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Exis ence, uniqueness and compa ison esul s o BSDEs wi h Lé y jumps in an ex ended
mono onic gene a o se ing
© The Au ho (s), 2018.
Published e sion
Geiss, Ch is el; S einicke, Alexande
Geiss, C., & S einicke, A. (2018). Exis ence, uniqueness and compa ison esul s o BSDEs wi h
Lé y jumps in an ex ended mono onic gene a o se ing. P obabili y, Unce ain y and
Quan i a i e Risk, 3(9), 1-33. h ps://doi.o g/10.1186/s41546-018-0034-y
2018
P
obabili y, Unce ain y
and Quan i a i e Risk
P obabili y, Unce ain y and Quan i a i e Risk (2018) 3:9
DOI 10.1186/s41546-018-0034-y
R E S E A RC H Open Access
Exis ence, uniqueness and compa ison esul s o
BSDEs wi h L´
e y jumps in an ex ended mono onic
gene a o se ing
Ch is el Geiss ·Alexande S einicke
Recei ed: 4 Janua y 2018 / Accep ed: 29 No embe 2018 /
© The Au ho (s). 2018 Open Access This a icle is dis ibu ed unde he e ms o he C ea i e Commons
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changes we e made.
Abs ac We show ha he compa ison esul s o a backwa d SDE wi h jumps
es ablished in Roye (S och. P ocess. Appl 116: 1358–1376, 2006) and Yin and
Mao (J. Ma h. Anal. Appl 346: 345–358, 2008) hold unde mo e simpli ied con-
di ions. Mo eo e , we p o e exis ence and uniqueness allowing he coe icien s in
he linea g ow h- and mono onici y-condi ion o he gene a o o be andom and
ime-dependen . In he L2-case wi h linea g ow h, his also gene alizes he esul s
o K use and Popie (S ochas ics 88: 491–539, 2016). Fo he p oo o he com-
pa ison esul , we in oduce an app oxima ion echnique: Gi en a BSDE d i en by
B ownian mo ion and Poisson andom measu e, we app oxima e i by BSDEs whe e
he Poisson andom measu e admi s only jumps o size la ge han 1/n.
Keywo ds Backwa d s ochas ic di e en ial equa ion ·L´
e y p ocess ·compa ison
heo em ·exis ence and uniqueness
Ma hema ics Subjec Classi ica ion: 60H10
C. Geiss
Uni e si y o Jy askyla, Depa men o Ma hema ics and S a is ics, P.O. Box 35, 40014 Jy askyla,
Finland
A. S einicke ()
Depa men o Ma hema ics and In o ma ion Technology, Mon anuni e si ae Leoben, Leoben,
Aus ia
e-mail: alexande [email p o ec ed]
Page 2 o 33 Geiss and S einicke
1 In oduc ion
In his pape , we s udy backwa d s ochas ic di e en ial equa ions (BSDEs) o he
o m
Y =ξ+T
(s,Ys,Z
s,U
s)ds −T
ZsdWs−] ,T]×(R {0})
Us(x) ˜
N(ds,dx),
(1)
whe e Wdeno es a one-dimensional B ownian mo ion and ˜
Na compensa ed Poisson
andom measu e belonging o a gi en L´
e y p ocess wi h L´
e y measu e ν.In
pa icula , ou ocus lies on compa ison esul s and exis ence and uniqueness o
solu ions.
Compa ison heo ems s a e ha —unde ce ain condi ions—i ξ≤ξand ≤ ,
hen he p ocess Yo he solu ion sa is ies Y ≤Y
o all ∈[0,T]. These ypes o
heo ems in he case o one-dimensional, B ownian BSDEs has been ea ed by Peng
(1992), El Ka oui e al. (1997,2009), and Cao and Yan (1999).
In (Ba les e al. (1997), Rema k 2.7) a coun e example was gi en, which shows
ha in he jump case he condi ions ξ≤ξand ≤ a e no su icien o gua an ee
Y≤Y. They p opose an addi ional su icien condi ion which has been gene alized
by K use and Popie (2016), Roye (2006),YinandMao(2008), Beche e e al.
(2018) (allowing mo e gene al jump p ocesses), and Cohen e al. (2010) ( o BSDEs
d i en by ma ingales). The condi ion o K use and Popie (2016) eads (in ou L2-
se ing) as ollows: o each s, y, z, u, u∈[0,T]×R×R×L2(ν) ×L2(ν) he e is
a p og essi ely measu able p ocess γy,z,u,u:×[0,T]×R {0}→Rsuch ha
(s,y,z,u)− s, 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,uγy,z,u,u
s∈L2(ν). (2)
One o he main esul s in he p esen pape is Theo em 3.5 which s a es ha (2)
can be eplaced by he simple condi ion
(s,y,z,u)− s, y, z, u≤R {0}u(x) −u(x)ν(dx), P⊗λ-a.e.
o all u, u∈L2(ν) wi h u≤u.(3)
No ice ha he .h.s. is in ini e o u(x) −u(x) /∈L1(ν). Clea ly, (3) is a weake
condi ion han (2), because one only needs o check he inequali y o hose u, u∈
L2(ν) o which u≤uholds. Mo eo e , we do no need any L2(ν) condi ion o
γy,z,u,u
sbu we choose γy,z,u,u
s(x) =−1.Unde he cons ain −1≤γy,z,u,u
s(x),
he choice γy,z,u,u
s(x) =−1 yields o u−u≥0 he la ges possible exp ession on
he .h.s. o (2), so ha (3) can be seen as he weakes possible condi ion which (2)
could impose on .
Fo a ini e L´
e y measu e ν, Theo em 3.5 can be shown using only elemen a y
means.
P obabili y, Unce ain y and Quan i a i e Risk (2018) 3:9 Page 3 o 33
Ano he main esul is a me hod o how o app oxima e a BSDE d i en by a
L´
e y p ocess wi h an in ini e measu e ν, by a sequence o BSDEs whe e he d i ing
p ocesses ha e a ini e L´
e y measu e. We apply his esul o show he compa i-
son heo em o BSDEs d i en by a gene al L´
e y p ocess. The p oo elies on he
Janko – on Neumann heo em on measu able sec ions/uni o miza ions ( his heo em
is also impo an o dynamic p og amming, see El Ka oui and Tan (2013). Unde
ce ain condi ions on he gene a o , he app oxima ing solu ions can be in e p e ed
as nonlinea condi ional expec a ions (in he sense o Peng (2010)), condi ioned on a
L´
e y p ocess whose jumps a e no o a bi a ily small size. (See he commen s a e
Theo em 3.4.)
S udying he exis ence, uniqueness, and compa ison esul s by Da ling and
Pa doux (1997), Pa doux and Zhang (1996), Pa doux (1997), Fan and Jiang (2012),
Roye (2006), Si u (1997), Yin and Mao (2008), K use and Popie (2016,2017),
Yao (2017), and Sow (2014), one no ices ha one can uni y and gene alize he
assump ions on .
Indeed, and his is ou hi d main esul , in he case o L2-solu ions, o a p o-
g essi ely measu able gene a o wi h linea g ow h, i su ices o assume (c .
Theo ems 3.1 and 3.5) he ollowing g ow h- and mono onici y condi ions wi h
ime-dependen , andom coe icien s:
•| (ω,s,y,z,u)|≤F(s,ω)+K1(s, ω)|y|+K2(s, ω)(|z|+u),
•y−y 1(ω,s,y,z,u)− 1ω,s,y,z
,u

≤α(s)ρ y−y
2+β(s,ω)y−yz−z+
u−u
,
wi h α∈L1([0,T])and Fbeing nonnega i e and p og essi ely measu able such
ha ET
0F(ω, )d 2<∞.The p ocesses K1,K
2,and βa e nonnega i e and
p og essi ely measu able such ha o a cons an c>0,
T
0K1(s) +K2(s)2+β(s)2ds < c, P-a.s.
The conca e unc ion ρin he mono onici y condi ion may g ow as e han linea
a ze o and sa is ies 0+1/ρ(x)dx =∞.This ype o unc ion al eady appea ed in
con ex wi h BSDEs in Mao (1995) in 1997.
These assump ions also ex end he mono onici y condi ion o K use and Popie
(2016,2017), o he L2-case wi h linea g ow h, since he coe icien s in ou se ing
ake andomness, he unc ion ρand ime-dependence in o accoun . BSDEs wi h
ime-dependen coe icien s appea , o example, in Gobe and Tu kedjie (2016).
The exis ence and uniqueness esul Theo em 3.1 and he compa ison esul
Theo em 3.5 a e basic ools in he o hcoming pape (Geiss and S einicke 2018)on
Mallia in di e en iabili y and boundedness o solu ions o BSDEs. To compu e he
Mallia in de i a i e o he jump pa o he L´
e y p ocess, mo e s uc u e om he
gene a o is equi ed in i s dependency on u, usually ia an in eg al w. . . ν(dx), o
example,
(s,u)=hs,R {0}
u(x)κ(s, x)ν(dx),
Page 4 o 33 Geiss and S einicke
whe e [0,T]×R(s, ) → h(s, ). One can ind hand κsuch ha he assump ions
o Theo em 3.5 a e sa is ied while condi on (2) does no hold: By he mean alue
heo em he e exis s a ζ∈]0,1[and
ζ:= R {0}ζu(x) +(1−ζ)u(x)κ(s,x)ν(dx),
such ha
(s,u)− s, u=∂ hs, ζR {0}u(x) −u(x)κ(s,x)ν(dx).
Assump ion (3) holds i γu,u
s(x) := ∂ hs, ζκ(s,x) ≥−1 o alls, u, u,x.
Choosing, o example, a bounded unc ion hsuch ha also sups, |∂ h(s, )|<∞,
bu ∂ h(s, ) = 0 o a.e. sand , and pu ing κ(s,x) =s−1
4(|x|∧1), hen (2) does
no hold since
sup
s,u,uγu,u
s/∈L2(ν).
Howe e , he Assump ions (A2),(A3) o Sec ion 3a e sa is ied o
K2(s) =β(s) =sup
|∂ h(s, )|κ(s,·)L2(ν) ≤cs−1
4.
The pape is s uc u ed as ollows: Sec ion 2con ains p elimina ies and basic de -
ini ions. In Sec ion 3, we p esen he main heo ems o his pape abou exis ence
and uniqueness o solu ions, he app oxima ion using BSDEs based on L´
e y p o-
cesses wi h ini e L´
e y measu e, and he compa ison esul . The la e we also p o e
he e. Ha ing s a ed and p o ed some auxilia y esul s in Sec ion 4, including an
a-p io i es ima e o ou ype o BSDEs, we a e able o p o e exis ence and unique-
ness and he app oxima ion esul om Sec ion 3. In he appendix, we ecall he
Biha i–LaSalle inequali y and he Janko – on Neumann heo em.
2 Se ing
Le X=(X ) ∈[0,T ]be a c`
adl`
ag L´
e y p ocess on a comple e p obabili y space
(, F,P)wi h L´
e y measu e ν. We will deno e he augmen ed na u al il a ion o
Xby (F ) ∈[0,T ]and assume ha F=FT.Fo 0 <p≤∞we use he no a ion
Lp,·p:= (Lp(, F,P), ·Lp). Equa ions o inequali ies o objec s o hese
spaces h oughou he pape a e conside ed up o P-null se s.
The L´
e y–I ˆ
o decomposi ion o a L´
e y p ocess Xcan be w i en as
X =a +σW +]0, ]×{|x|≤1}
x˜
N(ds,dx)+]0, ]×{|x|>1}
xN(ds, dx), (4)
whe e a∈R,σ≥0, Wis a B ownian mo ion and N(˜
N) is he (compensa ed)
Poisson andom measu e co esponding o X, see Applebaum (2004)o Sa o(1999).

P obabili y, Unce ain y and Quan i a i e Risk (2018) 3:9 Page 5 o 33
No a ion
•Le S2deno e he space o all (F )-p og essi ely measu able and c`
adl`
ag
p ocesses Y:×[0,T]→Rsuch ha
Y2
S2:= Esup
0≤ ≤T|Y |2<∞.
•We de ine L2(W ) as he space o all (F )-p og essi ely measu able p ocesses
Z:×[0,T]→Rsuch ha
Z2
L2(W) := ET
0|Zs|2ds < ∞.
•Le R0:= R {0}.Wede ineL2˜
Nas he space o all andom ields U:×
[0,T]×R0→Rwhich a e measu able wi h espec o P⊗B(R0)(whe e P
deno es he p edic able σ-algeb a on ×[0,T]gene a ed by he le -con inuous
(F )-adap ed p ocesses) such ha
U2
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) o p>0, whe e λis he Lebesgue
measu e on [0,T].
•Wi h a sligh abuse o he no a ion, we de ine
L2;L1([0,T])(5)
:=F∈L0( ×[0,T],F⊗B([0,T]), P⊗λ) :ET
0|F(ω, )|d 2
<∞.
Fo F∈L2;L1([0,T]),pu
IF(ω) := T
0
F(ω, )d and KF(ω, s) := F(ω,s)
IF(ω) .(6)
•Asolu ion o a BSDE wi h e minal condi ion ξand gene a o is a iple
(Y,Z,U)∈S2×L2(W) ×L2˜
Nwhich sa is ies o all ∈[0,T]:
Y =ξ+T
(s,Y
s,Z
s,U
s)ds−T
ZsdWs−] ,T]×R0
Us(x) ˜
N(ds,dx). (7)
The BSDE (7) i sel will be deno ed by (ξ, ).
3Main esul s
We s a wi h a esul abou exis ence and uniqueness which is p o ed in Sec ion 5.
Theo em 3.1 The e exis s a unique solu ion o he BSDE (ξ, ) wi h ξ∈L2and
gene a o :×[0,T]×R×R×L2(ν) →Rsa is ying he p ope ies
Page 6 o 33 Geiss and S einicke
(A1) Fo all (y,z,u):(ω, s) → (ω,s,y,z,u)is p og essi ely measu able.
(A2) The e a e nonnega i e, p og essi ely measu able p ocesses K1,K
2,and F
wi h
CK:= 


T
0K1(·,s)+K2(·,s)
2ds


∞
<∞(8)
and F∈L2;L1([0,T])(see (5)) such ha o all (y,z,u),
| (s,y,z,u)|≤F(s)+K1(s)|y|+K2(s)(|z|+u), P⊗λ-a.e.
(A3) Fo λ-almos all s, he mapping (y,z,u)→ (s,y,z,u)is P-a.s. con inuous.
Mo eo e , he e is a nonnega i e unc ion α∈L1([0,T]),c>0and a p o-
g essi ely measu able p ocess βwi h T
0β(ω,s)2ds < c,P-a.s. such ha o
all (y, z, u), y,z
,u
,
y−y (s,y,z,u)− s,y,z
,u

≤α(s)ρ |y−y|2+β(s)y−yz−z+
u−u
,P⊗λ-a.e.,
whe e ρis a nondec easing, con inuous and conca e unc ion om [0,∞[ o
i sel , sa is ying ρ(0)=0,and 0+1
ρ(x)dx =∞.
(A4) The unc ion ρin (A3) sa is ies lim supx↓0
ρ(x2)
x=0.
I sa is ies only (A1)–(A3), hen he e exis s a mos one solu ion.
Fo ρ(x) =x, we a e in he case o he o dina y mono onici y condi ion. Ano he
example o a unc ion ρis gi en by
ρ(x) =1−min x, 1
eminx,1
e,x≥0.
Rema k 3.2 .
1. Condi ion (A2) implies ha (s,y,z,u)is in eg able o a.e. s∈[0,T]since,
by Fubini’s heo em,
T
0
E| (s,y,z,u)|ds
≤ET
0[F(s)+K1(s)|y|+K2(s)(|z|+u))]ds < ∞.(9)
2. I lim supx↓0
ρ(x2)
x=0is sa is ied one can de i e Lipschi z con inui y o
(s,y,z,u)in z and u om he mono onici y condi ion in (A3). We equi e (A4)
since we la e wan o apply (Yin and Mao (2008), Theo em 2.1), whe e Lipschi z
con inui y in u is used o show uniqueness o solu ions. I only (A1)–(A3) a e
sa is ied bu no (A4), and a Lipschi z condi ion in z, u holds ne e heless, all
o he a icle’s heo ems emain alid. One can show ha (A4) does no ollow
om he o he condi ions imposed on ρin (A3): Assume a dec easing sequence
(xn)∞
n=0wi h x0=1and limn→∞ xn=0.De ine
P obabili y, Unce ain y and Quan i a i e Risk (2018) 3:9 Page 7 o 33
ρ(x) := √xni x=xn,n=0,1,2, ...
√xi x>1o x=0.
and le ρbe con inuous and piecewise linea on ]0,1].The so de ined ρis
a conca e unc ion wi h lim supx↓0
ρ(x)
√x=1.The sequence (xn)∞
n=0can be
cons uc ed such ha 1
0
1
ρ(x)dx =∞.Fo example, choose x1such ha
1
x1
1
ρ(x)dx ≥1,and i xnhas been chosen ind xn+1such ha
xn
xn+1
1
ρ(x)dx =1
2(log(xn)−log(xn+1))√xn+√xn+1≥1.
The nex esul shows how a solu ion o a BSDE can be app oxima ed by a
sequence o solu ions o BSDEs which a e d i en by L´
e y p ocesses wi h a ini e
L´
e y measu e. We do his by app oxima ing he unde lying L´
e y p ocess de ined
h ough
X =a +σW +]0, ]×{|x|>1}
xN(ds,dx) +]0, ]×{|x|≤1}
x˜
N(ds,dx)
o n≥1by
Xn
=a +σW +]0, ]×{|x|>1}
xN(ds,dx) +]0, ]×{1/n≤|x|≤1}
x˜
N(ds,dx).
The p ocess Xnhas a ini e L´
e y measu e νn. Fu he mo e, no e ha he
compensa ed Poisson andom measu e associa ed wi h Xncan be exp essed as
˜
Nn=χ{1/n≤|x|} ˜
N. Le
J0:= {,∅}∨N,
Jn:= σXn∨N,n≥1,(10)
whe e Ns ands o he null se s o F.No e ha (Jn)∞
n=0 o ms a il a ion. The
no a ion (Jn)∞
n=0was chosen o indica e ha his il a ion desc ibes he inclusion o
smalle and smalle jumps o he L´
e y p ocess. We will use
En·:=E·Jn
o he condi ional expec a ion.
The in ui i e idea now would be o wo k wi h a BSDE d i en by Xnwhe e one
uses he da a (Enξ,En ).The p oblem is ha he gene a o needs o be p og es-
si ely, and also join ly measu able w. . . (ω, ,y,z,u),bu i is no ob ious whe he
he condi ional expec a ion En p ese es his p ope y om . Fo BSDEs d i en
by a B ownian mo ion, his p oblem has been sol ed in (Ylinen (2017), P oposi ion
7.3), bu his p oposi ion does no apply o ou si u a ion. The e o e, we nex p o-
pose a me hod o he cons uc ion o a unique p og essi ely measu able and join ly
measu able w. . . (ω, ,y,z,u) e sion o En .
De ini ion 3.3 (De ini ion o n)Assume ha sa is ies (A1),(A2) and ha J:=
J[s]s∈[0,∞[ is buil using (10), whe e [·] deno es he loo unc ion. Le o,J
be he
op ional p ojec ion o he p ocess
Page 8 o 33 Geiss and S einicke
[0,∞[××[0,T]×R2×L2(ν) →R,
(s,ω, ,y,z,u)→ (ω, ,y,z,u)
in he a iables (s, ω) wi h espec o J,and wi h pa ame e s ( ,y,z,u). Fo each
n≥0, assume ha he il a ion Fn:= Fn
 ∈[0,T ]is gi en by Fn
:= F ∩Jn.Le
nbe he op ional p ojec ion o
(ω, ,y,z,u)→ o,J
(n,ω, ,y,z,u)
wi h espec o Fnwi h pa ame e s (y,z,u).
The eason o using he il a ion J[s]s∈[0,∞[ ins ead o he (Jn)∞
n=0 om (10)
is ha one can apply known measu abili y esul s w. . . igh con inuous il a ions
ins ead o p o ing measu abili y he e di ec ly. Indeed, he op ional p ojec ion o,J
de ined abo e is join ly measu able in (s,ω, ,y,z,u). Fo his we e e o Meye
(1979), whe e op ional and p edic able p ojec ions o andom p ocesses depending
on pa ame e s we e conside ed, and hei uniqueness up o indis inguishabili y was
shown.
I ollows ha o all ( ,y,z,u),
o,J
(n, ,y,z,u)=En ( , y, z, u), P-a.s.
Then, since is (F ) ∈[0,T ]-p og essi ely measu able, o all n≥0, ∈[0,T]and
all (y,z,u), i holds ha
n( ,y,z,u)=En ( , y, z, u), P-a.s. (11)
Hence, n( ,y,z,u) is a join ly measu able e sion o En ( ,y,z,u) which is
Fn
 ∈[0,T ]-op ional, so especially i is p og essi ely measu able.
We commen on he compa ibili y o he solu ions (Y n,Zn,Un) om he BSDE
co esponding o (Enξ, n),
Yn
=Enξ+T
ns,Yn
s,Zn
s,Un
sds −T
Zn
sdWs
−] ,T]×R0
Un
s(x) ˜
Nn(ds, dx)
wi h he space S2×L2(W) ×L2˜
N:
The iple (Yn,Zn,Un)∈S2×L2(W ) ×L2˜
Nncan be canonically embedded
in he space S2×L2(W)×L2˜
N, basically by ex ending Un
s(x) on o R0by de ining
Un
s(x) := 0 o |x|<1
n. Mo eo e , ecall ha ˜
Nn=χ{1/n≤|x|} ˜
N, so ha
] ,T]×R0
Un
s(x) ˜
Nn(ds, dx) =] ,T]×R0
Un
s(x)χ{1/n≤|x|} ˜
N(ds,dx).
The e o e, Yn,Zn,UnχR ]−1/n,1/n[sol es (Enξ, n)in S2×L2(W)×L2˜
N.
P obabili y, Unce ain y and Quan i a i e Risk (2018) 3:9 Page 15 o 33
We use his es ima e o R=2,and aking he expec a ion in (17), we ha e
ET
0
es
0η(τ)dτ η(s)|Ys|2+|Zs|2+Us2
ds≤EeT
0η(s)ds|ξ|2+ET
0
es
0η(τ)dτ |Zs|2+Us2
2ds
+2ET
0
es
0η(τ)dτ F(s)ds sup
∈[0,T ]|Y |
+ET
0
es
0η(τ)dτ 2K1(s) +2K2(s)2Y2
sds.
(19)
Then, we choose η(s) =2K1(s) +2K2(s)2and sub ac he e ms con aining
Y, Z, and U om he le hand side o (19). Mo eo e , we apply he i s inequali y
o (14) o he e m con aining he sup emum. I ollows ha
ET
0
es
0η(τ)dτ |Zs|2+Us2ds ≤2EeT
0η(s)ds|ξ|2+2RET
0
es
0η(τ)dτ F(s)ds2
+2
REsup
∈[0,T ]|Y |2.
(20)
No e ha
ET
0|Zs|2+Us2ds ≤ET
0
es
0η(τ)dτ |Zs|2+Us2ds.
Hence, by (20)andT
0η(τ)dτ ≤4CKa.s., we ha e
ET
0|Zs|2+Us2ds ≤2e4CKE|ξ|2+2Re8CKEI2
F+2
REsup
∈[0,T ]|Y |2.(21)
Now, we can plug in (21)in o(15) and ice e sa which yields o R:= 48c1 ha
Esup
∈[0,T ]|Y |2≤2c1+48c1e4CKE|ξ|2+2c1+(48c1)2e8CKEI2
F,
and
E
T
0|Zs|2+Us2ds ≤1
12 +4e4CKE|ξ|2+1
12 +192c1e8CKEI2
F.
Using (16) i is easy o see ha he e exis s a cons an C1>0 such ha each ac o
in on o he expec a ions on he igh side o he p e ious wo inequali ies is less
han eC1(1+CK)2.
Ou nex p oposi ion will be an L2a-p io i es ima e o BSDEs o ou ype. Fo he
B ownian case, Lpa-p io i es ima es a e done o p∈[1,∞[in B iand e al. (2003),
and o quad a ic BSDEs, o p∈[2,∞[ in Geiss and Ylinen (2018). Fo BSDEs
wi h jumps, o p∈]1,∞[,see K use and Popie (2016,2017); while Beche e e al.
(2018) con ains an a-p io i es ima e w. . . L∞.The ollowing asse ion is simila o
(Ba les e al. (1997), P oposi ion 2.2), bu i s ou ex ended se ing.

Page 16 o 33 Geiss and S einicke
P oposi ion 4.2 Le ξ,ξ∈L2and le , be wo gene a o unc ions sa is y-
ing (A1)–(A3), whe e he bounds in (A2) and he coe icien s in (A3) may di e o
and . The coe icien s o in (A3) will be e e ed o as αand β. Mo eo e ,
le he iple s (Y,Z,U)and (Y ,Z,U)∈L2(W ) ×L2(W ) ×L2˜
N, sa is y he
BSDEs (ξ, ) and (ξ, ), espec i ely.
Then,
Y−Y2
L2(W) +
Z−Z

2
L2(W) +
U−U

2
L2(˜
N)
≤ha,b,E|ξ−ξ|2+2ET
0|Y −Y
| ( ,Y
,Z
,U
)− ( , Y ,Z
,U
)d ,
whe e a=T
0α(s)ds, b =

T
0β(s)2ds

∞,and
h:]0,∞[×]0,∞[×[0,∞[→ [0,∞[
is a unc ion such ha h(a, b, x) →0=h(a, b, 0)i x→0.
P oo We s a wi h he ollowing obse a ion gained by I ˆ
o’s o mula o he di -
e ence o he BSDEs (ξ, ) and (ξ, ). We deno e di e ences o exp essions by
.I η=4β(s)2,we ha e analogously o (17)
e
0η(s)ds|Y |2+T
es
0η(τ)dτ η(s)|Ys|2+|Zs|2+Us2ds
=eT
0η(s)ds|ξ|2+M( )
+T
2es
0η(τ)dτ Ys( (s, Ys,Z
s,U
s)− (s, Y 
s,Z
s,U
s))ds, (22)
whe e
M( ) =−T
2es
0η(τ)dτ YsZsdWs
−] ,T]×R0
2es
0η(τ)dτ (Ys−+Us(x))2−Y 2
s−˜
N(ds,dx).
By he same easoning as o (18), we ha e EM( ) =0. We now p oceed wi h
he (s anda d) a gumen s simila o hose used o (17)–(19). By (A3) and he i s
inequali y om (14),
Ys( (s, Ys,Z
s,U
s)− (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+Us2
2.
(23)
P obabili y, Unce ain y and Quan i a i e Risk (2018) 3:9 Page 17 o 33
Taking he expec a ion in (22) and hen using (23) wi h R=1 (such ha we can
cancel ou he e ms wi h Zand Uon he le side), leads o
Ee
0η(s)ds|Y |2+ET
es
0η(τ)dτ η(s)|Ys|2ds ≤EeT
0η(s)ds|ξ|2
+ET
2es
0η(τ)dτ Ys·( )(s, Ys,Z
s,U
s)ds
+ET
es
0η(τ)dτ 2α(s)ρ |Ys|2+β(s)2|Ys|2ds.
The choice η(s) =4β(s)2and he ac ha T
0β(s)2ds ≤ba.s. leads o
E|Y |2≤e4bE|ξ|2+ET
2|Ys||( )(s, Ys,Z
s,U
s)|ds
+e4bT
2α(s)ρ E|Ys|2ds,
since ρis a conca e unc ion.
By P oposi ion 5.2, a backwa d e sion o he Biha i–LaSalle inequali y, shows
sup
∈[0,T ]
E|Y |2≤
G−1Ge4bE|ξ|2+ET
0
2|Ys||( )(s, Ys,Z
s,U
s)|ds+2e4bT
0
α(s)ds,
(24)
whe e G(x) =x
1
1
ρ(h)dh.
I we ake he expec a ion in (22) bu choose his ime (23) wi h R=1
2and omi
Ee
0η(s)ds|Y |2, hen
ET
es
0η(τ)dτ η(s)|Ys|2+|Zs|2+Us2ds
≤EeT
0η(s)ds|ξ|2+ET
2es
0η(τ)dτ Ys·( )(s, Ys,Z
s,U
s)ds
+ET
es
0η(τ)dτ 2α(s)ρ |Ys|2+4β(s)2|Ys|2+|Zs|2+Us2
2ds.
We sub ac he quad a ic e ms wi h Y, Z, and U which appea on he igh
hand side. This esul s in he inequali y
ET
es
0η(τ)dτ |Zs|2+Us2ds
≤2EeT
0η(s)ds|ξ|2+ET
2es
0η(τ)dτ |Ys|·|( )(s, Ys,Z
s,U
s)|ds
+ET
es
0η(τ)dτ 2α(s)ρ |Ys|2)ds.
Page 18 o 33 Geiss and S einicke
We con inue ou es ima e by
ET
es
0η(τ)dτ |Zs|2+Us2ds
≤2e4bE|ξ|2+ET
2|Ys|·|( )(s, Ys,Z
s,U
s)|ds (25)
+2T
α(s)ds ρ sup
s∈[0,T ]
E|Ys|2,
since η(s) =4β(s)2. We pu
H:= G−1Ge4bE|ξ|2+ET
02|Ys||( )(s, Ys,Z
s,U
s)|ds
+2e4bT
0α(s)ds
so ha (24) eads now as sup ∈[0,T ]E|Y |2≤H. I we add his inequali y o (25)
and no e ha ρsup ∈[0,T ]E|Ys|2≤ρ(H), we ha e
sup
s∈[0,T ]
E|Y |2+ET
0|Zs|2ds +ET
0Us2ds
≤2e4bE|ξ|2+ET
02|Ys|·|( )(s, Ys,Z
s,U
s)|ds
+2e4bT
0α(s)ds +1·(id +ρ)(H).
No e ha he in eg al condi ion on ρimplies ha , i he a gumen o Gapp oaches
ze o, hen he igh hand side anishes.
The ollowing Lemma will be used o es ima e he expec a ion o in eg als which
con ain |Ys|2.
Lemma 4.3 Le ξ∈L2and assume ha (A1) and (A2) hold. I (Y,Z,U) is a
solu ion o (ξ, ) and H is a nonnega i e, p og essi ely measu able p ocess wi h


T
0H(s)ds

∞<∞, hen
ET
0
H(s)|Ys|2ds ≤e2CKET
0
H(s)ds|ξ|2
+2e2CK


T
0
H(s)ds ·IF


2YS2.
(26)
P oo F om he ela ions (17), (18) and in eg a ion by pa s applied o he e m
T
0H(s)ds ·eT
0η(s)ds|YT|2,we ge
T
0
H(s)ds ·eT
0η( )d |YT|2=T
0
H(s)es
0η(τ)dτ |Ys|2ds −T
0s
0
H(τ)dτdM(s)
+T
0s
0
H( )d es
0η(τ)dτ η(s)|Ys|2+|Zs|2+Us2
−2Ys (s,Y
s,Z
s,U
s))ds.
P obabili y, Unce ain y and Quan i a i e Risk (2018) 3:9 Page 19 o 33
We ake expec a ions and ea ange he equa ion so ha
ET
0
H(s)es
0η(τ)dτ |Ys|2ds ≤ET
0
H(s)ds ·eT
0η(s)ds|ξ|2
+ET
0s
0
H(τ)dτes
0η(τ)dτ (2Ys (s,Y
s,Z
s,U
s)
−η(s)|Ys|2−|Zs|2−Us2ds.
By Assump ion (A2) and (14), we ha e
2Ys (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+Us2,
so ha o η(s) =2K1(s) +2K2(s)2i ollows
ET
0
H(s)|Ys|2ds ≤ET
0
H(s)ds ·eT
0η(s)ds|ξ|2
+2ET
0s
0
H(τ)dτes
0η(τ)dτ F(s)|Ys|ds
≤e2CKET
0
H(s)ds ·|ξ|2
+2e2CK


T
0
H(s)ds ·IF


2YS2.
(27)
5 P oo s o Theo ems 3.1 and 3.4
5.1 P oo o Theo em 3.1
S ep 1: Uniqueness
Uniqueness o he solu ion is a consequence o P oposi ion 4.2, since he e ms |ξ−
ξ|and | (s,Y
s,Z
s,U
s)− (s, Ys,Z
s,U
s)|a e ze o.
The p oo o exis ence will be spli up in u he s eps.
S ep 2:
In his s ep, we cons uc an app oxima ing sequence o gene a o s (n) o
and show se e al es ima es o he solu ion p ocesses (Y n,Zn,Un) o he BSDEs
ξ, (n).
Fo n≥1,de ine cn(z) := min(max(−n, z), n) and ˜cn(u) ∈L2(ν) o be he
p ojec ion o uon o { ∈L2(ν) : ≤n}.Le (Y n,Zn,Un)be he unique solu ion
o he BSDE ξ, (n), wi h he de ini ions
ˆ
(n)(ω,s,y,z,u):= (ω,s,y,c
n(z), ˜cn(u)),
Page 20 o 33 Geiss and S einicke
and
(n)(ω,s,y,z,u):= sign ˆ
(n)(ω,s,y,z,u)

×F(ω,s)∧n+(K1(ω, s) ∧n)|y|+(K2(ω, s) ∧n)(|cn(z)|+˜cn(u))
i |ˆ
(n)(ω,s,y,z,u)|>F(ω,s)∧n+(K1(ω, s) ∧n)|y|
+(K2(ω, s) ∧n)(|cn(z)|+˜cn(u)),
and
(n)(ω,s,y,z,u):= ˆ
(n)(ω,s,y,z,u) else.
No e ha (n) sa is ies (A1)–(A4), wi h he same coe icien s as . Mo eo e ,
by (A4), (n) sa is ies a Lipschi z condi ion wi h espec o u(see Rema k 3.2).
Thus, hanks o (Yin and Mao (2008), Theo em 2.1), ξ, (n)has a unique solu ion
(Y n,Zn,Un). Mo eo e , by P oposi ion 4.1, we ge ha
Yn2
S2+
Zn

2
L2(W) +
Un

2
L2(˜
N) ≤eC1(1+CK)2E|ξ|2+EI2
F<∞,(28)
uni o mly in n. This implies ha he amilies
sup
∈[0,T ]|Yn
|,n≥0,|Yn|,n≥0and |Zn|+Un,n≥0
a e uni o mly in eg able wi h espec o P,P⊗λand P⊗λ, espec i ely.
S ep 3:
The goal o his s ep is o use P oposi ion 4.2 o ge con e gence o (Y n,Zn,Un)n
in L2(W)×L2(W)×L2(˜
N) o a subsequence nk↑∞i δnk,nl→0 o k>l→∞,
whe e
δn,m := ET
0|Yn
s−Ym
s|| (n) s,Yn
s,Zn
s,Un
s− (m) s,Yn
s,Zn
s,Un
s|ds.
We obse e ha he di e ence o he gene a o s is ze o i wo condi ions a e sa -
is ied a he same ime: Fi s , i |Zn|,Un
s<n, and addi ionally, by he cu -o
p ocedu e o F,K1,K
2,i
n>max (F(ω,s),K
1(ω, s), K2(ω, s))=: k(ω,s).
Thus, pu ing
χn(s) := χ{|Zn
s|>n}∪{Un
s>n}∪{k(s)>n},(29)
we ha e
δn,m =ET
0|Yn
s−Ym
s|| (n) s,Yn
s,Zn
s,Un
s− (m) s,Yn
s,Zn
s,Un
s|χn(s)ds
≤ET
0
2|Yn
s−Ym
s|χn(s)×F(s)+K1(s)|Yn
s|+K2(s) |Zn
s|+Un
sds,

P obabili y, Unce ain y and Quan i a i e Risk (2018) 3:9 Page 21 o 33
due o he linea g ow h condi ion (A2). We es ima e his u he by
δn,m ≤ET
0
χn(s) F (s)ds sup
∈[0,T ]|Yn
|+ sup
∈[0,T ]|Ym
|
+ET
0
χn(s) K2(s) |Yn
s|+|Ym
s||Zn
s|+Un
sds
+ET
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)
Fo δ(1)
n,m,we use he Cauchy–Schwa z inequali y,
δ(1)
n,m ≤2ET
0
χn(s) F (s)ds
21
2
YnS2+YmS2.
Since supnYnS2<∞acco ding o (28), i emains o show ha he in eg al
e m con e ges o 0 o a subsequence.
Since |Zn
s|and Un
sa e uni o mly in eg able w. . . P⊗λ, we imply om (29)
ha χn→0inL1(P⊗λ). Hence, he e exis s a subsequence (nk)k≥1such ha
χnk→0k→∞,P⊗λ-a.e. (31)
By domina ed con e gence, we ha e ET
0χnk(s)F (s)ds
2→0 o k→∞
since F∈L2;L1([0,T]).
Fo δ(2)
n,m,we s a wi h he Cauchy–Schwa z inequali y and ge
δ(2)
n,m ≤2sup
k

Zk

L2(W) +

Uk

L2(˜
N)
×ET
0
χn(s) K2(s)2|Yn
s|2+|Ym
s|2ds
1
2
.
By Lemma 4.3,
ET
0
χn(s) K2(s)2|Yn
s|2+|Ym
s|2ds
≤2e2CKET
0
χn(s) K2(s)2ds|ξ|2
+2e2CK


T
0
χn(s) K2(s)2ds ·IF


2YnS2+YmS2.
(32)
Hence, (31) implies δ(2)
nk,m →0 o k→∞.
Finally,
δ(3)
n,m ≤2ET
02|Yn
s|2+|Ym
s|2χn(s) K1(s)ds,
Page 22 o 33 Geiss and S einicke
so ha we can a gue like in (32) o ge ha δ(3)
nk,m →0 o k→∞.
Thus (Y nk,Znk,Unk)k≥1con e ges o an objec (Y,Z,U)in L2(W ) ×L2(W ) ×
L2˜
N.
S ep 4:
In he inal s ep, we wan o show ha (Y,Z,U)sol es (ξ, ). Fo he app oxima -
ing sequence (Ynk,Znk,Unk)k≥1, he s ochas ic in eg als and he le hand side o
he BSDEs ξ, (nk)ob iously con e ge in L2 o he co esponding e ms o (ξ, ).
The e o e, his subsequence o T
(n) s,Yn
s,Zn
s,Un
sds∞
n=1con e ges o a an-
dom a iable V . We need o show ha V =T
(s,Y
s,Z
s,U
s)ds. To achie e his,
conside
δn:= ET
| (n) s,Yn
s,Zn
s,Un
s− s,Yn
s,Zn
s,Un
s|ds
+ET
| s,Yn
s,Zn
s,Un
s− (s,Ys,Z
s,U
s)|ds. (33)
We s a wi h he i s in eg and whe e, by he de ini ion o nand (29), and he
g ow h condi ion (A2),
| (n) s,Yn
s,Zn
s,Un
s− s,Yn
s,Zn
s,Un
s|
=| (n) s,Yn
s,Zn
s,Un
s− s,Yn
s,Zn
s,Un
s|χn
≤2F(s)χ
n(s) +K1(s)|Yn
s|χn(s) +K2(s)χn(s) |Zn
s|+Un
s
=: 2κ(1)
n(s) +κ(2)
n(s) +κ(3)
n(s).
The es ima es a e simila as in he p e ious s ep. Thanks o (31), we ha e
ET
κ(1)
nk(s)ds →0.Fo he nex e m, he Cauchy–Schwa z inequali y yields
ET
κ(2)
nk(s)ds ≤


T
0
χn(s) K1(s)ds


2
sup
lYlS2,
so ha by (31) he i s ac o con e ges o ze o along he subsequence (nk). The las
e m we es ima e using he Cauchy–Schwa z inequali y w. . . P⊗λ,
ET
κ(3)
nk(s)ds ≤ET
0
K2(s)2χn(s)ds
1
2
sup
l

Zl

L2(W)+

Ul

L2˜
N,
and again by (31), we ha e con e gence o ze o along he subsequence (nk).
We con inue showing he con e gence o he second e m in (33). We ex ac a
sub-subsequence o (nk)k≥1, which we call—sligh ly abusing he no a ion—again
(nk)k≥1such ha (Y nk,Znk,Unk), ega ded as a iple o measu able unc ions wi h
alues in R×R×L2(ν),con e ges o(Y,Z,U) o P⊗λ-a.a. (ω, s) . Then, o an
a bi a y K>0, we ha e
ET
 s, Ynk
s,Znk
s,Unk
s− (s,Ys,Z
s,U
s)ds
≤ET
 s, Ynk
s,Znk
s,Unk
s− (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>Kds.
P obabili y, Unce ain y and Quan i a i e Risk (2018) 3:9 Page 23 o 33
By domina ed con e gence and he con inui y o ,
ET
 s, Ynk
s,Znk
s,Unk
s− (s,Y
s,Z
s,U
s)χ|Ynk
s|≤K,|Znk
s|+Unk
s≤Kds →0,
since by (A2) we can bound he in eg and by
2F(s)+K1(s)(K +|Ys|)+K2(s)(2K+|Zs|+Us),
which is in eg able. We le
χK(nk,s):= χ|Ynk
s|>K+χ|Znk
s|+Unk
s>K.
Then, he emaining e ms o (34) a e bounded by
ET
0
(2F(s)+K1(s)|Ys|+K2(s)(|Zs|+Us))χK(nk,s)ds
+ET
0
K1(s)|Ynk
s|χK(nk,s)ds
+ET
0
K2(s) |Znk
s|+Unk
sχK(nk,s)ds
=: δ(1)
nk+δ(2)
nk+δ(3)
nk.
I we choose a Kla ge enough, hen δ(1)
nkcan be made a bi a ily small since
he amilies |Yn
s|,n≥0and |Zn
s|+Un
s,n≥0a e uni o mly in eg able wi h
espec o P⊗λ. The same holds o
δ(2)
nk2≤ET
0
K1(s)χK(nk,s)ds
2
sup
lYnl2
S2
≤


T
0
K1(s)ds


∞
ET
0
K1(s)χK(nk,s)dssup
lYnl2
S2,
and
δ(3)
nk2≤2ET
0
K2(s)2χK(nk,s)dssup
l
ET
0Znl
s
2+
Unl
s

2ds.
Hence, o δnde ined in (33), we ha e ha limk→∞ δnk=0,which implies
lim
k→∞
ET
(nk)s,Ynk
s,Znk
s,Unk
sds −T
(s,Y
s,Z
s,U
s)ds=0.
We in e ha o a sub-subsequence (nkl,l ≥0)we ge he a.s. con e gence
T
(nkl)s,Ynkl
s,Znkl
s,Unkl
sds →T
(s,Y
s,Z
s,U
s)ds.
Thus, o he o iginal sequence, a.s.
T
(nk)s,Ynk
s,Znk
s,Unk
sds →V =T
(s,Y
s,Z
s,U
s)ds,
and he e o e he iple (Y,Z,U)sa is ies he BSDE (ξ, ).
Page 24 o 33 Geiss and S einicke
5.2 P oo o Theo em 3.4
We s a wi h a p epa a o y lemma:
Lemma 5.1 I sa is ies (A1)–(A4), hen o all n≥0, ncons uc ed in
De ini ion 3.3 also sa is ies (A1)–(A4) (wi h di e en coe icien s).
P oo By de ini ion, (ω, ) → n( ,y,z,u) is p og essi ely measu able o all
(y,z,u), hus (A1) is sa is ied. The inequali ies in (A2) and (A3) a e a.s. sa is-
ied, wi h coe icien s EnF,EnK1,EnK2,Enβ. To ensu e ha hese coe icien s
ha e a Fn
 ∈[0,T ]-p og essi ely measu able e sion, one applies he p ocedu e om
De ini ion 3.3 o he inequali ies in (A2) and (A3) and no es ha an equa ion
analogous o (11) holds ue.
I emains o show a.s. con inui y o nin he (y,z,u)- a iables equi ed in (A3)
o a.e. . In (Ylinen (2017), P oposi ion 7.3), his was shown by he ac ha he
app oxima ion o he gene a o s appea ing he e can be done using spaces o con in-
uous unc ions. Howe e , since ou si ua ion in ol es L2(ν), a non-locally compac
space, we can no easily adap he p oo om Ylinen (2017) and he e o e we will
use di e en means.
Le D[0,T]be he space o c`
adl`
ag unc ions endowed by he Sko ohod me ic
(which makes his space a Polish space). The Bo el σ-algeb a B(D[0,T])is gene -
a ed by he coo dina e p ojec ions p :D[0,T]→R,x→ x(s) (see Theo em 12.5
o Billingsley (1968), o ins ance). On his σ-algeb a, le PXbe he image measu e
induced by he L´
e y p ocess X:→D[0,T],ω → X(ω).Wedeno ebyG he
comple ion wi h espec o PX.Fo ∈[0,T], he no a ion
x (s) := x( ∧s), o alls∈[0,T]
induces he na u al iden i ica ion
D[0, ]=x∈D[0,T]:x =x.
By his iden i ica ion, we de ine a il a ion on his space h ough
G =σ(B(D[0, ])∪NX[0,T]),0≤ ≤T,
whe e NX[0,T]deno es he null se s o B(D[0,T])wi h espec o he image mea-
su e PXo he L´
e y p ocess X. The same p ocedu e applied o he L´
e y p ocess Xn
yields a il a ion (Gn
) ∈[0,T ]de ined in he same way.
Acco ding o (S einicke (2016), Theo em 3.4), which is a gene aliza ion o Doob’s
ac o iza ion lemma o andom a iables depending on pa ame e s, he e is a G ⊗
B([0, ]×R2×L2(ν))-measu able unc ional
g :D[0, ]×[0, ]×R2×L2(ν) →R
and a Gn
⊗B([0, ]×R2×L2(ν))-measu able unc ional
g n:D[0, ]×[0, ]×R2×L2(ν) →R
such ha P-a.s.,
g (X(ω), ·)= (ω,·)and g n(Xn(ω), ·)= n(ω, ·). (35)
P obabili y, Unce ain y and Quan i a i e Risk (2018) 3:9 Page 31 o 33
Appendix
The Biha i–LaSalle inequali y. Fo he Biha i–LaSalle inequali y we e e o (Mao
(1997), pp. 45-46). He e, we o mula e a backwa d e sion o i which has been
appliedinYinandMao(2008). The p oo is analogous o ha in Mao (1997).
P oposi ion 5.2 Le c>0.Assume ha ρ:[0,∞[→ [0,∞[ is a con inuous and
non-dec easing unc ion such ha ρ(x) > 0 o all x>0.Le K be a non-nega i e,
in eg able Bo el unc ion on [0,T],and y a non-nega i e, bounded Bo el unc ion on
[0,T],such ha
y( ) ≤c+T
K(s)ρ(y(s))ds.
Then, i holds ha
y( ) ≤G−1G(c) +T
K(s)ds
o all ∈[0,T]such ha G(c) +T
K(s)ds ∈dom G−1.He e
G(x) := x
1
d
ρ( ),
and G−1is he in e se unc ion o G.
Especially, i ρ( ) = o ∈[0,∞[,i holds ha
y( ) ≤ceT
K(s)ds.(46)
The Janko – on Neumann heo em. I Xand Ya e se s and P⊆X×Y, hen
P∗⊆Pis called a uni o miza ion o Pi and only i P∗is he g aph o a unc ion
:p ojX(P ) →Y, i.e., P∗={(x, (x)) :x∈p ojX(P )}.Such a unc ion is
called a uni o mizing unc ion o P.Le 1
1(X) deno e he class o analy ic subse s
o X. The ollowing heo em can be ound, o example, in (Kech is (1994), Theo em
18.1).
Theo em 5.3 (Janko – on Neumann heo em) Assume ha X and Y a e s anda d
Bo el spaces and P⊆X×Yis an analy ic se . Then, P has a uni o mizing unc ion
ha is σ1
1(X)- measu able.
Acknowledgemen s The au ho s hank S e an Geiss and Juha Ylinen, Uni e si y o Jy ¨
askyl¨
a, o ui ul
discussions and aluable sugges ions.
Moe eo e , we a e since ly g a e ul o he anonymous e iewe s o hei help ul commen s and ques ions.
Ch is el Geiss would like o hank he E win Sch ¨
odinge Ins i u e, Vienna, o hospi ali y and suppo ,
whe e a pa o his wo k was w i en.
Funding
La ge pa s o his a icle we e w i en when Alexande S einicke was membe o he Ins i u e o Ma he-
ma ics and Scien i ic Compu ing, Uni e si y o G az, Aus ia, and suppo ed by he Aus ian Science Fund
(FWF): P ojec F5508-N26, which is pa o he Special Resea ch P og am “Quasi-Mon e Ca lo Me hods:
Theo y and Applica ions.”

Page 32 o 33 Geiss and S einicke
A ailabili y o da a and ma e ial
Da a sha ing is no applicable o his a icle as no da ase s we e gene a ed o analyzed du ing he cu en
s udy.
Au ho s’ con ibu ions
Bo h au ho s ead and app o ed he inal manusc ip .
E hics app o al and consen o pa icipa e
No applicable.
Consen o publica ion
No applicable.
Compe ing in e es s
The au ho s decla e ha hey ha e no compe ing in e es s.
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