y (1 ,M ) := Y and Y ( k,M ) t := Y − F t ( Y ( k − ,M ) , M ) − M t , k ≥ . Lemma 2.5.
If for given y ∈ L p ( F ) d and M ∈ M p , there exist a number k ∈ N and a constant C < such that (cid:13)(cid:13)(cid:13) F ( k ) ( Y, M ) − F ( k ) ( Y ′ , M ) (cid:13)(cid:13)(cid:13) S p ≤ C (cid:13)(cid:13) Y − Y ′ (cid:13)(cid:13) S p for all Y, Y ′ ∈ S p with Y = Y ′ = y, (2.4) then the SDE (2.2) has a unique solution Y ∈ S p .Proof. The mapping Y y − F ( k ) ( Y, M ) − M is a contraction on { Y ∈ S p : Y = y } . So it followsfrom Banach’s contraction mapping theorem that there exists a unique Y ∈ S p satisfying Y = y − F ( k ) ( Y, M ) − M = Y ( k +1 ,M ) . This implies Y (2 ,M ) = y − F t ( Y, M ) − M t = y − F t ( Y ( k +1 ,M ) , M ) − M t = Y ( k +2 ,M ) = y − F ( k ) ( Y (2 ,M ) , M ) − M, from which one deduces Y = Y (2 ,M ) = y − F ( Y, M ) − M . This shows that Y solves the SDE (2.2). If Y ′ ∈ S p is another solution of (2.2), then Y ′ = y − F ( k ) ( Y ′ , M ) − M , and one obtains Y ′ = Y .5 Existence and uniqueness of solutions under Lipschitz assump-tions
In this section we consider equations with Lipschitz coefficients and use Banach’s contraction map-ping theorem to show that they have unique solutions.
We start with a result for general Lipschitz BSEs. Let us denote c = 15 , c ∞ = 14 and c p = p − p − for p ∈ (1 , ∞ ) \ { } . Then the following holds:
Theorem 3.1.
Let ξ ∈ L p ( F T ) d for some p ∈ (1 , ∞ ] . If there exist a number k ∈ N and a constant C < c p such that (cid:13)(cid:13)(cid:13) F ( k ) ( Y, M ) − F ( k ) ( Y ′ , M ′ ) (cid:13)(cid:13)(cid:13) S p ≤ C (cid:0)(cid:13)(cid:13) Y − Y ′ (cid:13)(cid:13) S p + (cid:13)(cid:13) M − M ′ (cid:13)(cid:13) S p (cid:1) for all Y, Y ′ ∈ S p and M, M ′ ∈ M p , then the BSE (2.1) has a unique solution ( Y, M ) in S p × M p .Proof. Since
C < , it follows from Lemma 2.5 that F satisfies (S). So by Theorem 2.3, it is enoughto prove that G has a unique fixed point in L p ( F T ) d . This follows from Banach’s contraction mappingtheorem if we can show that G is a contraction on L p ( F T ) d . Since for V ∈ L p ( F T ) d , Y V is the uniquefixed point of the mapping Y E V − F ( Y, M V ) − M V , it follows from the definition of F ( k ) that F ( Y V , M V ) = F ( k ) ( Y V , M V ) . Hence, one has for all V, V ′ ∈ L p ( F T ) d , Y Vt − Y V ′ t = y V − y V ′ − n F ( k ) t ( Y V , M V ) − F ( k ) t ( Y V ′ , M V ′ ) o − ( M Vt − M V ′ t )= E t ( V − V ′ ) − n F ( k ) t ( Y V , M V ) − F ( k ) t ( Y V ′ , M V ′ ) o . Therefore, sup ≤ t ≤ T | Y Vt − Y V ′ t | ≤ sup ≤ t ≤ T | E t ( V − V ′ ) | + sup ≤ t ≤ T | F ( k ) t ( Y V , M V ) − F ( k ) t ( Y V ′ , M V ′ ) | , and it follows that (cid:13)(cid:13)(cid:13) Y V − Y V ′ (cid:13)(cid:13)(cid:13) S p ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup ≤ t ≤ T | E t ( V − V ′ ) | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p + (cid:13)(cid:13)(cid:13) F ( k ) ( Y V , M V ) − F ( k ) ( Y V ′ , M V ′ ) (cid:13)(cid:13)(cid:13) S p ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup ≤ t ≤ T | E t ( V − V ′ ) | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p + C (cid:16)(cid:13)(cid:13)(cid:13) Y V − Y V ′ (cid:13)(cid:13)(cid:13) S p + (cid:13)(cid:13)(cid:13) M V − M V ′ (cid:13)(cid:13)(cid:13) S p (cid:17) . In particular, (cid:13)(cid:13)(cid:13) Y V − Y V ′ (cid:13)(cid:13)(cid:13) S p ≤ − C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup ≤ t ≤ T | E t ( V − V ′ ) | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p + C (cid:13)(cid:13)(cid:13) M V − M V ′ (cid:13)(cid:13)(cid:13) S p , (cid:13)(cid:13) G ( V ) − G ( V ′ ) (cid:13)(cid:13) p = (cid:13)(cid:13)(cid:13) F ( k ) T ( Y V , M V ) − F ( k ) T ( Y V ′ , M V ′ ) (cid:13)(cid:13)(cid:13) p ≤ C (cid:16)(cid:13)(cid:13)(cid:13) Y V − Y V ′ (cid:13)(cid:13)(cid:13) S p + (cid:13)(cid:13)(cid:13) M V − M V ′ (cid:13)(cid:13)(cid:13) S p (cid:17) ≤ C − C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup ≤ t ≤ T | E t ( V − V ′ ) | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p + C (cid:13)(cid:13)(cid:13) M V − M V ′ (cid:13)(cid:13)(cid:13) S p + C (cid:13)(cid:13)(cid:13) M V − M V ′ (cid:13)(cid:13)(cid:13) S p = C − C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup ≤ t ≤ T | E t ( V − V ′ ) | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p + (cid:13)(cid:13)(cid:13) M V − M V ′ (cid:13)(cid:13)(cid:13) S p . By Doob’s L p -maximal inequality, if we let C p = p/ ( p − for p ∈ (1 , ∞ ) and C ∞ = 1 , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup ≤ t ≤ T | E t ( V − V ′ ) − E ( V − V ′ ) | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p ≤ C p (cid:13)(cid:13) V − V ′ − E ( V − V ′ ) (cid:13)(cid:13) p , and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup ≤ t ≤ T | E t ( V − V ′ ) | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p ≤ C p (cid:13)(cid:13) V − V ′ (cid:13)(cid:13) p . Hence, (cid:13)(cid:13)(cid:13) M V − M V ′ (cid:13)(cid:13)(cid:13) S p ≤ (cid:26) k V − V ′ − E ( V − V ′ ) k ≤ k V − V ′ k for p = 2 C p k V − V ′ − E ( V − V ′ ) k p ≤ C p k V − V ′ k p for p = 2 , and (cid:13)(cid:13) G ( V ) − G ( V ′ ) (cid:13)(cid:13) p ≤ (cid:26) C − C k V − V ′ k for p = 23 C p C − C k V − V ′ k p for p = 2 . This shows that G is a contraction. Remark 3.2.
One cannot hope to obtain a general existence and uniqueness result like Theorem 3.1for equations with path-dependent coefficients without the assumption that the Lipschitz constant C is sufficiently small. For instance, if the generator is given by F t ( Y, M ) = atY for a constant a , theBSE (2.1) takes the form Y t − a ( T − t ) Y = ξ + M T − M t . (3.1)This is a variant of the equation studied in Example 3.1 of Delong and Imkeller (2010a), who noticedthat time-delayed BSDEs with Lipschtitz coefficients are not always well-posed. Obviously, F ( Y, M ) is Lipschitz in ( Y, M ) . But if one sets t = 0 and takes expectation on both sides of (3.1), one obtains (1 − aT ) Y = E ξ . This shows that for aT = 1 and E ξ = 0 , (3.1) cannot have a solution. On the otherhand, if aT = 1 and E ξ = 0 then Y t = (1 − t/T ) Y + E t ξ and M t = − E t ξ defines a solution for anyinitial value Y ∈ L p ( F ) d . So in this case, (3.1) has infinitely many solutions in S p × M p .If the generator is of integral form F t ( Y, M ) = R t f ( s, Y, M ) ds for a driver f : [0 , T ] × Ω × S p × M p → R d , (3.2)7he BSE (2.1) becomes a BSDE of the general form Y t = ξ + Z Tt f ( s, Y, M ) ds + M T − M t . (3.3)If for a RCLL measurable processe X , one denotes k X k S p [0 ,t ] := (cid:13)(cid:13) sup ≤ s ≤ t | X t | (cid:13)(cid:13) p , the following holds: Proposition 3.3.
Let ξ ∈ L p ( F T ) d for some p ∈ (1 , ∞ ] . Then the BSDE (3.3) has a unique solution ( Y, M ) ∈ S p × M p for every driver of the form (3.2) satisfying the following conditions: (i) For all ( Y, M ) ∈ S p × M p , f ( · , Y, M ) is progressively measurable with R T k f ( t, , k p dt < ∞ . (ii) There exist nonnegative constants C > and C < c p C e C T − such that (cid:13)(cid:13) f ( t, Y, M ) − f ( t, Y ′ , M ′ ) (cid:13)(cid:13) p ≤ C (cid:13)(cid:13) Y − Y + M − ( Y ′ − Y ′ + M ′ ) (cid:13)(cid:13) S p [0 ,t ] + C (cid:16)(cid:13)(cid:13) Y − Y ′ (cid:13)(cid:13) p + (cid:13)(cid:13) M − M ′ (cid:13)(cid:13) S p (cid:17) for all ( Y, M ) , ( Y ′ , M ′ ) ∈ S p × M p .Proof. Let q = p/ ( p − ∈ [1 , ∞ ) . It follows from the assumptions that for all ( Y, M ) ∈ S p × M p , (cid:13)(cid:13)(cid:13)(cid:13)Z T | f ( t, Y, M ) | dt (cid:13)(cid:13)(cid:13)(cid:13) p = sup k X k q ≤ Z T E [ | f ( t, Y, M ) || X | ] dt ≤ sup k X k q ≤ Z T k f ( t, Y, M ) k p k X k q dt = Z T k f ( t, Y, M ) k p dt ≤ Z T k f ( t, , k p dt + T C k Y − Y + M k S p + T C (cid:16) k Y k p + k M k S p (cid:17) < ∞ . So F t ( Y, M ) := R t f ( s, Y, M ) ds is a well-defined mapping from S p × M p to S p for all p ∈ (1 , ∞ ] .For given Y, Y ′ ∈ S p and M, M ′ ∈ M p , set δ := C C (cid:16)(cid:13)(cid:13) Y − Y ′ (cid:13)(cid:13) p + (cid:13)(cid:13) M − M ′ (cid:13)(cid:13) S p (cid:17) H t := H := 2 (cid:0)(cid:13)(cid:13) Y − Y ′ (cid:13)(cid:13) S p + (cid:13)(cid:13) M − M ′ (cid:13)(cid:13) S p (cid:1) H kt := (cid:13)(cid:13)(cid:13) F ( k ) ( Y, M ) − F ( k ) ( Y ′ , M ′ ) (cid:13)(cid:13)(cid:13) S p [0 ,t ] . Then H kt ≤ Z t (cid:13)(cid:13)(cid:13) f ( s, Y ( k,M ) , M ) − f ( s, ( Y ′ ) ( k,M ′ ) , M ′ ) (cid:13)(cid:13)(cid:13) p ds ≤ Z t (cid:16) C H k − s + C (cid:16)(cid:13)(cid:13) Y − Y ′ (cid:13)(cid:13) p + (cid:13)(cid:13) M − M ′ (cid:13)(cid:13) S p (cid:17)(cid:17) ds ≤ C Z t ( H k − s + δ ) ds, H kt ≤ ( C t ) k k ! H + (cid:18) C t + · · · + ( C t ) k k ! (cid:19) δ. In particular, (cid:13)(cid:13)(cid:13) F ( k ) ( Y, M ) − F ( k ) ( Y ′ , M ) (cid:13)(cid:13)(cid:13) S p ≤ C T ) k k ! (cid:0)(cid:13)(cid:13) Y − Y ′ (cid:13)(cid:13) S p + (cid:13)(cid:13) M − M ′ (cid:13)(cid:13) S p (cid:1) + (cid:0) e C T − (cid:1) C C (cid:16)(cid:13)(cid:13) Y − Y ′ (cid:13)(cid:13) p + (cid:13)(cid:13) M − M ′ (cid:13)(cid:13) S p (cid:17) . So for k large enough, there exists a constant C < c p such that (cid:13)(cid:13)(cid:13) F ( k ) ( Y, M ) − F ( k ) ( Y ′ , M ′ ) (cid:13)(cid:13)(cid:13) S p ≤ C (cid:0)(cid:13)(cid:13) Y − Y ′ (cid:13)(cid:13) S p + (cid:13)(cid:13) M − M ′ (cid:13)(cid:13) S p (cid:1) , and the proposition follows from Theorem 3.1. Remark 3.4.
The backward stochastic dynamics Y t = Z Tt f ( s, Y s , L ( M ) s ) ds + Z Tt f ( s, Y s ) dB s − ( M T − M t ) studied by Liang et al. (2011) can be viewed as a BSE with generator F t ( Y, M ) = Z t f ( s, Y s , L ( M ) s ) ds + Z t f ( s, Y s ) dB s . But it also fits into the framework (3.3) if the transformation ˜ M t = Z t f ( s, Y s ) dB s − M t and ˜ f ( t, Y, ˜ M ) = f (cid:18) t, Y t , L (cid:18)Z f ( s, Y s ) dB s − ˜ M (cid:19) t (cid:19) is applied. In addition, (3.3) includes BSDEs with drivers depending on the past or future of theprocesses Y and M , such as the time-delayed BSDEs of Delong and Imkeller (2010a, 2010b) or theanticipating BSDEs of Peng and Yang (2009). Previous existence and uniqueness results like Theorem3.3 of Liang et al. (2011), Theorem 2.1 of Delong and Imkeller (2010a) or Theorem 2.1 of Delong andImkeller (2010b), can all be recovered as special cases of Proposition 3.3. Remark 3.5.
Let f : [0 , T ] × Ω × S p × M p → R d be a driver satisfying condition (i) of Proposition 3.3for some p ∈ (1 , ∞ ] . If there exist nonnegative constants D , D such that (cid:13)(cid:13) f ( t, Y, M ) − f ( t, Y ′ , M ′ ) (cid:13)(cid:13) p ≤ D (cid:13)(cid:13) Y − Y ′ (cid:13)(cid:13) S p [0 ,t ] + D (cid:13)(cid:13) M − M ′ (cid:13)(cid:13) S p for all Y, Y ′ ∈ S p and M, M ′ ∈ M p , then (cid:13)(cid:13) f ( t, Y, M ) − f ( t, Y ′ , M ′ ) (cid:13)(cid:13) p ≤ D (cid:13)(cid:13) Y − Y + M − ( Y ′ − Y ′ + M ′ ) (cid:13)(cid:13) S p [0 ,t ] + D (cid:13)(cid:13) Y − Y ′ (cid:13)(cid:13) p + ( D + D ) (cid:13)(cid:13) M − M ′ (cid:13)(cid:13) S p . So the assumptions of Proposition 3.3 only hold if the constants D and D are small enough, oralternatively, the maturity T is sufficiently short. This is in line with Remark 3.2 above (note that(3.1) is a path-dependent BSDE of the form (3.3) with f ( t, Y, M ) = aY ).9he following corollary gives conditions under which it directly follows from Proposition 3.3 thatthe BSDE (3.3) has a unique solution for arbitrary Lipschitz constant and maturity. More examplesof (3.3) admitting solutions under general Lipschitz assumptions are given in Section 3.2 below. Corollary 3.6.
Let p ∈ (1 , ∞ ] and consider a terminal condition ξ ∈ L p ( F T ) d together with a driver f of the form (3.2) fulfilling condition (i) of Proposition 3.3 such that f ( t, Y, M ) = h ( t, Y − Y + M ) for amapping h : [0 , T ] × Ω × S p → R d . If (cid:13)(cid:13) h ( t, X ) − h ( t, X ′ ) (cid:13)(cid:13) p ≤ C (cid:13)(cid:13) X − X ′ (cid:13)(cid:13) S p [0 ,t ] , X, X ′ ∈ S p for a constant C ≥ , then the BSDE (3.3) has a unique solution ( Y, M ) ∈ S p × M p . Let W be an n -dimensional Brownian motion and N an independent Poisson random measure on [0 , T ] × E for E = R m \ { } with an intensity measure of the form dtµ ( dx ) for a measure µ over theBorel σ -algebra B ( E ) of E satisfying Z E (1 ∧ | x | ) µ ( dx ) < ∞ . Denote by ˜ N the compensated random measure N ( dt, dx ) − dtµ ( dx ) , and assume that, for A ∈ B ( E ) with µ ( A ) < ∞ , ˜ N ([0 , t ] × A ) and W are martingales with respect to F . We need the following spacesof integrands: • H : all R d × n -valued predictable processes Z satisfying k Z k H := (cid:18)Z T E | Z t | dt (cid:19) / < ∞ . • L ( ˜ N ) : all P ⊗ B ( E ) -measurable mappings U : [0 , T ] × Ω × E → R d such that k U k L ( ˜ N ) := (cid:18)Z T Z E E | U t ( x ) | µ ( dx ) dt (cid:19) / < ∞ , where P is the σ -algebra of F -predictable subsets of [0 , T ] × Ω .Any square-integrable F -martingale M ∈ M has a unique representation of the form M t = Z t Z Ms dW s + Z t Z E U Ms ( x ) ˜ N ( ds, dx ) + K Mt (3.4)for a triple ( Z M , U M , K M ) ∈ H × L ( ˜ N ) × M such that K M is strongly orthogonal to W and ˜ N (seee.g. Jacod, 1979). This makes it possible to consider BSDEs Y t = ξ + Z Tt f ( s, Y, Z M , U M ) ds + M T − M t (3.5)10or terminal conditions ξ ∈ L ( F T ) d and drivers f : [0 , T ] × Ω × S × H × L ( ˜ N ) → R d . (3.6)In the special case where the filtration F is generated by W and N , the orthogonal part K M in therepresentation (3.4) vanishes (see e.g. Ikeda and Watanabe, 1989), and as a result, (3.5) can be writtenas Y t = ξ + Z Tt f ( s, Y, Z M , U M ) ds + Z Tt Z Ms dW s + Z Tt Z E U Ms ( x ) ˜ N ( ds, dx ) . (3.7)This generalizes the classical BSDEs of Pardoux and Peng (1990) and Tang and Li (1994), whichhave drivers that at time s only depend on the realizations Y s ( ω ) , Z Ms ( ω ) , U Ms ( ω ) , to equations withfunctional drivers that can depend on the full processes Y , Z M and U M .In the rest of this subsection, we consider different specifications of (3.5) with drivers dependingon the future, present or past of the processes Y , Z M and U M . In all instances, we are able to derivethe existence of a unique solution for an arbitrary Lipschitz constant and maturity. In the followingproposition, the driver can depend on the present and future of Y , Z M and U M , but not on their past– this is ruled out by condition (ii). For its proof, we need the isometry E | M t | = Z t E | Z Ms | ds + Z t Z E E | U Ms ( x ) | µ ( dx ) ds + E | K Mt | (3.8)(see e.g. Jacod, 1979). Proposition 3.7.
The BSDE (3.5) has a unique solution ( Y, M ) ∈ S × M for every terminal condition ξ ∈ L ( F T ) d and driver f : [0 , T ] × Ω × S × H × L ( ˜ N ) → R d satisfying the following two conditions: (i) For all ( Y, Z, U ) ∈ S × H × L ( ˜ N ) , f ( t, Y, Z, U ) is progressively measurable with R T k f ( t, , , k dt < ∞ . (ii) There exists a constant C ≥ such that Z Tt (cid:13)(cid:13) f ( s, Y, Z, U ) − f ( s, Y ′ , Z ′ , U ′ ) (cid:13)(cid:13) ds ≤ C Z Tt (cid:13)(cid:13) Y s − Y ′ s (cid:13)(cid:13) + (cid:13)(cid:13) Z s − Z ′ s (cid:13)(cid:13) + (cid:13)(cid:13) U s − U ′ s (cid:13)(cid:13) L ( P ⊗ µ ) ds for all t ∈ [0 , T ] and ( Y, Z, U ) , ( Y ′ , Z ′ , U ′ ) ∈ S × H × L ( ˜ N ) .Proof. Choose δ > so that C p δ ( δ + 1) < and k := T /δ ∈ N . By (3.8), one has for every M ∈ M , (cid:18)Z t (cid:13)(cid:13) Z Ms (cid:13)(cid:13) + (cid:13)(cid:13) U Ms (cid:13)(cid:13) L ( P ⊗ µ ) ds (cid:19) ≤ t Z t (cid:16)(cid:13)(cid:13) Z Ms (cid:13)(cid:13) + (cid:13)(cid:13) U Ms (cid:13)(cid:13) L ( P ⊗ µ ) (cid:17) ds ≤ t Z t (cid:13)(cid:13) Z Ms (cid:13)(cid:13) + (cid:13)(cid:13) U Ms (cid:13)(cid:13) L ( P ⊗ µ ) ds ≤ t k M t k . ( Y, M ) ∈ S × M , (cid:13)(cid:13)(cid:13)(cid:13)Z TT − δ | f ( s, Y, Z M , U M ) | ds (cid:13)(cid:13)(cid:13)(cid:13) ≤ Z TT − δ (cid:13)(cid:13) f ( s, Y, Z M , U M ) (cid:13)(cid:13) ds ≤ Z TT − δ k f ( s, , , k ds + C Z TT − δ (cid:16) k Y s k + (cid:13)(cid:13) Z Ms (cid:13)(cid:13) + (cid:13)(cid:13) U Ms (cid:13)(cid:13) L ( P ⊗ µ ) (cid:17) ds < ∞ , where the first inequality follows from the same argument as in the proof of Proposition 3.3. Inparticular, for every pair ( Y, M ) ∈ S × M , F t ( Y, M ) := Z t f ( s, Y, Z M , U M )1 [ T − δ,T ] ( s ) ds defines a process in S . Furthermore, one has (cid:13)(cid:13) F ( Y, M ) − F ( Y ′ , M ′ ) (cid:13)(cid:13) S ≤ (cid:13)(cid:13)(cid:13)(cid:13)Z TT − δ (cid:12)(cid:12)(cid:12) f ( s, Y, Z M , U M ) − f ( s, Y ′ , Z M ′ , U M ′ ) (cid:12)(cid:12)(cid:12) ds (cid:13)(cid:13)(cid:13)(cid:13) ≤ Z TT − δ (cid:13)(cid:13)(cid:13) f ( s, Y, Z M , U M ) − f ( s, Y ′ , Z M ′ , U M ′ ) (cid:13)(cid:13)(cid:13) ds ≤ C Z TT − δ (cid:13)(cid:13) Y s − Y ′ s (cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) Z Ms − Z M ′ s (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) U Ms − U M ′ s (cid:13)(cid:13)(cid:13) L ( P ⊗ µ ) ds ≤ C s δ Z TT − δ (cid:16) k Y s − Y ′ s k + k Z Ms − Z M ′ s k + k U Ms − U M ′ s k L ( P ⊗ µ ) (cid:17) ds ≤ C s δ Z TT − δ k Y s − Y ′ s k + k Z Ms − Z M ′ s k + k U Ms − U M ′ s k L ( P ⊗ µ ) ds ≤ C q δ k Y − Y ′ k S + 3 δ k M − M ′ k S ≤ C p δ ( δ + 1)( (cid:13)(cid:13) Y − Y ′ (cid:13)(cid:13) S + (cid:13)(cid:13) M − M ′ (cid:13)(cid:13) S ) for all ( Y, M ) , ( Y ′ , M ′ ) ∈ S × M . Since C p δ ( δ + 1) < / , one obtains from Theorem 3.1 that theBSDE Y t = ξ + Z Tt f ( s, Y, Z M , U M )1 [ T − δ,T ] ( s ) ds + M T − M t has a unique solution ( Y ( k ) , M ( k ) ) in S × M . Now, consider the BSDE Y t = Y ( k ) T − δ + Z T − δt f ( k − ( s, Y, Z M , U M )1 [ T − δ,T − δ ] ( s ) ds + M T − δ − M t (3.9)on the time interval [0 , T − δ ] , where f ( k − is given by f ( k − ( s, Y, Z, U ) := f (cid:0) s, ( Y, Z, U )1 [0 ,T − δ ) + (cid:0) Y ( k ) , Z M ( k ) , U M ( k ) (cid:1) [ T − δ,T ] (cid:1) . ( Y ( k − , M ( k − ) in S × M over thetime interval [0 , T − δ ] . Repeating the same argument, one obtains solutions ( Y ( j ) , M ( j ) ) , j = 1 , . . . , k .If one sets Y t := Y (1) t , M t := M (1) t for ≤ t ≤ δ and Y t := Y ( j ) t , M t − M ( j − δ := M ( j ) t − M ( j )( j − δ for ( j − δ < t ≤ jδ , j = 2 , . . . , k , then ( Z Mt , U Mt ) = ( Z M ( j ) t , U M ( j ) t ) for ( j − δ < t ≤ jδ . Since thisconstruction is backwards in time and by condition (ii), f ( t, Y, Z M , U M ) cannot depend on the past ofthe processes Y, Z M and U M , the pair ( Y, M ) forms a unique solution of (3.5) in S × M . Remark 3.8.
The assumptions of Proposition 3.7 allow for drivers f such that f ( t, Y, Z, U ) dependson the future of the processes Y, Z, U in a general F t -measurable way. This covers BSDEs with antic-ipating drivers of the form − dY t = f ( t, Y t , Z t , E t Y t + δ ( t ) , E t Z t + ζ ( t ) ) dt + Z t dW t , t ∈ [0 , T ]( Y t , Z t ) = ( ξ t , η t ) , t ∈ [ T, T + K ] or more generally, − dY t = f ( t, Y t , Z t , Y t + δ ( t ) , Z t + ζ ( t ) ) dt + Z t dW t , t ∈ [0 , T ]( Y t , Z t ) = ( ξ t , η t ) , t ∈ [ T, T + K ] (3.10)for a Brownian motion ( W t ) t ∈ [0 ,T ] , continuous functions δ, ζ : [0 , T ] → R + , and stochastic processes ( ξ t ) t ∈ [ T,T + K ] , ( η t ) t ∈ [ T,T + K ] . Equations of the form (3.10) were introduced by Peng and Yang (2009) asduals of time-delayed forward SDEs. Their existence and uniqueness result, Theorem 4.2, as well asextensions for equations with jumps, can easily be derived from Proposition 3.7.As an immediate consequence of Proposition 3.7 one obtains the following result for BSDEs withfunctional drivers depending on Y s , Z Ms and U Ms . Corollary 3.9.
The BSDE Y t = ξ + Z Tt f ( s, Y s , Z Ms , U Ms ) ds + M T − M t (3.11) has a unique solution ( Y, M ) ∈ S × M for every terminal condition ξ ∈ L ( F T ) d and driver f : [0 , T ] × Ω × L ( F T ) d × L ( F T ) d × n × L (Ω × E, F T ⊗ B ( E ) , P ⊗ µ ; R d ) → R d satisfying the following two conditions: (i) For all ( Y, Z, U ) ∈ S × H × L ( ˜ N ) , f ( t, Y t , Z t , U t ) is progressively measurable with R T k f ( t, , , k dt < ∞ . (ii) There exists a constant C ≥ such that (cid:13)(cid:13) f ( t, Y t , Z t , U t ) − f ( t, Y ′ t , Z ′ t , U ′ t ) (cid:13)(cid:13) ≤ C (cid:16)(cid:13)(cid:13) Y t − Y ′ t (cid:13)(cid:13) + (cid:13)(cid:13) Z t − Z ′ t (cid:13)(cid:13) + (cid:13)(cid:13) U t − U ′ t (cid:13)(cid:13) L ( P × µ ) (cid:17) for all t ∈ [0 , T ] and ( Y, Z, U ) , ( Y ′ , Z ′ , U ′ ) ∈ S × H × L ( ˜ N ) . Proposition 3.10.
Let ξ ∈ L ( F T ) d and ν be a finite Borel measure on [0 , T ] . Then the BSDE Y t = ξ + Z Tt Z [0 ,s ] g ( s − r, Z Ms − r , U Ms − r ) ν ( dr ) ds + M T − M t (3.12) has a unique solution ( Y, M ) ∈ S × M for every mapping g : [0 , T ] × Ω × L ( F T ) d × n × L (Ω × E, F T ⊗ B ( E ) , P ⊗ µ ; R d ) → R d satisfying the following two conditions: (i) For all ( Z, U ) ∈ H × L ( ˜ N ) , g ( t, Z t , U t ) is progressively measurable, and R T k g ( t, , k dt < ∞ . (ii) There exists a constant C ≥ such that (cid:13)(cid:13) g ( t, Z t , U t ) − g ( t, Z ′ t , U ′ t ) (cid:13)(cid:13) ≤ C (cid:16)(cid:13)(cid:13) Z t − Z ′ t (cid:13)(cid:13) + (cid:13)(cid:13) U t − U ′ t (cid:13)(cid:13) L ( P ⊗ µ ) (cid:17) for all t ∈ [0 , T ] and ( Z, U ) , ( Z ′ , U ′ ) ∈ H × L ( ˜ N ) .Proof. The generator corresponding to the BSDE (3.12) is given by F t ( M ) = Z t Z [0 ,s ] g ( s − r, Z Ms − r , U Ms − r ) ν ( dr ) ds. Since it does not depend on Y , it satisfies condition (S). So, by Theorem 2.3, it is enough to show thatthere exists a unique V ∈ L ( F T ) d such that V = G ( V ) = ξ + Z T Z [0 ,s ] g ( s − r, Z M V s − r , U M V s − r ) ν ( dr ) ds. (3.13)From Fubini’s theorem and a change of variable, one obtains Z T Z [0 ,s ] g ( s − r, Z M V s − r , U M V s − r ) ν ( dr ) ds = Z T ν ([0 , T − s ]) g ( s, Z M V s , U M V s ) ds. Since the driver h ( s, Z s , U s ) = ν ([0 , T − s ]) g ( s, Z s , U s ) satisfies the conditions of Corollary 3.9, the BSDE Y t = ξ + Z Tt h ( s, Z Ms , U Ms ) ds + M T − M t has a unique solution in S × M . The associated generator, ˜ F t ( M ) = R T h ( s, Z Ms , U Ms ) ds , does notdepend on Y either. So it also satisfies condition (S), and one obtains from Theorem 2.3 that thereexists a unique V ∈ L ( F T ) d satisfying (3.13). This completes the proof.14s special cases of Corollary 3.9 and Proposition 3.10, one obtains existence and uniqueness re-sults for McKean–Vlasov type BSDEs with drivers depending on the realizations Y s ( ω ) , Z Ms ( ω ) , U Ms ( ω ) as well as the distributions L ( Y s ) , L ( Z Ms ) , L ( U Ms ) of Y s , Z Ms and U Ms . We recall that if M ( X ) is theset of all probability measures defined on the Borel σ -algebra of a normed vector space ( X , k·k ) , the p -Wasserstein metric on M p ( X ) := (cid:8) η ∈ M ( X ) : R X k x k p η ( dx ) < ∞ (cid:9) is given by W p ( η, η ′ ) := inf (cid:26)Z X ×X k x − x ′ k p ψ ( dx, dx ′ ) : ψ ∈ M p ( X × X ) with marginals η and η ′ (cid:27) /p . The following is a consequence of Corollary 3.9 and generalizes the existence and uniquenessresult for mean-field BSDEs of Buckdahn et al. (2009).
Corollary 3.11.
Consider a BSDE of the form Y t = ξ + Z Tt f ( s, Y s , Z Ms , U Ms , L ( Y s ) , L ( Z Ms ) , L ( U Ms )) ds + M T − M t (3.14) for a terminal condition ξ ∈ L ( F T ) d and a driver f from [0 , T ] × Ω × R d × R d × n × L ( E, B ( E ) , µ ; R d ) ×M ( R d ) ×M ( R d × n ) ×M ( L ( E, B ( E ) , µ ; R d )) to R d . Then (3.14) has a unique solution ( Y, M ) in S × M if for fixed ( y, z, u, η, ζ, κ ) ∈ R d × R d × n × L ( E, B ( E ) , µ ; R d ) × M ( R d ) × M ( R d × n ) × M ( L ( E, B ( E ) , µ ; R d )) ,f ( · , y, z, u, η, ζ, κ ) is progressively measurable, and the following two conditions hold: (i) R T k f ( t, , , , L (0) , L (0)) , L (0) k dt < ∞ (ii) There exists a constant C ≥ such that | f ( t, y, z, u, η, ζ, κ ) − f ( t, y ′ , z ′ , u ′ , η ′ , ζ ′ , κ ′ ) |≤ C (cid:16) | y − y ′ | + | z − z ′ | + (cid:13)(cid:13) u − u ′ (cid:13)(cid:13) L ( µ ) + W ( η, η ′ ) + W ( ζ, ζ ′ ) + W ( κ, κ ′ ) (cid:17) . Proof.
It follows from the assumptions that the driver f is progressively measurable in ( t, ω ) andcontinuous in ( y, z, u, η, ζ, κ ) . Since R d × R d × n × L ( E, B ( E ) , µ ; R d ) × M ( R d ) × M ( R d × n ) × M ( L ( E, B ( E ) , µ ; R d )) is a separable metric space, one obtains from Lemma 4.51 of Aliprantis and Border (2006) that f isjointly measurable in all its arguments. This implies that f ( t, Y t , Z t , U t , L ( Y t ) , L ( Z t ) , L ( U t )) is proges-sively measurable for every triple ( Y, Z, U ) ∈ S × H × U ∈ L ( ˜ N ) . It follows that condition (i) ofCorollary 3.9 holds, and it just remains to show that (cid:13)(cid:13) f ( t, Y t , Z t , U t , L ( Y t ) , L ( Z t ) , L ( U t )) − f ( t, Y ′ t , Z ′ t , U ′ t , L ( Y ′ t ) , L ( Z ′ t ) , L ( U ′ t ) (cid:13)(cid:13) ≤ D (cid:16)(cid:13)(cid:13) Y t − Y ′ t (cid:13)(cid:13) + (cid:13)(cid:13) Z t − Z ′ t (cid:13)(cid:13) + (cid:13)(cid:13) U t − U ′ t (cid:13)(cid:13) L ( P × µ ) (cid:17) for some constant D . But this is a consequence of condition (ii) since one has W ( L ( Y t ) , L ( Y ′ t )) ≤ Z R d × R d | y − y ′ | L ( Y t , Y ′ t )( dy, dy ′ ) = (cid:13)(cid:13) Y t − Y ′ t (cid:13)(cid:13) , W ( L ( Z t ) , L ( Z ′ t )) ≤ (cid:13)(cid:13) Z t − Z ′ t (cid:13)(cid:13) , W ( L ( U t ) , L ( U t )) ≤ (cid:13)(cid:13) U t − U ′ t (cid:13)(cid:13) L ( P × µ ) . Using the same arguments as in the proof of Corollary 3.11, one obtains from Proposition 3.10 thefollowing result for time-delayed McKean–Vlasov type BSDEs.
Corollary 3.12.
Consider a BSDE of the form Y t = ξ + Z Tt Z s g ( s − r, Z Ms − r , U Ms − r , L ( Z Ms − r ) , L ( U Ms − r )) ν ( dr ) ds + M T − M t (3.15) for a terminal condition ξ ∈ L ( F T ) d , a finite Borel measure ν on [0 , T ] and a mapping g : [0 , T ] × Ω × R d × n × L ( E, B ( E ) , µ ; R d ) × M ( R d × n ) × M ( L ( E, B ( E ) , µ ; R d )) → R d . Then (3.15) has a unique solution ( Y, M ) in S × M if for fixed ( z, u, ζ, κ ) ∈ R d × n × L ( E, B ( E ) , µ ; R d ) × M ( R d × n ) × M ( L ( E, B ( E ) , µ ; R d )) ,g ( · , z, u, ζ, κ ) is progressively measurable, and the following two conditions hold: (i) R T k g ( t, , , L (0)) , L (0) k dt < ∞ (ii) There exists a constant C ≥ such that | g ( t, z, u, ζ, κ ) − g ( t, z ′ , u ′ , ζ ′ , κ ′ ) | ≤ C (cid:16) | z − z ′ | + (cid:13)(cid:13) u − u ′ (cid:13)(cid:13) L ( µ ) + W ( ζ, ζ ′ ) + W ( κ, κ ′ ) (cid:17) . In this section we use compactness assumptions to derive existence results for different BSEs andBSDEs with non-Lipschitz coefficients. To find compact sets in the space L ( F T ) d , we assume in allof Section 4 that the sample space Ω is an infinite-dimensional separable Hilbert space with innerproduct h· , ·i and corresponding norm k·k . We fix a complete orthonormal system e j , j ∈ N , of Ω together with positive numbers λ j , j ∈ N satisfying P j ∈ N λ j < ∞ . Then Qe j := λ j e j defines a positiveself-adjoint trace class operator Q : Ω → Ω . The mean zero Gaussian measure P with covariance Q is the unique probability measure on the Borel σ -algebra B (Ω) of Ω under which the functions φ j ( ω ) = h ω, e j i , j ∈ N , are independent normal random variables with mean zero and variance λ j , j ∈ N ; see Da Prato (2006) for details. The map e j φ j / p λ j has a unique continuous linear extension W : Ω → L (Ω) , called white noise mapping. It is an isometry between Ω and the closed subspace of L (Ω) generated by φ j , j ∈ N .To define the Sobolev space W , (Ω) in L (Ω) , let E (Ω) be the linear span of all real and imaginaryparts of functions of the form ω e i h ω,η i for some η ∈ Ω . For ϕ ∈ E (Ω) , we denote by D j ϕ the derivativeof ϕ in the direction of e j : D j ϕ ( ω ) = lim ε → ϕ ( ω + εe j ) − ϕ ( ω ) ε . D : E (Ω) ⊆ L (Ω) → L (Ω; Ω) , ϕ Dϕ := P j ∈ N D j ϕe j is closable. We maintain thenotation D for the closure of D and denote its domain by W , (Ω) . Endowed with the inner product h ϕ, ψ i W , := E ( ϕψ + h Dϕ, Dψ i ) , the Sobolev space W , (Ω) becomes a Hilbert space. For ϕ ∈ L (Ω) d and ψ ∈ W , (Ω) d , we set k ϕ k := d X i =1 E ϕ i , k Dψ k := d X i =1 E h Dψ i , Dψ i i and k ψ k W , := k ψ k + k Dψ k . Theorem 10.25 of Da Prato (2006) shows that every ϕ ∈ W , (Ω) d satisfies the Poincar´e inequality E | ϕ − E ϕ | ≤ λ k Dϕ k for λ := max j λ j . (4.1)Moreover, by Theorem 10.16 of Da Prato (2006), every bounded set in W , (Ω) d is relatively compactin L (Ω) d .We say a function ϕ : Ω → R d is ω -Lipschitz with constant L ≥ if (cid:12)(cid:12) ϕ ( ω ) − ϕ ( ω ′ ) (cid:12)(cid:12) ≤ L (cid:13)(cid:13) ω − ω ′ (cid:13)(cid:13) for all ω, ω ′ ∈ Ω . It follows from Propositon 10.11 of Da Prato (2006) that every ω -Lipschitz function ϕ : Ω → R d withconstant L belongs to W , (Ω) d with k Dϕ k ≤ L . In particular, one obtains that for given numbers K, L ≥ , the set of all ω -Lipschitz ϕ : Ω → R d with constant L satisfying | E ϕ | ≤ K is compact in L (Ω) d . Moreover, the following holds: Lemma 4.1.
Let h : l → R d be a mapping satisfying | h ( x ) − h ( y ) | ≤ K k x − y k for some constant K ≥ . Then for any x ∈ l , ϕ = h (cid:16)p λ j x j W ( e j ) , j ∈ N (cid:17) is an ω -Lipschitz random variable with constant K k x k .Proof. One has | ϕ ( ω ) − ϕ ( ω ′ ) | ≤ K k x j h ω − ω ′ , e j i , j ∈ N k ≤ K k x k k ω − ω ′ k . Remark 4.2.
The assumptions on Ω in this section are not restrictive for the purpose of studyingBSEs and BSDEs. For instance, they allow for probability spaces rich enough to support an n -dimensional Brownian motion together with an independent Poisson random measure on [0 , T ] × R m \ { } . For an explicit construction, one can e.g., choose Ω to be of the form Ω = L ([0 , T ]; R n ) ⊕ l ,where L ([0 , T ]; R n ) is the space of square-integrable measurable functions from [0 , T ] to R n and l thespace of square-summable sequences. The inner product on L ([0 , T ]; R n ) ⊕ l is given by (cid:10) ( h, x ) , ( h ′ , x ′ ) (cid:11) = Z T h ( s ) · h ′ ( s ) ds + X j ∈ N x j x ′ j , where · denotes the standard scalar product on R n . Let P be a mean zero Gaussian measure corre-sponding to a positive self-adjoint trace class operator given by Qe j = λ j e j for a complete orthonormalsystem ( e j ) of Ω and positive numbers ( λ j ) satisfying P j ∈ N λ j < ∞ . If W : Ω → L (Ω) is the corre-sponding white noise mapping, b i denotes the i -th unit vector in R n and ( c j ) is a complete orthonormal17ystem in l , then W it := W ( b i [0 ,t ] , defines an n -dimensional Brownian motion independent of thesequence ζ j := W (0 , c j ) of independent standard normals. For a given σ -finite measure µ on the Borel σ -algebra of R m \ { } , a Poisson random measure N on [0 , T ] × R m \ { } with intensity measure dtµ ( dx ) can be realized as a function of ζ j , j ∈ N . Alternatively, N can be realized with only ζ j − , j ∈ N , and ζ j , j ∈ N , can be used to model additional noise. Denote by F the completion of the Borel σ -algebra B (Ω) with respect to P , and let F = ( F t ) t ∈ [0 ,T ] be ageneral filtration satisfying the usual conditions. The following theorem provides a general existenceresult for non-Lipschitz BSEs. It uses the theorem of Krasnoselskii (1964), which combines the fixedpoint results of Banach and Schauder; for a textbook treatment, see e.g., Smart (1974). Theorem 4.3.
Let ξ ∈ L ( F T ) d and assume F is of the form F = F + F for mappings F , F : S × M → S . Then the BSE (2.1) has a solution ( Y, M ) ∈ S × M if there exist constants C < and R , R , R ≥ such that the following hold: (i) k F ( Y, M ) − F ( Y ′ , M ) k S ≤ C k Y − Y ′ k S and F ( Y, M ) ∈ S is continuous in M ∈ M (ii) (cid:13)(cid:13) F T ( Y, M ) − F T ( Y ′ , M ′ ) (cid:13)(cid:13) ≤ C q k Y − Y ′ k + k M − M ′ k S / (iii) For all ( Y, M ) ∈ S × M satisfying q k Y k + k M k S / ≤ R , one has F T ( Y, M ) ∈ W , (Ω) d with (cid:13)(cid:13) F T ( Y, M ) (cid:13)(cid:13) ≤ R and (cid:13)(cid:13) DF T ( Y, M ) (cid:13)(cid:13) ≤ R (iv) k ξ k + (cid:13)(cid:13) F T (0 , (cid:13)(cid:13) + CR + R ≤ R .Proof. By Lemma 2.5, it follows from condition (i) that F satisfies (S). So by Theorem 2.3, it is enoughto show that the mapping V G ( V ) = ξ + F T ( Y V , M V ) has a fixed point in L ( F T ) d . To do that wedefine C := (cid:8) V ∈ L ( F T ) d : k V k ≤ R (cid:9) , G ( V ) := ξ + F T ( Y V , M V ) , G ( V ) := F T ( Y V , M V ) and showthe following: 1) G is a contraction on L ( F T ) d ; 2) G is continuous with respect to k . k ; 3) G maps C into a compact subset of L ( F T ) d ; and 4) G ( V ) + G ( V ′ ) ∈ C for all V, V ′ ∈ C . Then it follows fromKrasnoselskii’s theorem that G has a fixed point.Step 1: G : L ( F T ) d → L ( F T ) d is a contraction with respect to k . k :It follows from (ii) that (cid:13)(cid:13) G ( V ) − G ( V ′ ) (cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) F T ( Y V , M V ) − F T ( Y V ′ , M V ′ ) (cid:13)(cid:13)(cid:13) ≤ C (cid:18)(cid:13)(cid:13)(cid:13) Y V − Y V ′ (cid:13)(cid:13)(cid:13) + 14 (cid:13)(cid:13)(cid:13) M V − M V ′ (cid:13)(cid:13)(cid:13) S (cid:19) . By Doob’s L -maximal inequality, one has (cid:13)(cid:13)(cid:13) M V − M V ′ (cid:13)(cid:13)(cid:13) S ≤ (cid:13)(cid:13)(cid:13) M VT − M V ′ T (cid:13)(cid:13)(cid:13) . Therefore, (cid:13)(cid:13) G ( V ) − G ( V ′ ) (cid:13)(cid:13) ≤ C (cid:18)(cid:13)(cid:13) E ( V − V ′ ) (cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) M VT − M V ′ T (cid:13)(cid:13)(cid:13) (cid:19) ≤ C (cid:13)(cid:13) V − V ′ (cid:13)(cid:13) , which shows that G is a contraction. 18tep 2: G : L ( F T ) d → L ( F T ) d is continuous with respect to k . k :By Doob’s L -maximal inequality, V M V is a continuous mapping from L ( F T ) d to M . Moreover,since Y Vt = ˆ M Vt − F t ( Y V , M V ) for ˆ M Vt := E t V = E V − M Vt , one obtains from the first part of condition (i) that (cid:13)(cid:13)(cid:13) Y V − Y V ′ (cid:13)(cid:13)(cid:13) S ≤ (cid:13)(cid:13)(cid:13) ˆ M V − ˆ M V ′ (cid:13)(cid:13)(cid:13) S + (cid:13)(cid:13)(cid:13) F ( Y V , M V ) − F ( Y V ′ , M V ′ ) (cid:13)(cid:13)(cid:13) S ≤ (cid:13)(cid:13) V − V ′ (cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) F ( Y V , M V ) − F ( Y V , M V ′ ) (cid:13)(cid:13)(cid:13) S + (cid:13)(cid:13)(cid:13) F ( Y V , M V ′ ) − F ( Y V ′ , M V ′ ) (cid:13)(cid:13)(cid:13) S ≤ (cid:13)(cid:13) V − V ′ (cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) F ( Y V , M V ) − F ( Y V , M V ′ ) (cid:13)(cid:13)(cid:13) S + C (cid:13)(cid:13)(cid:13) Y V − Y V ′ (cid:13)(cid:13)(cid:13) S . Therefore, (1 − C ) (cid:13)(cid:13)(cid:13) Y V − Y V ′ (cid:13)(cid:13)(cid:13) S ≤ (cid:13)(cid:13) V − V ′ (cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) F ( Y V , M V ) − F ( Y V , M V ′ ) (cid:13)(cid:13)(cid:13) S , and it follows from the second part of (i) that V Y V is continuous from L ( F T ) d to S . Since F = F − F , one obtains from (i) and (ii) that ( Y V , M V ) F T ( Y V , M V ) is continuous from S × M to L ( F T ) d . This proves the continuity of G .Step 3: G ( C ) is contained in a compact subset of L ( F T ) d :For V ∈ C , one has (cid:13)(cid:13) Y V (cid:13)(cid:13) + 14 (cid:13)(cid:13) M V (cid:13)(cid:13) S ≤ k E V k + (cid:13)(cid:13) M VT (cid:13)(cid:13) = k V k ≤ R . (4.2)So it follows from (iii) that F T ( Y V , M V ) is in W , (Ω) d with (cid:13)(cid:13) F T ( Y V , M V ) (cid:13)(cid:13) ≤ R and (cid:13)(cid:13) DF T ( Y V , M V ) (cid:13)(cid:13) ≤ R . Since bounded subsets of W , (Ω) d are relatively compact in L (Ω) d , thisshows that G ( C ) is contained in a compact subset of L ( F T ) d .Step 4: G ( V ) + G ( V ′ ) ∈ C for all V, V ′ ∈ C :If V ∈ C , one obtains from (4.2) that (cid:13)(cid:13) Y V (cid:13)(cid:13) + (cid:13)(cid:13) M V (cid:13)(cid:13) S / ≤ R . So it follows from (ii) that (cid:13)(cid:13) G ( V ) (cid:13)(cid:13) ≤ k ξ k + (cid:13)(cid:13) F T ( Y V , M V ) (cid:13)(cid:13) ≤ k ξ k + (cid:13)(cid:13) F T (0 , (cid:13)(cid:13) + C ( (cid:13)(cid:13) Y V (cid:13)(cid:13) + (cid:13)(cid:13) M V (cid:13)(cid:13) S / / ≤ k ξ k + (cid:13)(cid:13) F T (0 , (cid:13)(cid:13) + CR . By (iii), one has (cid:13)(cid:13) G ( V ′ ) (cid:13)(cid:13) ≤ R . Therefore, one obtains from (iv) that (cid:13)(cid:13) G ( V ) + G ( V ′ ) (cid:13)(cid:13) ≤ R .So Krasnoselskii’s theorem applies, and one can conclude that G has a fixed point in L ( F T ) d .Assumption (i) of Theorem 4.3 is needed to ensure that condition (S) holds and F T ( Y, M ) is contin-uous in ( Y, M ) . In the following special case it is not needed. Proposition 4.4.
Let ξ ∈ L ( F T ) d and assume F is of the form F ( Y, M ) = F ( Y , M ) + F ( Y , M ) formappings F , F : L ( F ) d × M → S . Then the BSE (2.1) has a solution ( Y, M ) ∈ S × M if there exista constant C < and a nondecreasing function ρ : R + → R + satisfying lim sup x →∞ ρ ( x ) x < − C (4.3) such that the following two conditions hold: (cid:13)(cid:13) F T ( Y , M ) − F T ( Y ′ , M ′ ) (cid:13)(cid:13) ≤ C q k Y − Y ′ k + k M − M ′ k S / (ii) F T : L ( F ) d × M → L ( F T ) d is continuous and takes values in W , (Ω) d with | E F T ( Y , M ) | + λ (cid:13)(cid:13) DF T ( Y , M ) (cid:13)(cid:13) ≤ ρ (cid:18)q k Y k + k M k S / (cid:19) . Proof.
Since F only depends on Y and M , condition (S) holds trivially. By Theorem 2.3, the proposi-tion follows if we can show that V G ( V ) = ξ + F T ( Y V , M V ) has a fixed point in L ( F T ) d . To do that,we fix a constant R ≥ and define C , G and G as in the proof of Theorem 4.3. Then one obtainsfrom (i) like in the proof of Theorem 4.3 that G is a contraction on L ( F T ) d . Condition (ii) impliesthat G is continuous with respect to k . k , and since ρ (cid:18)q(cid:13)(cid:13) Y V (cid:13)(cid:13) + k M V k S / (cid:19) ≤ ρ (cid:18)q(cid:13)(cid:13) Y V (cid:13)(cid:13) + (cid:13)(cid:13) M VT (cid:13)(cid:13) (cid:19) = ρ ( k V k ) , that G ( C ) is relatively compact in L ( F T ) d . Due to (4.3), one has k ξ k + (cid:13)(cid:13) F T (0 , (cid:13)(cid:13) + CR + ρ ( R ) ≤ R if R is chosen large enough. Then for V, V ′ ∈ C , (cid:13)(cid:13) G ( V ) (cid:13)(cid:13) ≤ k ξ k + (cid:13)(cid:13) F T ( Y V , M V ) (cid:13)(cid:13) ≤ k ξ k + (cid:13)(cid:13) F T (0 , (cid:13)(cid:13) + C ( (cid:13)(cid:13) Y V (cid:13)(cid:13) + (cid:13)(cid:13) M VT (cid:13)(cid:13) ) / ≤ k ξ k + (cid:13)(cid:13) F T (0 , (cid:13)(cid:13) + CR , and, by Poincar´e’s inequality, (cid:13)(cid:13) G ( V ′ ) (cid:13)(cid:13) ≤ | E F T ( Y V ′ , M V ′ ) | + λ (cid:13)(cid:13)(cid:13) DF T ( Y V ′ , M V ′ ) (cid:13)(cid:13)(cid:13) ≤ ρ (cid:18)q k Y V ′ k + k M V ′ k S / (cid:19) ≤ ρ (cid:18)q k Y V ′ k + k M V ′ T k (cid:19) = ρ ( (cid:13)(cid:13) V ′ (cid:13)(cid:13) ) . Therefore, (cid:13)(cid:13) G ( V ) + G ( V ′ ) (cid:13)(cid:13) ≤ k ξ k + (cid:13)(cid:13) F T (0 , (cid:13)(cid:13) + CR + ρ ( R ) ≤ R , and it follows from Krasnoselskii’s theorem that G has a fixed point in L ( F T ) d .As a consequence of Proposition 4.4 one obtains an existence result for BSDEs Y t = ξ + Z Tt f ( s, Y , M ) ds + M T − M t (4.4)with drivers f depending on Y and the whole martingale M . Corollary 4.5.
Let ξ ∈ L ( F T ) d and assume f to be of the form f = f + f for mappings f , f :[0 , T ] × Ω × L ( F ) d × M → R d . Then the BSDE (4.4) has a solution ( Y, M ) ∈ S × M if there exist aconstant C < T − and a nondecreasing function ρ : R + → R + satisfying lim sup x →∞ ρ ( x ) x < − CT such that the following two conditions hold: For all ( Y , M ) ∈ L ( F ) d × M , f ( · , Y , M ) is progressively measurable with R T | f ( t, , | dt ∈ L ( F T ) , and (cid:13)(cid:13) f ( t, Y , M ) − f ( t, Y ′ , M ′ ) (cid:13)(cid:13) ≤ C q k Y − Y ′ k + k M − M ′ k S / . (ii) For all ( Y , M ) ∈ L ( F ) d × M , f ( ., Y , M ) is progressively measurable with R T | f ( t, Y , M ) | dt ∈ L ( F T ) , and J ( Y , M ) := R T f ( t, Y , M ) dt defines a continuous mapping J : L ( F ) d × M → L ( F T ) d with values in W , (Ω) d such that | E J ( Y , M ) | + λ k DJ ( Y , M ) k ≤ ρ (cid:18)q k Y k + k M T k S / (cid:19) . Proof.
It follows from the assumptions that for all Y and M , F it ( Y , M ) = R t f i ( s, Y , M ) ds belongs to S for i = 1 , , and E (cid:12)(cid:12) F T ( Y , M ) − F T ( Y ′ , M ′ ) (cid:12)(cid:12) ≤ C T (cid:16)(cid:13)(cid:13) Y − Y ′ (cid:13)(cid:13) + (cid:13)(cid:13) M − M ′ (cid:13)(cid:13) S / (cid:17) . So the conditions of Proposition 4.4 hold with CT instead of C , and the corollary follows.If F does not depend on Y , the assumptions of Theorem 4.3 can be relaxed further, and one obtainsthe following Theorem 4.6.
Let ξ ∈ L ( F T ) d and assume F is of the form F ( Y, M ) = F ( M ) + F ( M ) for mappings F , F : M → S . Then the BSE (2.1) has a solution ( Y, M ) ∈ S × M if there exist a constant C < / and a nondecreasing function ρ : R + → R + satisfying lim sup x →∞ ρ ( x ) x < / − C √ λ (4.5) such that the following two conditions hold: (i) (cid:13)(cid:13) F T ( M ) − E F T ( M ) − ( F T ( M ′ ) − E F T ( M ′ )) (cid:13)(cid:13) ≤ C k M − M ′ k S (ii) F T : M → L ( F T ) d is continuous and takes values in W , (Ω) d with (cid:13)(cid:13) DF T ( M ) (cid:13)(cid:13) ≤ ρ ( k M k S ) .Proof. By Corollary 2.4, it is enough to show that the mapping V G ( V ) = ξ − E ξ + F T ( M V ) − E F T ( M V ) has a fixed point in L ( F T ) d . For a given constant R ≥ , define C := (cid:8) V ∈ L ( F T ) d : k V k ≤ R (cid:9) , G ( V ) := ξ − E ξ + F T ( M V ) − E F T ( M V ) and G ( V ) := F T ( M V ) − E F T ( M V ) . By (i) and Doob’s L -maximal inequality, one has (cid:13)(cid:13) G ( V ) − G ( V ′ ) (cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) F T ( M V ) − E F T ( M V ) − ( F T ( M V ′ ) − E F T ( M V ′ )) (cid:13)(cid:13)(cid:13) ≤ C (cid:13)(cid:13)(cid:13) M V − M V ′ (cid:13)(cid:13)(cid:13) S ≤ C (cid:13)(cid:13)(cid:13) M VT − M V ′ T (cid:13)(cid:13)(cid:13) ≤ C (cid:13)(cid:13) V − V ′ (cid:13)(cid:13) . G is a contraction on L ( F T ) d . Moreover, it follows from (ii) that G : L ( F T ) d → L ( F T ) d iscontinuous and G ( C ) is relatively compact in L ( F T ) d . Finally, let V, V ′ ∈ C . Then (cid:13)(cid:13) G ( V ) (cid:13)(cid:13) ≤ k ξ − E ξ k + (cid:13)(cid:13) F T (0) − E F T (0) (cid:13)(cid:13) + 2 CR, and (cid:13)(cid:13) G ( V ′ ) (cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) F T ( M V ′ ) − E F T ( M V ′ ) (cid:13)(cid:13)(cid:13) ≤ √ λ (cid:13)(cid:13)(cid:13) DF T ( M V ′ ) (cid:13)(cid:13)(cid:13) ≤ √ λρ (cid:16)(cid:13)(cid:13)(cid:13) M V ′ (cid:13)(cid:13)(cid:13) S (cid:17) ≤ √ λρ (2 R ) . By (4.5), one has G ( V ) + G ( V ′ ) ∈ C for R large enough. So it follows like in the proof of Theorem 4.3from Krasnoselskii’s theorem that G = G + G has a fixed point in L ( F T ) d . Corollary 4.7.
A BSDE of the form Y t = ξ + Z Tt ( f ( s, M ) + f ( s, M )) ds + M T − M t for a terminal condition ξ ∈ L ( F T ) d and mappings f , f : [0 , T ] × Ω × M → R d has a solution ( Y, M ) ∈ S × M if there exist a constant C < (2 T ) − and a nondecreasing function ρ : R + → R + satisfying lim sup x →∞ ρ ( x ) x < / − CT √ λ such that the following two conditions hold: (i) For all M ∈ M , f ( ., M ) is progressively measurable with R T | f ( t, | dt ∈ L ( F T ) , and (cid:13)(cid:13) f ( t, M ) − f ( t, M ′ ) (cid:13)(cid:13) ≤ C (cid:13)(cid:13) M − M ′ (cid:13)(cid:13) S (ii) For all M ∈ M , f ( ., M ) is progressively measurable with R T | f ( t, M ) | dt ∈ L ( F T ) , and J ( M ) := R T f ( t, M ) dt defines a continuous map J : M → L ( F T ) d such that for all M ∈ M , J ( M ) is ω -Lipschitz with constant ρ ( k M k S ) .Proof. As in Corollary 4.5, it follows from the assumptions that F it ( M ) = R t f i ( s, M ) ds is in S for i = 1 , and all M ∈ M . Moreover, E (cid:12)(cid:12) F T ( M ) − F T ( M ′ ) (cid:12)(cid:12) ≤ C T (cid:13)(cid:13) M − M ′ (cid:13)(cid:13) S , and since R T f ( s, M ) ds is ω -Lipschitz with constant ρ ( k M k S ) , one has (cid:13)(cid:13) DF T ( M ) (cid:13)(cid:13) ≤ ρ ( k M k S ) . Sothe conditions of Theorem 4.6 hold with CT instead of C , and the corollary follows as a consequence. Remark 4.8.
As a special case of Corollary 4.7, one obtains that the BSDE Y t = ξ + Z Tt f ( s, M ) ds + M T − M t has a solution for every terminal condition ξ ∈ L ( F T ) d and driver f satisfying condition (ii) of Corol-lary 4.7. This provides an existence result for multidimensional BSDEs with drivers exhibiting gen-eral dependence on the whole process M . In contrast to the BSDE results in Section 3, here the driveris not required to be Lipschitz in M . On the other hand, it is supposed to satisfy the ω -Lipschitznessassumption contained in condition (ii) of Corollary 4.7.22 .2 Non-Lipschitz BSDEs based on a Brownian motion and a Poisson randommeasure We now focus on BSDEs with non-Lipschtiz coefficients that depend on an n -dimensional Brownianmotion W and an independent Poisson random measure N on [0 , T ] × E , where E = R m \ { } , with anintensity measure of the form dtµ ( dx ) for a measure µ over the Borel σ -algebra B ( E ) of E satisfying Z E (1 ∧ | x | ) µ ( dx ) < ∞ (see Remark 4.2 above for a construction of W and N in the case where P is a mean zero Gaussianmeasure on the infinite-dimensional separable Hilbert space Ω ).As in Subsection 4.1, we denote by F the completed Borel σ -algebra on Ω and let F = ( F t ) ≤ t ≤ T be afiltration satisfying the usual conditions. Let ˜ N be the compensated random measure N ( dt, dx ) − dtµ ( dx ) , and assume that, for A ∈ B ( E ) with µ ( A ) < ∞ , ˜ N ([0 , t ] × A ) and W are mar-tingales with respect to F . The next proposition gives an existence result for BSDEs with functionaldrivers of the form Y t = ξ + Z Tt f ( s, Z Ms , U Ms ) ds + M T − M t . (4.6) Proposition 4.9.
Let ξ ∈ L ( F T ) d and assume the driver is of the form f = f + f for mappings f , f : [0 , T ] × Ω × L ( F T ) d × n × L (Ω × E, F T ⊗ B ( E ) , P ⊗ µ ) d → R d . Then the BSDE (4.6) has a solution ( Y, M ) ∈ S × M if there exist a constant C ≥ and a nondecreasingfunction ρ : R + → R + such that for all M, M ′ ∈ M , the following two conditions hold: (i) f ( t, Z Mt , U Mt ) is progressively measurable with R T (cid:13)(cid:13) f ( t, , (cid:13)(cid:13) dt < ∞ , and (cid:13)(cid:13)(cid:13) f ( t, Z Mt , U Mt ) − f ( t, Z M ′ t , U M ′ t ) (cid:13)(cid:13)(cid:13) ≤ C (cid:18)(cid:13)(cid:13)(cid:13) Z Mt − Z M ′ t (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) U Mt − U M ′ t (cid:13)(cid:13)(cid:13) L ( P ⊗ µ ) (cid:19) (ii) f ( t, Z Mt , U Mt ) is progressively measurable with R T (cid:13)(cid:13) f ( t, , (cid:13)(cid:13) dt < ∞ , and (cid:13)(cid:13)(cid:13)(cid:13)Z T (cid:12)(cid:12)(cid:12) f ( t, Z Mt , U Mt ) − f ( t, Z M ′ t , U M ′ t ) (cid:12)(cid:12)(cid:12) dt (cid:13)(cid:13)(cid:13)(cid:13) ≤ ρ (cid:18)(cid:13)(cid:13) Z M (cid:13)(cid:13) H + (cid:13)(cid:13)(cid:13) Z M ′ (cid:13)(cid:13)(cid:13) H + (cid:13)(cid:13) U M (cid:13)(cid:13) L ( ˜ N ) + (cid:13)(cid:13)(cid:13) U M ′ (cid:13)(cid:13)(cid:13) L ( ˜ N ) (cid:19) (cid:18)(cid:13)(cid:13)(cid:13) Z M − Z M ′ (cid:13)(cid:13)(cid:13) H + (cid:13)(cid:13)(cid:13) U M − U M ′ (cid:13)(cid:13)(cid:13) L ( ˜ N ) (cid:19) , and f ( t, Z Mt , U Mt ) is ω -Lipschitz with constant C (cid:16) (cid:13)(cid:13) Z Mt (cid:13)(cid:13) + (cid:13)(cid:13) U Mt (cid:13)(cid:13) L ( P ⊗ µ ) (cid:17) .Proof. Choose δ > so that √ δC (cid:16) √ λ (cid:17) < and k := T /δ ∈ N . F it ( M ) = R t f i ( s, Z Ms , U Ms )1 [ T − δ,T ] ( s ) ds . It follows from the assumptions that F i ( M ) ∈ S for i = 1 , and all M ∈ M . Moreover, (cid:13)(cid:13) F T ( M ) − E F T ( M ) − ( F T ( M ′ ) − E F T ( M ′ )) (cid:13)(cid:13) ≤ (cid:13)(cid:13) F T ( M ) − F T ( M ′ ) (cid:13)(cid:13) ≤ δC Z TT − δ (cid:18)(cid:13)(cid:13)(cid:13) Z Ms − Z M ′ s (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) U Ms − U M ′ s (cid:13)(cid:13)(cid:13) L ( P ⊗ µ ) (cid:19) ds ≤ δC (cid:13)(cid:13) M − M ′ (cid:13)(cid:13) S . From condition (ii) one obtains that M ∈ M F T ( M ) ∈ L ( F T ) d is continuous, and (cid:12)(cid:12)(cid:12)(cid:12)Z TT − δ f ( s, Z Ms , U Ms )( ω ) − f ( s, Z Ms , U Ms )( ω ′ ) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z TT − δ (cid:12)(cid:12) f ( s, Z Ms , U Ms )( ω ) − f ( s, Z Ms , U Ms )( ω ′ ) (cid:12)(cid:12) ds ≤ C (cid:18)Z TT − δ (cid:16) (cid:13)(cid:13) Z Ms (cid:13)(cid:13) + (cid:13)(cid:13) U Ms (cid:13)(cid:13) L ( P ⊗ µ ) (cid:17) ds (cid:19) (cid:13)(cid:13) ω − ω ′ (cid:13)(cid:13) ≤ δC + √ δC sZ TT − δ (cid:16) k Z Ms k + k U Ms k L ( P ⊗ µ ) (cid:17) ds (cid:13)(cid:13) ω − ω ′ (cid:13)(cid:13) ≤ (cid:16) δC + √ δC k M k S (cid:17) (cid:13)(cid:13) ω − ω ′ (cid:13)(cid:13) . It follows that for all M ∈ M , F T ( M ) is in W , (Ω) d with (cid:13)(cid:13) DF T ( M ) (cid:13)(cid:13) ≤ δC + √ δC k M k S . So theconditions of Theorem 4.6 hold with √ δC instead of C and ρ ( x ) = δC + √ δCx . Therefore, Y t = ξ + Z Tt ( f ( s, Z Ms , U Ms ) + f ( s, Z Ms , U Ms ))1 [ T − δ,T ] ( s ) ds + M T − M t has a solution ( Y ( k ) , M ( k ) ) ∈ S × M . From the same argument one obtains that, for t ≤ T − δ , Y t = Y ( k ) T − δ + Z T − δt ( f ( s, Z s ) + f ( s, Z s ))1 [ T − δ,T − δ ] ( s ) ds + M T − δ − M t has a solution ( Y ( k − , Z ( k − ) ∈ S × M . Iterating this procedure, one obtains ( Y ( j ) , Z ( j ) ) , j = 1 , . . . , k .Now, define Y t := Y (1) t , M t := M (1) t for ≤ t ≤ δ and Y t := Y ( j ) t , M t − M ( j − δ := M ( j ) t − M ( j )( j − δ for ( j − δ < t ≤ jδ , j = 2 , . . . , k . Then ( Z Mt , U Mt ) = ( Z M ( j ) t , U M ( j ) t ) for ( j − δ < t ≤ jδ . So ( Y, M ) is asolution of (4.6) in S × M .As a consequence of Proposition 4.9, one obtains the following existence result for multidimen-sional mean-field BSDEs with drivers of quadratic growth and square integrable terminal conditions.While there exist general existence and uniqueness results for one-dimensional BSDEs with driversof quadratic growth (see e.g., Kobylanski, 2000, Briand and Hu, 2006, 2008, or Delbaen et al., 2011),multidimensional quadratic BSDEs do not always admit solutions (see Peng, 1999, or Frei and dosReis, 2011). An existence and uniqueness result for multidimensional BSDEs with general drivers ofquadratic growth was given by Tevzadze (2008). But it only holds for terminal conditions with small L ∞ -norm. Other results, such as the ones in Cheridito and Nam (2015), require the driver to havespecial structure. 24 orollary 4.10. Let ξ ∈ L ( F T ) d and assume the driver is of the form f ( t, Z t , U t ) = ˜ E a ( t, Z t , ˜ Z t , U t , ˜ U t ) + B ( t, E b ( t, Z t , U t )) for mappings a : [0 , T ] × Ω × ( R d × n ) × ( L ( µ )) → R d , b : [0 , T ] × Ω × R d × n × L ( µ ) → R l and B :[0 , T ] × Ω × R l → R d , where ( ˜ Z t , ˜ U t ) is a copy of ( Z t , U t ) living on a separate probability space ( ˜Ω , ˜ F , ˜ P ) ,and ˜ E a ( t, Z t , ˜ Z t , U t , ˜ U t ) means R ˜Ω a ( t, Z t , ˜ Z t , U t , ˜ U t ) d ˜ P .Then the BSDE (4.6) has a solution ( Y, M ) ∈ S × M if there exists a constant C ≥ such that for all z, ˜ z, z ′ , ˜ z ′ ∈ R d × n , u, ˜ u, u ′ , ˜ u ′ ∈ L ( µ ) and x, x ′ ∈ R k , a ( ., z, ˜ z, u, ˜ u ) , b ( ., z, u ) and B ( ., x ) are progressivelymeasurable and the following hold: (i) a ( ., , , , ∈ H and | a ( t, z, ˜ z, u, ˜ u ) − a ( t, z ′ , ˜ z ′ , u ′ , ˜ u ′ ) | ≤ C (cid:16) | z − z ′ | + | ˜ z − ˜ z ′ | + (cid:13)(cid:13) u − u ′ (cid:13)(cid:13) L ( µ ) + (cid:13)(cid:13) ˜ u − ˜ u ′ (cid:13)(cid:13) L ( µ ) (cid:17) (ii) | b ( t, , | , | B ( t, | ≤ C and at least one of the following two conditions is satisfied:(a) For any given t ∈ [0 , T ] , x, x ′ ∈ R l , z, z ′ ∈ R d × n , and u, u ′ ∈ L ( µ ) , B ( t, x ) is ω -Lipschitz withconstant C (1 + p | x | ) , and (cid:12)(cid:12) b ( t, z, u ) − b (cid:0) t, z ′ , u ′ (cid:1)(cid:12)(cid:12) ≤ C (cid:0) | z | + (cid:12)(cid:12) z ′ (cid:12)(cid:12) + k u k L ( µ ) + (cid:13)(cid:13) u ′ (cid:13)(cid:13) L ( µ ) (cid:1)(cid:0)(cid:12)(cid:12) z − z ′ (cid:12)(cid:12) + (cid:13)(cid:13) u − u ′ (cid:13)(cid:13) L ( µ ) (cid:1) , | B ( t, x ) − B ( t, x ′ ) | ≤ C | x − x ′ | . (b) For any given t ∈ [0 , T ] , x, x ′ ∈ R l , z, z ′ ∈ R d × n , and u, u ′ ∈ L ( µ ) , B ( t, x ) is ω -Lipschitz withconstant C (1 + | x | ) , and | b ( t, z, u ) − b ( t, z ′ , u ′ ) | ≤ C (cid:0)(cid:12)(cid:12) z − z ′ (cid:12)(cid:12) + (cid:13)(cid:13) u − u ′ (cid:13)(cid:13) L ( µ ) (cid:1) , | B ( t, x ) − B ( t, x ′ ) | ≤ C (1 + | x | + | x ′ | ) | x − x ′ | . Proof.
It is enough to show that f ( t, Z t , U t ) := ˜ E a ( t, Z t , ˜ Z t , U t , ˜ U t ) and f ( t, Z t , U t ) := B ( t, E b ( t, Z t , U t )) satisfy the conditions of Proposition 4.9. As in the proof of Corollary 3.11, one can deduce fromLemma 4.51 of Aliprantis and Border (2006) that f i ( t, Z t , U t ) is progressively measurable and satisfies R T (cid:13)(cid:13) f i ( t, , (cid:13)(cid:13) dt < ∞ for i = 1 , and all Z ∈ H and U ∈ L ( ˜ N ) .Now consider Z, Z ′ ∈ H , U, U ′ ∈ L ( ˜ N ) , and let ( ˜ Z, ˜ U , ˜ Z ′ , ˜ U ′ ) be a copy of ( Z, U, Z ′ , U ′ ) on ˜Ω . Then,for fixed t ∈ [0 , T ] , E | ˜ E a ( t, Z t , ˜ Z t , U t , ˜ U t ) − ˜ E a ( t, Z ′ t , ˜ Z ′ t , U ′ t , ˜ U ′ t ) | ≤ E ˜ E | a ( t, Z t , ˜ Z t , U t , ˜ U t ) − a ( t, Z ′ t , ˜ Z ′ t , U ′ t , ˜ U ′ t ) | ≤ C (cid:18) E | Z t − Z ′ t | + ˜ E | ˜ Z t − ˜ Z ′ t | + E (cid:13)(cid:13) U t − U ′ t (cid:13)(cid:13) L ( µ ) + ˜ E (cid:13)(cid:13)(cid:13) ˜ U t − ˜ U ′ t (cid:13)(cid:13)(cid:13) L ( µ ) (cid:19) = 8 C (cid:16)(cid:13)(cid:13) Z t − Z ′ t (cid:13)(cid:13) + (cid:13)(cid:13) U t − U ′ t (cid:13)(cid:13) L ( P ⊗ µ ) (cid:17) .
25n the other hand, if condition (ii.a) holds, then (cid:13)(cid:13)(cid:13)(cid:13)Z T (cid:12)(cid:12) B ( t, E b ( t, Z t , U t )) − B ( t, E b ( t, Z ′ t , U ′ t )) (cid:12)(cid:12) dt (cid:13)(cid:13)(cid:13)(cid:13) ≤ C Z T (cid:12)(cid:12) E b ( t, Z t , U t ) − E b ( t, Z ′ t , U ′ t ) (cid:12)(cid:12) dt ≤ C E Z T (cid:12)(cid:12) b ( t, Z t , U t ) − b ( t, Z ′ t , U ′ t ) (cid:12)(cid:12) dt ≤ C E Z T (cid:16) | Z t | + | Z ′ t | + k U t k L ( µ ) + (cid:13)(cid:13) U ′ t (cid:13)(cid:13) L ( µ ) (cid:17) (cid:16) | Z t − Z ′ t | + (cid:13)(cid:13) U t − U ′ t (cid:13)(cid:13) L ( µ ) (cid:17) dt ≤ C s E Z T (cid:16) | Z t | + | Z ′ t | + k U t k L ( µ ) + k U ′ t k L ( µ ) (cid:17) dt s E Z T (cid:16) | Z t − Z ′ t | + k U t − U ′ t k L ( µ ) (cid:17) dt ≤ C √ q T + k Z k H + k Z ′ k H + k U k L ( ˜ N ) + k U ′ k L ( ˜ N ) q k Z − Z ′ k H + k U − U ′ k L ( ˜ N ) ≤ C √ (cid:16) √ T + k Z k H + (cid:13)(cid:13) Z ′ (cid:13)(cid:13) H + k U k L ( ˜ N ) + (cid:13)(cid:13) U ′ (cid:13)(cid:13) L ( ˜ N ) (cid:17) (cid:16)(cid:13)(cid:13) Z − Z ′ (cid:13)(cid:13) H + (cid:13)(cid:13) U − U ′ (cid:13)(cid:13) L ( ˜ N ) (cid:17) . Moreover, B ( t, E b ( t, Z t , U t )) is ω -Lipschitz with constant C (1 + p | E b ( t, Z t , U t ) | ) , and | E b ( t, Z t , U t ) | ≤ E | b ( t, Z t , U t ) |≤ C E (cid:16) | Z t | + k U t k L ( µ ) (cid:17) (cid:16) | Z t | + k U t k L ( µ ) (cid:17) ≤ C (cid:16) k Z t k + k U t k L ( P ⊗ µ ) (cid:17) (cid:16) k Z t k + k U t k L ( P ⊗ µ ) (cid:17) , from which one obtains that B ( t, E b ( t, Z t , U t )) is ω -Lipschitz with constant C (1 + √ C (1 + k Z t k + k U t k L ( P ⊗ µ ) )) . Similarly, if condition (ii.b) holds, one has | B ( t, E b ( t, Z t , U t )) − B ( t, E b ( t, Z ′ t , U ′ t )) |≤ C (cid:0) | E b ( t, Z t , U t ) | + (cid:12)(cid:12) E b ( t, Z ′ t , U ′ t ) (cid:12)(cid:12)(cid:1) (cid:12)(cid:12) E b ( t, Z t , U t ) − E b ( t, Z ′ t , U ′ t ) (cid:12)(cid:12) ≤ C (cid:0) E | b ( t, Z t , U t ) | + E (cid:12)(cid:12) b ( t, Z ′ t , U ′ t ) (cid:12)(cid:12)(cid:1) E (cid:12)(cid:12) b ( t, Z t , U t ) − b ( t, Z ′ t , U ′ t ) (cid:12)(cid:12) ≤ C (cid:16) E | b ( t, , | + C E (cid:16) | Z t | + | Z ′ t | + k U t k L ( µ ) + (cid:13)(cid:13) U ′ t (cid:13)(cid:13) L ( µ ) (cid:17)(cid:17) E (cid:16) | Z t − Z ′ t | + (cid:13)(cid:13) U t − U ′ t (cid:13)(cid:13) L ( µ ) (cid:17) . (cid:13)(cid:13)(cid:13)(cid:13)Z T (cid:12)(cid:12) B ( t, E b ( t, Z t , U t )) − B ( t, E b ( t, Z ′ t , U ′ t )) (cid:12)(cid:12) dt (cid:13)(cid:13)(cid:13)(cid:13) ≤ C sZ T (cid:16) C + C E (cid:16) | Z t | + | Z ′ t | + k U t k L ( µ ) + k U ′ t k L ( µ ) (cid:17)(cid:17) dt × sZ T (cid:16) E | Z t − Z ′ t | + E k U t − U ′ t k L ( µ ) (cid:17) dt ≤ C sZ T (cid:16) C − + 4 + k Z t k + k Z ′ t k + k U t k L ( P ⊗ µ ) + k U ′ t k L ( P ⊗ µ ) (cid:17) dt × sZ T (cid:16) k Z t − Z ′ t k + k U t − U ′ t k L ( P ⊗ µ ) (cid:17) dt ≤ C √ (cid:16)p T ( C − + 4) + k Z k H + (cid:13)(cid:13) Z ′ (cid:13)(cid:13) H + k U k L ( ˜ N ) + (cid:13)(cid:13) U ′ (cid:13)(cid:13) L ( ˜ N ) (cid:17) (cid:16)(cid:13)(cid:13) Z − Z ′ (cid:13)(cid:13) H + (cid:13)(cid:13) U − U ′ (cid:13)(cid:13) L ( ˜ N ) (cid:17) . Moreover, B ( t, E b ( s, Z t , U t )) is ω -Lipschitz with constant C (1 + | E b ( t, Z t , U t ) | ) . So since | E b ( t, Z t , U t ) | ≤ E | b ( t, Z t , U t ) | ≤ C (cid:16) E (cid:16) | Z t | + k U t k L ( µ ) (cid:17)(cid:17) ≤ C (cid:16) k Z t k + k U t k P ⊗ L ( µ ) (cid:17) ,B ( t, E b ( t, Z t , U t )) is ω -Lipschitz with constant C (cid:16) C (cid:16) k Z t k + k U t k P ⊗ L ( µ ) (cid:17)(cid:17) . This shows thatthe conditions of Proposition 4.9 hold, and the corollary follows. Example 4.11.
A simple example of a driver satisfying the conditions of Corollary 4.10 is given by f ( Z t ) = f ( Z t ) + f ( Z t ) , for a Lipschitz function f : R d × n → R d and a mapping f : L ( F T ) d × n → R d of the form f ( Z t ) := α + E ( Z t | Z t | ) β with constant vectors α ∈ R d × and β ∈ R n × . In particular, if W is an n -dimensional Brownianmotion generating the filtration F , the BSDE Y t = ξ + Z Tt f ( Z s ) ds + Z Tt Z s dW s has a solution ( Y, Z ) ∈ S × H for every terminal condition ξ ∈ L ( F T ) d .Since f has quadratic growth, the contraction mapping principle used by Buckdahn et al. (2009)cannot be applied here. Also, if d > and f were a function with quadratic growth of the realizations Z t ( ω ) , the existence of a global solution could not be guaranteed; see Frei and dos Reis (2011) for acounterexample. 27 eferenceseferences