Existence, uniqueness and comparison results for BSDEs with Lévy jumps in an extended monotonic generator setting
Full text
This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Existence, uniqueness and comparison results for BSDEs with Lévy jumps in an extended monotonic generator setting © The Author(s), 2018. Published version Geiss, Christel; Steinicke, Alexander Geiss, C., & Steinicke, A. (2018). Existence, uniqueness and comparison results for BSDEs with Lévy jumps in an extended monotonic generator setting. Probability, Uncertainty and Quantitative Risk, 3(9), 1-33. https://doi.org/10.1186/s41546-018-0034-y 2018
P robability, Uncertainty and Quantitative Risk Probability, Uncertainty and Quantitative Risk (2018) 3:9 DOI 10.1186/s41546-018-0034-y R E S E A RC H Open Access Existence, uniqueness and comparison results for BSDEs with L´ evy jumps in an extended monotonic generator setting Christel Geiss ·Alexander Steinicke Received: 4 January 2018 / Accepted: 29 November 2018 / © The Author(s). 2018 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Abstract We show that the comparison results for a backward SDE with jumps established in Royer (Stoch. Process. Appl 116: 1358–1376, 2006) and Yin and Mao (J. Math. Anal. Appl 346: 345–358, 2008) hold under more simplified conditions. Moreover, we prove existence and uniqueness allowing the coefficients in the linear growthand monotonicity-condition for the generator to be random and time-dependent. In the L2-case with linear growth, this also generalizes the results of Kruse and Popier (Stochastics 88: 491–539, 2016). For the proof of the comparison result, we introduce an approximation technique: Given a BSDE driven by Brownian motion and Poisson random measure, we approximate it by BSDEs where the Poisson random measure admits only jumps of size larger than 1/n. Keywords Backward stochastic differential equation ·L´ evy process ·comparison theorem ·existence and uniqueness Mathematics Subject Classification: 60H10 C. Geiss University of Jyvaskyla, Department of Mathematics and Statistics, P.O. Box 35, 40014 Jyvaskyla, Finland A. Steinicke () Department of Mathematics and Information Technology, Montanuniversitaet Leoben, Leoben, Austria e-mail: alexander[email protected]
Page 2 of 33 Geiss and Steinicke 1 Introduction In this paper, we study backward stochastic differential equations (BSDEs) of the form Yt=ξ+T t f(s,Ys,Z s,U s)ds −T t ZsdWs−]t,T]×(R\{0}) Us(x) ˜ N(ds,dx), (1) where Wdenotes a one-dimensional Brownian motion and ˜ Na compensated Poisson random measure belonging to a given L´ evy process with L´ evy measure ν.In particular, our focus lies on comparison results and existence and uniqueness of solutions. Comparison theorems state that—under certain conditions—if ξ≤ξand f≤f, then the process Yof the solution satisfies Yt≤Y tfor all t∈[0,T]. These types of theorems in the case of one-dimensional, Brownian BSDEs has been treated by Peng (1992), El Karoui et al. (1997,2009), and Cao and Yan (1999). In (Barles et al. (1997), Remark 2.7) a counterexample was given, which shows that in the jump case the conditions ξ≤ξand f≤fare not sufficient to guarantee Y≤Y. They propose an additional sufficient condition which has been generalized by Kruse and Popier (2016), Royer (2006),YinandMao(2008), Becherer et al. (2018) (allowing more general jump processes), and Cohen et al. (2010) (for BSDEs driven by martingales). The condition of Kruse and Popier (2016) reads (in our L2setting) as follows: for each s, y, z, u, u∈[0,T]×R×R×L2(ν) ×L2(ν) there is a progressively measurable process γy,z,u,u:×[0,T]×R\{0}→Rsuch that f(s,y,z,u)−fs, y, z, u≤R\{0}u(x) −u(x)γy,z,u,u s(x)ν(dx), −1≤γy,z,u,u s(x) and sup s,ω,y,z,u,uγy,z,u,u s∈L2(ν). (2) One of the main results in the present paper is Theorem 3.5 which states that (2) can be replaced by the simpler condition f(s,y,z,u)−fs, y, z, u≤R\{0}u(x) −u(x)ν(dx), P⊗λ-a.e. for all u, u∈L2(ν) with u≤u.(3) Notice that the r.h.s. is infinite for u(x) −u(x) /∈L1(ν). Clearly, (3) is a weaker condition than (2), because one only needs to check the inequality for those u, u∈ L2(ν) for which u≤uholds. Moreover, we do not need any L2(ν) condition for γy,z,u,u sbut we choose γy,z,u,u s(x) =−1.Under the constraint −1≤γy,z,u,u s(x), the choice γy,z,u,u s(x) =−1 yields for u−u≥0 the largest possible expression on the r.h.s. of (2), so that (3) can be seen as the weakest possible condition which (2) could impose on f. For a finite L´ evy measure ν, Theorem 3.5 can be shown using only elementary means.
Probability, Uncertainty and Quantitative Risk (2018) 3:9 Page 3 of 33 Another main result is a method of how to approximate a BSDE driven by a L´ evy process with an infinite measure ν, by a sequence of BSDEs where the driving processes have a finite L´ evy measure. We apply this result to show the comparison theorem for BSDEs driven by a general L´ evy process. The proof relies on the Jankov–von Neumann theorem on measurable sections/uniformizations (this theorem is also important for dynamic programming, see El Karoui and Tan (2013). Under certain conditions on the generator, the approximating solutions can be interpreted as nonlinear conditional expectations (in the sense of Peng (2010)), conditioned on a L´ evy process whose jumps are not of arbitrarily small size. (See the comments after Theorem 3.4.) Studying the existence, uniqueness, and comparison results by Darling and Pardoux (1997), Pardoux and Zhang (1996), Pardoux (1997), Fan and Jiang (2012), Royer (2006), Situ (1997), Yin and Mao (2008), Kruse and Popier (2016,2017), Yao (2017), and Sow (2014), one notices that one can unify and generalize the assumptions on f. Indeed, and this is our third main result, in the case of L2-solutions, for a progressively measurable generator fwith linear growth, it suffices to assume (cf. Theorems 3.1 and 3.5) the following growthand monotonicity conditions with time-dependent, random coefficients: •|f(ω,s,y,z,u)|≤F(s,ω)+K1(s, ω)|y|+K2(s, ω)(|z|+u), •y−yf1(ω,s,y,z,u)−f1ω,s,y,z ,u ≤α(s)ρ y−y 2+β(s,ω)y−yz−z+ u−u , with α∈L1([0,T])and Fbeing nonnegative and progressively measurable such that ET 0F(ω,t)dt2<∞.The processes K1,K 2,and βare nonnegative and progressively measurable such that for a constant c>0, T 0K1(s) +K2(s)2+β(s)2ds < c, P-a.s. The concave function ρin the monotonicity condition may grow faster than linear at zero and satisfies 0+1/ρ(x)dx =∞.This type of function already appeared in context with BSDEs in Mao (1995) in 1997. These assumptions also extend the monotonicity condition of Kruse and Popier (2016,2017), for the L2-case with linear growth, since the coefficients in our setting take randomness, the function ρand time-dependence into account. BSDEs with time-dependent coefficients appear, for example, in Gobet and Turkedjiev (2016). The existence and uniqueness result Theorem 3.1 and the comparison result Theorem 3.5 are basic tools in the forthcoming paper (Geiss and Steinicke 2018)on Malliavin differentiability and boundedness of solutions to BSDEs. To compute the Malliavin derivative for the jump part of the L´ evy process, more structure from the generator is required in its dependency on u, usually via an integral w.r.t. ν(dx), for example, f(s,u)=hs,R\{0} u(x)κ(s, x)ν(dx),
Page 4 of 33 Geiss and Steinicke where [0,T]×R(s, v) → h(s, v). One can find hand κsuch that the assumptions of Theorem 3.5 are satisfied while conditon (2) does not hold: By the mean value theorem there exists a ζ∈]0,1[and vζ:= R\{0}ζu(x) +(1−ζ)u(x)κ(s,x)ν(dx), such that f(s,u)−fs, u=∂vhs,vζR\{0}u(x) −u(x)κ(s,x)ν(dx). Assumption (3) holds if γu,u s(x) := ∂vhs, vζκ(s,x) ≥−1foralls, u, u,x. Choosing, for example, a bounded function hsuch that also sups,v |∂vh(s, v)|<∞, but ∂vh(s, v) = 0 for a.e. sand v, and putting κ(s,x) =s−1 4(|x|∧1), then (2) does not hold since sup s,u,uγu,u s/∈L2(ν). However, the Assumptions (A2),(A3) of Section 3are satisfied for K2(s) =β(s) =sup v|∂vh(s, v)|κ(s,·)L2(ν) ≤cs−1 4. The paper is structured as follows: Section 2contains preliminaries and basic definitions. In Section 3, we present the main theorems of this paper about existence and uniqueness of solutions, the approximation using BSDEs based on L´ evy processes with finite L´ evy measure, and the comparison result. The latter we also prove there. Having stated and proved some auxiliary results in Section 4, including an a-priori estimate for our type of BSDEs, we are able to prove existence and uniqueness and the approximation result from Section 3. In the appendix, we recall the Bihari–LaSalle inequality and the Jankov–von Neumann theorem. 2 Setting Let X=(Xt)t∈[0,T ]be a c` adl` ag L´ evy process on a complete probability space (, F,P)with L´ evy measure ν. We will denote the augmented natural filtration of Xby (Ft)t∈[0,T ]and assume that F=FT.For 0 <p≤∞we use the notation Lp,·p:= (Lp(, F,P), ·Lp). Equations or inequalities for objects of these spaces throughout the paper are considered up to P-null sets. The L´ evy–Itˆ o decomposition of a L´ evy process Xcan be written as Xt=at +σWt+]0,t]×{|x|≤1} x˜ N(ds,dx)+]0,t]×{|x|>1} xN(ds, dx), (4) where a∈R,σ≥0, Wis a Brownian motion and N(˜ N) is the (compensated) Poisson random measure corresponding to X, see Applebaum (2004)orSato(1999).
Probability, Uncertainty and Quantitative Risk (2018) 3:9 Page 5 of 33 Notation •Let S2denote the space of all (Ft)-progressively measurable and c` adl` ag processes Y:×[0,T]→Rsuch that Y2 S2:= Esup 0≤t≤T|Yt|2<∞. •We define L2(W ) as the space of all (Ft)-progressively measurable processes Z:×[0,T]→Rsuch that Z2 L2(W) := ET 0|Zs|2ds < ∞. •Let R0:= R\{0}.WedefineL2˜ Nas the space of all random fields U:× [0,T]×R0→Rwhich are measurable with respect to P⊗B(R0)(where P denotes the predictable σ-algebra on ×[0,T]generated by the left-continuous (Ft)-adapted processes) such that U2 L2˜ N:= E[0,T ]×R0|Us(x)|2ds ν(dx) < ∞. •L2(ν) := L2(R0,B(R0),ν),·:=·L2(ν). •Lp([0,T]):= Lp([0,T],B([0,T]), λ) for p>0, where λis the Lebesgue measure on [0,T]. •With a slight abuse of the notation, we define L2;L1([0,T])(5) :=F∈L0( ×[0,T],F⊗B([0,T]), P⊗λ) :ET 0|F(ω,t)|dt2 <∞. For F∈L2;L1([0,T]),put IF(ω) := T 0 F(ω,t)dt and KF(ω, s) := F(ω,s) IF(ω) .(6) •Asolution to a BSDE with terminal condition ξand generator fis a triplet (Y,Z,U)∈S2×L2(W) ×L2˜ Nwhich satisfies for all t∈[0,T]: Yt=ξ+T t f(s,Y s,Z s,U s)ds−T t ZsdWs−]t,T]×R0 Us(x) ˜ N(ds,dx). (7) The BSDE (7) itself will be denoted by (ξ, f ). 3Mainresults We start with a result about existence and uniqueness which is proved in Section 5. Theorem 3.1 There exists a unique solution to the BSDE (ξ, f ) with ξ∈L2and generator f:×[0,T]×R×R×L2(ν) →Rsatisfying the properties
Page 6 of 33 Geiss and Steinicke (A1) For all (y,z,u):(ω, s) → f(ω,s,y,z,u)is progressively measurable. (A2) There are nonnegative, progressively measurable processes K1,K 2,and F with CK:= T 0K1(·,s)+K2(·,s) 2ds ∞ <∞(8) and F∈L2;L1([0,T])(see (5)) such that for all (y,z,u), |f(s,y,z,u)|≤F(s)+K1(s)|y|+K2(s)(|z|+u), P⊗λ-a.e. (A3) For λ-almost all s, the mapping (y,z,u)→ f(s,y,z,u)is P-a.s. continuous. Moreover, there is a nonnegative function α∈L1([0,T]),c>0and a progressively measurable process βwith T 0β(ω,s)2ds < c,P-a.s. such that for all (y, z, u), y,z ,u , y−yf(s,y,z,u)−fs,y,z ,u ≤α(s)ρ |y−y|2+β(s)y−yz−z+ u−u ,P⊗λ-a.e., where ρis a nondecreasing, continuous and concave function from [0,∞[ to itself, satisfying ρ(0)=0,and 0+1 ρ(x)dx =∞. (A4) The function ρin (A3) satisfies lim supx↓0 ρ(x2) x=0. If f satisfies only (A1)–(A3), then there exists at most one solution. For ρ(x) =x, we are in the case of the ordinary monotonicity condition. Another example for a function ρis given by ρ(x) =1−min x, 1 eminx,1 e,x≥0. Remark 3.2 . 1. Condition (A2) implies that f(s,y,z,u)is integrable for a.e. s∈[0,T]since, by Fubini’s theorem, T 0 E|f(s,y,z,u)|ds ≤ET 0[F(s)+K1(s)|y|+K2(s)(|z|+u))]ds < ∞.(9) 2. If lim supx↓0 ρ(x2) x=0is satisfied one can derive Lipschitz continuity of f(s,y,z,u)in z and u from the monotonicity condition in (A3). We require (A4) since we later want to apply (Yin and Mao (2008), Theorem 2.1), where Lipschitz continuity in u is used to show uniqueness of solutions. If only (A1)–(A3) are satisfied but not (A4), and a Lipschitz condition in z, u holds nevertheless, all of the article’s theorems remain valid. One can show that (A4) does not follow from the other conditions imposed on ρin (A3): Assume a decreasing sequence (xn)∞ n=0with x0=1and limn→∞ xn=0.Define
Probability, Uncertainty and Quantitative Risk (2018) 3:9 Page 7 of 33 ρ(x) := √xnif x=xn,n=0,1,2, ... √xif x>1orx=0. and let ρbe continuous and piecewise linear on ]0,1].The so defined ρis a concave function with lim supx↓0 ρ(x) √x=1.The sequence (xn)∞ n=0can be constructed such that 1 0 1 ρ(x)dx =∞.For example, choose x1such that 1 x1 1 ρ(x)dx ≥1,and if xnhas been chosen find xn+1such that xn xn+1 1 ρ(x)dx =1 2(log(xn)−log(xn+1))√xn+√xn+1≥1. The next result shows how a solution to a BSDE can be approximated by a sequence of solutions of BSDEs which are driven by L´ evy processes with a finite L´ evy measure. We do this by approximating the underlying L´ evy process defined through Xt=at +σWt+]0,t]×{|x|>1} xN(ds,dx) +]0,t]×{|x|≤1} x˜ N(ds,dx) for n≥1by Xn t=at +σWt+]0,t]×{|x|>1} xN(ds,dx) +]0,t]×{1/n≤|x|≤1} x˜ N(ds,dx). The process Xnhas a finite L´ evy measure νn. Furthermore, note that the compensated Poisson random measure associated with Xncan be expressed as ˜ Nn=χ{1/n≤|x|} ˜ N. Let J0:= {,∅}∨N, Jn:= σXn∨N,n≥1,(10) where Nstands for the null sets of F.Note that (Jn)∞ n=0forms a filtration. The notation (Jn)∞ n=0was chosen to indicate that this filtration describes the inclusion of smaller and smaller jumps of the L´ evy process. We will use En·:=E·Jn for the conditional expectation. The intuitive idea now would be to work with a BSDE driven by Xnwhere one uses the data (Enξ,Enf).The problem is that the generator fneeds to be progressively, and also jointly measurable w.r.t. (ω,t,y,z,u),but it is not obvious whether the conditional expectation Enfpreserves this property from f. For BSDEs driven by a Brownian motion, this problem has been solved in (Ylinen (2017), Proposition 7.3), but this proposition does not apply to our situtation. Therefore, we next propose a method for the construction of a unique progressively measurable and jointly measurable w.r.t. (ω,t,y,z,u)version of Enf. Definition 3.3 (Definition of fn)Assume that f satisfies (A1),(A2) and that J:= J[s]s∈[0,∞[ is built using (10), where [·] denotes the floor function. Let o,J fbe the optional projection of the process
Page 8 of 33 Geiss and Steinicke [0,∞[××[0,T]×R2×L2(ν) →R, (s,ω,t,y,z,u)→f(ω,t,y,z,u) in the variables (s, ω) with respect to J,and with parameters (t,y,z,u). For each n≥0, assume that the filtration Fn:= Fn tt∈[0,T ]is given by Fn t:= Ft∩Jn.Let fnbe the optional projection of (ω,t,y,z,u)→ o,J f(n,ω,t,y,z,u) with respect to Fnwith parameters (y,z,u). The reason for using the filtration J[s]s∈[0,∞[ instead of the (Jn)∞ n=0from (10) is that one can apply known measurability results w.r.t. right continuous filtrations instead of proving measurability here directly. Indeed, the optional projection o,J f defined above is jointly measurable in (s,ω,t,y,z,u). For this we refer to Meyer (1979), where optional and predictable projections of random processes depending on parameters were considered, and their uniqueness up to indistinguishability was shown. It follows that for all (t,y,z,u), o,J f(n,t,y,z,u)=Enf (t, y, z, u), P-a.s. Then, since fis (Ft)t∈[0,T ]-progressively measurable, for all n≥0, t∈[0,T]and all (y,z,u), it holds that fn(t,y,z,u)=Enf (t, y, z, u), P-a.s. (11) Hence, fn(t,y,z,u) is a jointly measurable version of Enf(t,y,z,u) which is Fn tt∈[0,T ]-optional, so especially it is progressively measurable. We comment on the compatibility of the solutions (Y n,Zn,Un)from the BSDE corresponding to (Enξ,fn), Yn t=Enξ+T t fns,Yn s,Zn s,Un sds −T t Zn sdWs −]t,T]×R0 Un s(x) ˜ Nn(ds, dx) with the space S2×L2(W) ×L2˜ N: The triplet (Yn,Zn,Un)∈S2×L2(W ) ×L2˜ Nncan be canonically embedded in the space S2×L2(W)×L2˜ N, basically by extending Un s(x) onto R0by defining Un s(x) := 0for|x|<1 n. Moreover, recall that ˜ Nn=χ{1/n≤|x|} ˜ N, so that ]t,T]×R0 Un s(x) ˜ Nn(ds, dx) =]t,T]×R0 Un s(x)χ{1/n≤|x|} ˜ N(ds,dx). Therefore, Yn,Zn,UnχR\]−1/n,1/n[solves (Enξ,fn)in S2×L2(W)×L2˜ N.
Probability, Uncertainty and Quantitative Risk (2018) 3:9 Page 15 of 33 We use this estimate for R=2,and taking the expectation in (17), we have ET 0 es 0η(τ)dτ η(s)|Ys|2+|Zs|2+Us2 ds≤EeT 0η(s)ds|ξ|2+ET 0 es 0η(τ)dτ |Zs|2+Us2 2ds +2ET 0 es 0η(τ)dτ F(s)ds sup t∈[0,T ]|Yt| +ET 0 es 0η(τ)dτ 2K1(s) +2K2(s)2Y2 sds. (19) Then, we choose η(s) =2K1(s) +2K2(s)2and subtract the terms containing Y, Z, and Ufrom the left hand side of (19). Moreover, we apply the first inequality of (14) to the term containing the supremum. It follows that ET 0 es 0η(τ)dτ |Zs|2+Us2ds ≤2EeT 0η(s)ds|ξ|2+2RET 0 es 0η(τ)dτ F(s)ds2 +2 REsup t∈[0,T ]|Yt|2. (20) Note that ET 0|Zs|2+Us2ds ≤ET 0 es 0η(τ)dτ |Zs|2+Us2ds. Hence, by (20)andT 0η(τ)dτ ≤4CKa.s., we have ET 0|Zs|2+Us2ds ≤2e4CKE|ξ|2+2Re8CKEI2 F+2 REsup t∈[0,T ]|Yt|2.(21) Now, we can plug in (21)into(15) and vice versa which yields for R:= 48c1that Esup t∈[0,T ]|Yt|2≤2c1+48c1e4CKE|ξ|2+2c1+(48c1)2e8CKEI2 F, and E T 0|Zs|2+Us2ds ≤1 12 +4e4CKE|ξ|2+1 12 +192c1e8CKEI2 F. Using (16) it is easy to see that there exists a constant C1>0 such that each factor in front of the expectations on the right side of the previous two inequalities is less than eC1(1+CK)2. Our next proposition will be an L2a-priori estimate for BSDEs of our type. For the Brownian case, Lpa-priori estimates are done for p∈[1,∞[in Briand et al. (2003), and for quadratic BSDEs, for p∈[2,∞[ in Geiss and Ylinen (2018). For BSDEs with jumps, for p∈]1,∞[,see Kruse and Popier (2016,2017); while Becherer et al. (2018) contains an a-priori estimate w.r.t. L∞.The following assertion is similar to (Barles et al. (1997), Proposition 2.2), but fits our extended setting.
Page 16 of 33 Geiss and Steinicke Proposition 4.2 Let ξ,ξ∈L2and let f, f be two generator functions satisfying (A1)–(A3), where the bounds in (A2) and the coefficients in (A3) may differ for f and f. The coefficients of fin (A3) will be referred to as αand β. Moreover, let the triplets (Y,Z,U)and (Y ,Z,U)∈L2(W ) ×L2(W ) ×L2˜ N, satisfy the BSDEs (ξ, f ) and (ξ,f), respectively. Then, Y−Y2 L2(W) + Z−Z 2 L2(W) + U−U 2 L2(˜ N) ≤ha,b,E|ξ−ξ|2+2ET 0|Yt−Y t|f(t,Y t,Z t,U t)−f(t, Yt,Z t,U t)dt, where a=T 0α(s)ds, b = T 0β(s)2ds ∞,and h:]0,∞[×]0,∞[×[0,∞[→ [0,∞[ is a function such that h(a, b, x) →0=h(a, b, 0)if x→0. Proof We start with the following observation gained by Itˆ o’s formula for the difference of the BSDEs (ξ, f ) and (ξ,f). We denote differences of expressions by .Ifη=4β(s)2,we have analogously to (17) et 0η(s)ds|Yt|2+T tes 0η(τ)dτ η(s)|Ys|2+|Zs|2+Us2ds =eT 0η(s)ds|ξ|2+M(t) +T t2es 0η(τ)dτ Ys(f (s, Ys,Z s,U s)−f(s, Y s,Z s,U s))ds, (22) where M(t) =−T t 2es 0η(τ)dτ YsZsdWs −]t,T]×R0 2es 0η(τ)dτ (Ys−+Us(x))2−Y 2 s−˜ N(ds,dx). By the same reasoning as for (18), we have EM(t) =0. We now proceed with the (standard) arguments similar to those used for (17)–(19). By (A3) and the first inequality from (14), Ys(f (s, Ys,Z s,U s)−f(s, Y s,Z s,U s)) ≤α(s)ρ |Ys|2+β(s)|Ys|(|Zs| +Us)≤α(s)ρ |Ys|2 +β(s)2|Ys|2 R+R|Zs|2+Us2 2. (23)
Probability, Uncertainty and Quantitative Risk (2018) 3:9 Page 17 of 33 Taking the expectation in (22) and then using (23) with R=1 (such that we can cancel out the terms with Zand Uon the left side), leads to Eet 0η(s)ds|Yt|2+ET t es 0η(τ)dτ η(s)|Ys|2ds ≤EeT 0η(s)ds|ξ|2 +ET t 2es 0η(τ)dτ Ys·(f )(s, Ys,Z s,U s)ds +ET t es 0η(τ)dτ 2α(s)ρ |Ys|2+β(s)2|Ys|2ds. The choice η(s) =4β(s)2and the fact that T 0β(s)2ds ≤ba.s. leads to E|Yt|2≤e4bE|ξ|2+ET t 2|Ys||(f )(s, Ys,Z s,U s)|ds +e4bT t 2α(s)ρ E|Ys|2ds, since ρis a concave function. By Proposition 5.2, a backward version of the Bihari–LaSalle inequality, shows sup t∈[0,T ] E|Yt|2≤ G−1Ge4bE|ξ|2+ET 0 2|Ys||(f )(s, Ys,Z s,U s)|ds+2e4bT 0 α(s)ds, (24) where G(x) =x 1 1 ρ(h)dh. If we take the expectation in (22) but choose this time (23) with R=1 2and omit Eet 0η(s)ds|Yt|2,then ET t es 0η(τ)dτ η(s)|Ys|2+|Zs|2+Us2ds ≤EeT 0η(s)ds|ξ|2+ET t 2es 0η(τ)dτ Ys·(f )(s, Ys,Z s,U s)ds +ET t es 0η(τ)dτ 2α(s)ρ |Ys|2+4β(s)2|Ys|2+|Zs|2+Us2 2ds. We subtract the quadratic terms with Y, Z, and U which appear on the right hand side. This results in the inequality ET tes 0η(τ)dτ |Zs|2+Us2ds ≤2EeT 0η(s)ds|ξ|2+ET t2es 0η(τ)dτ |Ys|·|(f )(s, Ys,Z s,U s)|ds +ET tes 0η(τ)dτ 2α(s)ρ |Ys|2)ds.
Page 18 of 33 Geiss and Steinicke We continue our estimate by ET tes 0η(τ)dτ |Zs|2+Us2ds ≤2e4bE|ξ|2+ET t2|Ys|·|(f )(s, Ys,Z s,U s)|ds (25) +2T tα(s)ds ρ sup s∈[0,T ] E|Ys|2, since η(s) =4β(s)2. We put H:= G−1Ge4bE|ξ|2+ET 02|Ys||(f )(s, Ys,Z s,U s)|ds +2e4bT 0α(s)ds so that (24) reads now as supt∈[0,T ]E|Yt|2≤H. If we add this inequality to (25) and note that ρsupt∈[0,T ]E|Ys|2≤ρ(H), we have sup s∈[0,T ] E|Yt|2+ET 0|Zs|2ds +ET 0Us2ds ≤2e4bE|ξ|2+ET 02|Ys|·|(f )(s, Ys,Z s,U s)|ds +2e4bT 0α(s)ds +1·(id +ρ)(H). Note that the integral condition on ρimplies that, if the argument of Gapproaches zero, then the right hand side vanishes. The following Lemma will be used to estimate the expectation of integrals which contain |Ys|2. Lemma 4.3 Let ξ∈L2and assume that (A1) and (A2) hold. If (Y,Z,U) is a solution to (ξ, f ) and H is a nonnegative, progressively measurable process with T 0H(s)ds ∞<∞,then ET 0 H(s)|Ys|2ds ≤e2CKET 0 H(s)ds|ξ|2 +2e2CK T 0 H(s)ds ·IF 2YS2. (26) Proof From the relations (17), (18) and integration by parts applied to the term T 0H(s)ds ·eT 0η(s)ds|YT|2,we get T 0 H(s)ds ·eT 0η(r)dr |YT|2=T 0 H(s)es 0η(τ)dτ |Ys|2ds −T 0s 0 H(τ)dτdM(s) +T 0s 0 H(r)dres 0η(τ)dτ η(s)|Ys|2+|Zs|2+Us2 −2Ysf(s,Y s,Z s,U s))ds.
Probability, Uncertainty and Quantitative Risk (2018) 3:9 Page 19 of 33 We take expectations and rearrange the equation so that ET 0 H(s)es 0η(τ)dτ |Ys|2ds ≤ET 0 H(s)ds ·eT 0η(s)ds|ξ|2 +ET 0s 0 H(τ)dτes 0η(τ)dτ (2Ysf(s,Y s,Z s,U s) −η(s)|Ys|2−|Zs|2−Us2ds. By Assumption (A2) and (14), we have 2Ysf(s,Y s,Z s,U s)≤2|Ys|F(s)+2K1(s)|Ys|2 +2K2(s)|Ys|(|Zs|+Us)≤2|Ys|F(s) +2K1(s)|Ys|2+2K2(s)2|Ys|2+|Zs|2+Us2, so that for η(s) =2K1(s) +2K2(s)2it follows ET 0 H(s)|Ys|2ds ≤ET 0 H(s)ds ·eT 0η(s)ds|ξ|2 +2ET 0s 0 H(τ)dτes 0η(τ)dτ F(s)|Ys|ds ≤e2CKET 0 H(s)ds ·|ξ|2 +2e2CK T 0 H(s)ds ·IF 2YS2. (27) 5 Proofs of Theorems 3.1 and 3.4 5.1 Proof of Theorem 3.1 Step 1: Uniqueness Uniqueness of the solution is a consequence of Proposition 4.2, since the terms |ξ− ξ|and |f(s,Y s,Z s,U s)−f(s, Ys,Z s,U s)|are zero. The proof of existence will be split up in further steps. Step 2: In this step, we construct an approximating sequence of generators f(n) for f and show several estimates for the solution processes (Y n,Zn,Un)to the BSDEs ξ,f(n). For n≥1,define cn(z) := min(max(−n, z), n) and ˜cn(u) ∈L2(ν) to be the projection of uonto {v∈L2(ν) :v≤n}.Let(Y n,Zn,Un)be the unique solution of the BSDE ξ,f(n), with the definitions ˆ f(n)(ω,s,y,z,u):= f(ω,s,y,c n(z), ˜cn(u)),
Page 20 of 33 Geiss and Steinicke and f(n)(ω,s,y,z,u):= sign ˆ f(n)(ω,s,y,z,u) ×F(ω,s)∧n+(K1(ω, s) ∧n)|y|+(K2(ω, s) ∧n)(|cn(z)|+˜cn(u)) if |ˆ f(n)(ω,s,y,z,u)|>F(ω,s)∧n+(K1(ω, s) ∧n)|y| +(K2(ω, s) ∧n)(|cn(z)|+˜cn(u)), and f(n)(ω,s,y,z,u):= ˆ f(n)(ω,s,y,z,u) else. Note that f(n) satisfies (A1)–(A4), with the same coefficients as f. Moreover, by (A4),f(n) satisfies a Lipschitz condition with respect to u(see Remark 3.2). Thus, thanks to (Yin and Mao (2008), Theorem 2.1), ξ,f(n)has a unique solution (Y n,Zn,Un). Moreover, by Proposition 4.1, we get that Yn2 S2+ Zn 2 L2(W) + Un 2 L2(˜ N) ≤eC1(1+CK)2E|ξ|2+EI2 F<∞,(28) uniformly in n. This implies that the families sup t∈[0,T ]|Yn t|,n≥0,|Yn|,n≥0and |Zn|+Un,n≥0 are uniformly integrable with respect to P,P⊗λand P⊗λ, respectively. Step 3: The goal of this step is to use Proposition 4.2 to get convergence of (Y n,Zn,Un)n in L2(W)×L2(W)×L2(˜ N)for a subsequence nk↑∞if δnk,nl→0fork>l→∞, where δn,m := ET 0|Yn s−Ym s||f(n) s,Yn s,Zn s,Un s−f(m) s,Yn s,Zn s,Un s|ds. We observe that the difference of the generators is zero if two conditions are satisfied at the same time: First, if |Zn|,Un s<n, and additionally, by the cut-off procedure for F,K1,K 2,if n>max (F(ω,s),K 1(ω, s), K2(ω, s))=: k(ω,s). Thus, putting χn(s) := χ{|Zn s|>n}∪{Un s>n}∪{k(s)>n},(29) we have δn,m =ET 0|Yn s−Ym s||f(n) s,Yn s,Zn s,Un s−f(m) s,Yn s,Zn s,Un s|χn(s)ds ≤ET 0 2|Yn s−Ym s|χn(s)×F(s)+K1(s)|Yn s|+K2(s) |Zn s|+Un sds,
Probability, Uncertainty and Quantitative Risk (2018) 3:9 Page 21 of 33 due to the linear growth condition (A2). We estimate this further by δn,m ≤ET 0 χn(s) F (s)ds sup r∈[0,T ]|Yn r|+ sup r∈[0,T ]|Ym r| +ET 0 χn(s) K2(s) |Yn s|+|Ym s||Zn s|+Un sds +ET 0 2|Yn s−Ym s||Yn s|χn(s) K1(s)ds =: δ(1) n,m +δ(2) n,m +δ(3) n,m. (30) For δ(1) n,m,we use the Cauchy–Schwarz inequality, δ(1) n,m ≤2ET 0 χn(s) F (s)ds 21 2 YnS2+YmS2. Since supnYnS2<∞according to (28), it remains to show that the integral term converges to 0 for a subsequence. Since |Zn s|and Un sare uniformly integrable w.r.t. P⊗λ, we imply from (29) that χn→0inL1(P⊗λ). Hence, there exists a subsequence (nk)k≥1such that χnk→0k→∞,P⊗λ-a.e. (31) By dominated convergence, we have ET 0χnk(s)F (s)ds 2→0fork→∞ since F∈L2;L1([0,T]). For δ(2) n,m,we start with the Cauchy–Schwarz inequality and get δ(2) n,m ≤2sup k Zk L2(W) + Uk L2(˜ N) ×ET 0 χn(s) K2(s)2|Yn s|2+|Ym s|2ds 1 2 . By Lemma 4.3, ET 0 χn(s) K2(s)2|Yn s|2+|Ym s|2ds ≤2e2CKET 0 χn(s) K2(s)2ds|ξ|2 +2e2CK T 0 χn(s) K2(s)2ds ·IF 2YnS2+YmS2. (32) Hence, (31) implies δ(2) nk,m →0fork→∞. Finally, δ(3) n,m ≤2ET 02|Yn s|2+|Ym s|2χn(s) K1(s)ds,
Page 22 of 33 Geiss and Steinicke so that we can argue like in (32) to get that δ(3) nk,m →0fork→∞. Thus (Y nk,Znk,Unk)k≥1converges to an object (Y,Z,U)in L2(W ) ×L2(W ) × L2˜ N. Step 4: In the final step, we want to show that (Y,Z,U)solves (ξ, f ). For the approximating sequence (Ynk,Znk,Unk)k≥1,the stochastic integrals and the left hand side of the BSDEs ξ,f(nk)obviously converge in L2to the corresponding terms of (ξ, f ). Therefore, this subsequence of T tf(n) s,Yn s,Zn s,Un sds∞ n=1converges to a random variable Vt. We need to show that Vt=T tf(s,Y s,Z s,U s)ds. To achieve this, consider δn:= ET t|f(n) s,Yn s,Zn s,Un s−fs,Yn s,Zn s,Un s|ds +ET t|fs,Yn s,Zn s,Un s−f(s,Ys,Z s,U s)|ds. (33) We start with the first integrand where, by the definition of fnand (29), and the growth condition (A2), |f(n) s,Yn s,Zn s,Un s−fs,Yn s,Zn s,Un s| =|f(n) s,Yn s,Zn s,Un s−fs,Yn s,Zn s,Un s|χn ≤2F(s)χ n(s) +K1(s)|Yn s|χn(s) +K2(s)χn(s) |Zn s|+Un s =: 2κ(1) n(s) +κ(2) n(s) +κ(3) n(s). The estimates are similar as in the previous step. Thanks to (31), we have ET tκ(1) nk(s)ds →0.For the next term, the Cauchy–Schwarz inequality yields ET t κ(2) nk(s)ds ≤ T 0 χn(s) K1(s)ds 2 sup lYlS2, so that by (31) the first factor converges to zero along the subsequence (nk). The last term we estimate using the Cauchy–Schwarz inequality w.r.t. P⊗λ, ET t κ(3) nk(s)ds ≤ET 0 K2(s)2χn(s)ds 1 2 sup l Zl L2(W)+ Ul L2˜ N, and again by (31), we have convergence to zero along the subsequence (nk). We continue showing the convergence of the second term in (33). We extract a sub-subsequence of (nk)k≥1, which we call—slightly abusing the notation—again (nk)k≥1such that (Y nk,Znk,Unk), regarded as a triplet of measurable functions with values in R×R×L2(ν),convergesto(Y,Z,U)for P⊗λ-a.a. (ω, s) . Then, for an arbitrary K>0, we have ET tfs, Ynk s,Znk s,Unk s−f(s,Ys,Z s,U s)ds ≤ET tfs, Ynk s,Znk s,Unk s−f(s,Y s,Z s,U s)(34) ×χ|Ynk s|≤K,|Znk s|+Unk s≤K+χ|Ynk s|>K+χ|Znk s|+Unk s>Kds.
Probability, Uncertainty and Quantitative Risk (2018) 3:9 Page 23 of 33 By dominated convergence and the continuity of f, ET tfs, Ynk s,Znk s,Unk s−f(s,Y s,Z s,U s)χ|Ynk s|≤K,|Znk s|+Unk s≤Kds →0, since by (A2) we can bound the integrand by 2F(s)+K1(s)(K +|Ys|)+K2(s)(2K+|Zs|+Us), which is integrable. We let χK(nk,s):= χ|Ynk s|>K+χ|Znk s|+Unk s>K. Then, the remaining terms of (34) are bounded by ET 0 (2F(s)+K1(s)|Ys|+K2(s)(|Zs|+Us))χK(nk,s)ds +ET 0 K1(s)|Ynk s|χK(nk,s)ds +ET 0 K2(s) |Znk s|+Unk sχK(nk,s)ds =: δ(1) nk+δ(2) nk+δ(3) nk. If we choose a Klarge enough, then δ(1) nkcan be made arbitrarily small since the families |Yn s|,n≥0and |Zn s|+Un s,n≥0are uniformly integrable with respect to P⊗λ. The same holds for δ(2) nk2≤ET 0 K1(s)χK(nk,s)ds 2 sup lYnl2 S2 ≤ T 0 K1(s)ds ∞ ET 0 K1(s)χK(nk,s)dssup lYnl2 S2, and δ(3) nk2≤2ET 0 K2(s)2χK(nk,s)dssup l ET 0Znl s 2+ Unl s 2ds. Hence, for δndefined in (33), we have that limk→∞ δnk=0,which implies lim k→∞ ET t f(nk)s,Ynk s,Znk s,Unk sds −T t f(s,Y s,Z s,U s)ds=0. We infer that for a sub-subsequence (nkl,l ≥0)we get the a.s. convergence T t f(nkl)s,Ynkl s,Znkl s,Unkl sds →T t f(s,Y s,Z s,U s)ds. Thus, for the original sequence, a.s. T t f(nk)s,Ynk s,Znk s,Unk sds →Vt=T t f(s,Y s,Z s,U s)ds, and therefore the triplet (Y,Z,U)satisfies the BSDE (ξ, f ).
Page 24 of 33 Geiss and Steinicke 5.2 Proof of Theorem 3.4 We start with a preparatory lemma: Lemma 5.1 If f satisfies (A1)–(A4), then for all n≥0,fnconstructed in Definition 3.3 also satisfies (A1)–(A4) (with different coefficients). Proof By definition, (ω, t) → fn(t,y,z,u) is progressively measurable for all (y,z,u), thus (A1) is satisfied. The inequalities in (A2) and (A3) are a.s. satisfied, with coefficients EnF,EnK1,EnK2,Enβ. To ensure that these coefficients have a Fn tt∈[0,T ]-progressively measurable version, one applies the procedure from Definition 3.3 to the inequalities in (A2) and (A3) and notes that an equation analogous to (11) holds true. It remains to show a.s. continuity of fnin the (y,z,u)-variables required in (A3) for a.e. t. In (Ylinen (2017), Proposition 7.3), this was shown by the fact that the approximation of the generators appearing there can be done using spaces of continuous functions. However, since our situation involves L2(ν), a non-locally compact space, we can not easily adapt the proof from Ylinen (2017) and therefore we will use different means. Let D[0,T]be the space of c` adl` ag functions endowed by the Skorohod metric (which makes this space a Polish space). The Borel σ-algebra B(D[0,T])is generated by the coordinate projections pt:D[0,T]→R,x→ x(s) (see Theorem 12.5 of Billingsley (1968), for instance). On this σ-algebra, let PXbe the image measure induced by the L´ evy process X:→D[0,T],ω → X(ω).WedenotebyGthe completion with respect to PX.Fort∈[0,T],the notation xt(s) := x(t ∧s), for alls∈[0,T] induces the natural identification D[0,t]=x∈D[0,T]:xt=x. By this identification, we define a filtration on this space through Gt=σ(B(D[0,t])∪NX[0,T]),0≤t≤T, where NX[0,T]denotes the null sets of B(D[0,T])with respect to the image measure PXof the L´ evy process X. The same procedure applied to the L´ evy process Xn yields a filtration (Gn t)t∈[0,T ]defined in the same way. According to (Steinicke (2016), Theorem 3.4), which is a generalization of Doob’s factorization lemma to random variables depending on parameters, there is a Gt⊗ B([0,t]×R2×L2(ν))-measurable functional gf:D[0,t]×[0,t]×R2×L2(ν) →R and a Gn t⊗B([0,t]×R2×L2(ν))-measurable functional gfn:D[0,t]×[0,t]×R2×L2(ν) →R such that P-a.s., gf(X(ω), ·)=f(ω,·)and gfn(Xn(ω), ·)=fn(ω, ·). (35)
Probability, Uncertainty and Quantitative Risk (2018) 3:9 Page 31 of 33 Appendix The Bihari–LaSalle inequality. For the Bihari–LaSalle inequality we refer to (Mao (1997), pp. 45-46). Here, we formulate a backward version of it which has been appliedinYinandMao(2008). The proof is analogous to that in Mao (1997). Proposition 5.2 Let c>0.Assume that ρ:[0,∞[→ [0,∞[ is a continuous and non-decreasing function such that ρ(x) > 0for all x>0.Let K be a non-negative, integrable Borel function on [0,T],and y a non-negative, bounded Borel function on [0,T],such that y(t) ≤c+T tK(s)ρ(y(s))ds. Then, it holds that y(t) ≤G−1G(c) +T t K(s)ds for all t∈[0,T]such that G(c) +T tK(s)ds ∈dom G−1.Here G(x) := x 1 dr ρ(r), and G−1is the inverse function of G. Especially, if ρ(r) =rfor r∈[0,∞[,it holds that y(t) ≤ceT tK(s)ds.(46) The Jankov–von Neumann theorem. If Xand Yare sets and P⊆X×Y, then P∗⊆Pis called a uniformization of Pif and only if P∗is the graph of a function f:projX(P ) →Y, i.e., P∗={(x, f (x)) :x∈projX(P )}.Such a function fis called a uniformizing function for P.Let1 1(X) denote the class of analytic subsets of X. The following theorem can be found, for example, in (Kechris (1994), Theorem 18.1). Theorem 5.3 (Jankov–von Neumann theorem) Assume that X and Y are standard Borel spaces and P⊆X×Yis an analytic set. Then, P has a uniformizing function that is σ1 1(X)- measurable. Acknowledgements The authors thank Stefan Geiss and Juha Ylinen, University of Jyv¨ askyl¨ a, for fruitful discussions and valuable suggestions. Moereover, we are sincerly grateful to the anonymous reviewers for their helpful comments and questions. Christel Geiss would like to thank the Erwin Schr¨ odinger Institute, Vienna, for hospitality and support, where a part of this work was written. Funding Large parts of this article were written when Alexander Steinicke was member of the Institute of Mathematics and Scientific Computing, University of Graz, Austria, and supported by the Austrian Science Fund (FWF): Project F5508-N26, which is part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications.”
Page 32 of 33 Geiss and Steinicke Availability of data and material Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study. Authors’ contributions Both authors read and approved the final manuscript. Ethics approval and consent to participate Not applicable. Consent for publication Not applicable. Competing interests The authors declare that they have no competing interests. References Applebaum, D: L´ evy Processes and Stochastic Calculus. Cambridge University Press, Cambridge (2004) Barles, G, Buckdahn, R, Pardoux, ´ E: Backward stochastic differential equations and integral-partial differential equations. Stoch. Stoch. Rep. 60(1-2), 57–83 (1997) Becherer, D, B¨ uttner, M, Kentia, K: On the monotone stability approach to BSDEs with jumps: Extensions, concrete criteria and examples (2018). https://arxiv.org/abs/1607.06644 Billingsley, P: Convergence of probability measures. Wiley, New York (1968) Briand, P, Delyon, B, Hu, Y, Pardoux, E, Stoica, L: Lpsolutions of backward stochastic differential equations. Stoch. Proc. Appl. 108, 109–129 (2003) Cao, Z, Yan, J: A comparison theorem for solutions of backward stochastic differential equations. Adv. Math. 28(4), 304–308 (1999) Cohen, S, Elliott, R, Pearce, C: A General Comparison Theorem for Backward Stochastic Differential Equations. Adv. Appl. Probab. 42(3), 878–898 (2010) Darling, R, Pardoux, ´ E: Backwards sde with random terminal time and applications to semilinear elliptic pde. Ann. Probab. 25(3), 1135–1159 (1997) Delzeith, O: On Skorohod spaces as universal sample path spaces (2004). https://arxiv.org/abs/math/ 0412092v1 El Karoui, N, Hamad` ene, S, Matoussi, A: Backward stochastic differential equations. In: Carmona, R (ed.) Indifference Hedging: Theory and Applications, pp. 267–320. Princeton University Press (2009) El Karoui, N, Tan, X: Capacities, measurable selection and dynamic programming part I: abstract framework (2013). https://arxiv.org/abs/1310.3363 El Karoui, N, Peng, S, Quenez, M: Backward Stochastic Differential Equations in Finance. Math. Financ. 7(1), 1–71 (1997) Fan, S, Jiang, L: A Generalized Comparison Theorem for BSDEs and Its Applications. J. Theor. Probab. 25, 50–61 (2012) Geiss, C, Steinicke, A: Existence, Uniqueness and Malliavin Differentiability of L´ evy-driven BSDEs with locally Lipschitz Driver (2018). https://arxiv.org/abs/1805.05851 Geiss, S, Ylinen, J: Decoupling on the Wiener Space, Related Besov Spaces, and Applications to BSDEs (2018). is to appear in: Memoirs AMS, 2018 Gobet, E, Turkedjiev, P: Linear regression MDP scheme for discrete backward stochastic differential equations under general conditions. Math. Comp. 85, 1359–1391 (2016)
Probability, Uncertainty and Quantitative Risk (2018) 3:9 Page 33 of 33 He, SW, Wang, JG, Yan, JA: Semimartingale Theory and Stochastic Calculus. CRC Press, Boca Raton (1992) Kechris, A: Classical Descriptive Set Theory. Springer, New York (1994) Kruse, T, Popier, A: BSDEs with monotone generator driven by Brownian and Poisson noises in a general filtration. Stochastics. 88(4), 491–539 (2016) Kruse, T, Popier, A: Lp-solution for BSDEs with jumps in the case p<2. Stochastics (2017). http://dx. doi.org/10.1080/17442508.2017.1290095 Mao, X: Adapted solutions of backward stochastic differential equations with non-Lipschitz coefficients. Stoch. Process. Appl. 58, 281–292 (1995) Mao, X: Stochastic Differential Equations and Applications. Woodhead Publishing Limited, Cambridge (1997) Meyer, PA: Une remarque sur le calcul stochastique d´ ependant d’un param` etre. S´ eminaire probabilit´ es (Strasbourg), tome. 13, 199–203 (1979) Pardoux, ´ E: Generalized discontinuous backward stochastic differential equations. In: El Karoui, N, Mazliak, L (eds.) Backward Stochastic Differential Equations, Pitman Res. Notes Math., vol. 364, pp. 207–219. Longman, Harlow (1997) Pardoux, ´ E, Zhang, S: Generalized BSDEs and nonlinear Neumann boundary value problems. Probab. Theory Relat. Fields. 110, 535–558 (1996) Peng, S: A generalized dynamic programming principle and hamilton-jacobi-bellman equation. Stoch. Stoch. Rep. 38, 119–134 (1992) Peng, S: Backward stochastic differential equation, nonlinear expectation and their applications. Proceedings of the International Congress of Mathematicians, Volume I, pp. 393–432. Hindustan Book Agency, New Delhi (2010) Protter, P: Stochastic Integration and Differential Equations. Springer, Berlin (2004) Royer, M: Backward stochastic differential equations with jumps and related non-linear expectations. Stoch. Process. Appl. 116(10), 1358–1376 (2006) Sato, K: L´ evy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999) Situ, R: On solutions of backward stochastic differential equations with jumps and applications. Stoch. Process. Appl. 66, 209–236 (1997) Steinicke, A: Functionals of a L´ evy Process on Canonical and Generic Probability Spaces. J. Theoret. Probab. 29, 443–458 (2016) Sow, AB: BSDE with jumps and non-Lipschitz coefficients: Application to large deviations. Braz. J. Probab. Stat. 28(1), 96–108 (2014) Yao, S: Lp-solutions of Backward Stochastic Differential Equations with Jumps. Stoch. Proc. Appl. 127(11), 3465–3511 (2017) Yin, J, Mao, X: The adapted solution and comparison theorem for backward stochastic differential equations with Poisson jumps and applications. J. Math. Anal. Appl. 346, 345–358 (2008) Ylinen, J: Weighted Bounded Mean Oscillation applied to Backward Stochastic Differential Equations (2017). https://arxiv.org/abs/1501.01183