Junike, Ge o; S ie , Hauke; Ch is iansen, Ma cus
A icle — Published Ve sion
P o i and loss decomposi ion in con inuous ime and
app oxima ions
Finance and S ochas ics
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Junike, Ge o; S ie , Hauke; Ch is iansen, Ma cus (2025) : P o i and loss
decomposi ion in con inuous ime and app oxima ions, Finance and S ochas ics, ISSN 1432-1122,
Sp inge , Be lin, Heidelbe g, Vol. 29, Iss. 4, pp. 1075-1107,
h ps://doi.o g/10.1007/s00780-025-00571-7
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Finance and S ochas ics (2025) 29:1075–1107
h ps://doi.o g/10.1007/s00780-025-00571-7
P ofi and loss decomposi ion in con inuous ime and
app oxima ions
Ge o Junike1·Hauke S ie 1·Ma cus Ch is iansen1
Recei ed: 4 July 2024 / Accep ed: 19 Decembe 2024 / Published online: 18 July 2025
© The Au ho (s) 2025
Abs ac
Financial ins i u ions and insu ance companies ha analyse he e olu ion and sou ces
o p o i s and losses o en look a isk ac o s only a disc e e epo ing da es, igno -
ing he de ailed pa hs. Con inuous- ime decomposi ions a oid his weakness and also
make decomposi ions consis en ac oss di e en epo ing g ids. We cons uc a la ge
class o con inuous- ime decomposi ions om a ea anged e sion o I ô’s o mula,
and uniquely iden i y a p e e ed decomposi ion om he axioms o exac ness, sym-
me y and no malisa ion. This unique decomposi ion u ns ou o be a s ochas ic limi
o ecu si e Shapley alues, bu i su e s om a cu se o dimensionali y as he num-
be o isk ac o s inc eases. We de elop an app oxima ion ha b eaks his cu se when
he isk ac o s almos su ely ha e no simul aneous jumps.
Keywo ds P o i and loss a ibu ion ·Sequen ial decomposi ions ·Change
analysis ·Risk decomposi ion ·I ô’s o mula
Ma hema ics Subjec Classifica ion 60H05 ·60H30 ·91G10 ·91G60 ·91G30 ·
91G40
JEL Classifica ion C02 ·C30 ·C63 ·G10 ·G12
1In oduc ion
P o i and loss (P&L) a ibu ion, also known as change analysis, has a long his o y
in isk managemen . P&L a ibu ion is he p ocess o analysing he change be ween
✉G. Junike
[email p o ec ed]
H. S ie
[email p o ec ed]
M. Ch is iansen
[email p o ec ed]
1Ca l on Ossie zky Uni e si ä , Ins i u ü Ma hema ik, 26129 Oldenbu g, Ge many
1076 G. Junike e al.
wo alua ion da es and explaining he e olu ion o he P&L by he mo emen o
he sou ces ( isk ac o s) be ween he wo da es; see Candland and Lo z [4]. In o he
wo ds, he change in he P&L o e ime is decomposed in e ms o he di e en isk
ac o s o explain how each ac o con ibu es o he P&L. In he li e a u e, he e
a e many ways o ob ain a P&L a ibu ion. Fo example, conside a po olio in EUR
consis ing o a long posi ion in he S&P 500, Y o sho . The P&L o such a po olio
is d i en by wo isk ac o s, namely Yand he USD/EUR exchange a e, X o sho .
To decompose he P&L o e one yea , we look o wo andom a iables DXand DY
such ha
X1Y1−X0Y0=DX+DY.
The numbe s DXand DYa e in e p e ed as he con ibu ions o Xand Y o he P&L.
In he li e a u e, we can ind many desi able p ope ies ha a decomposi ion should
possess; see Shubik [28], F iedman and Moulin [12] and Sho ocks [26] among many
o he s. The au ho s a gue ha a decomposi ion should be symme ic, i.e., he con i-
bu ions o he isk ac o s should be independen o he way in which he isk ac o s
a e labelled o o de ed. These au ho s also equi e ha he sum o all con ibu ions
equals he P&L; such decomposi ions a e called exac . Fu he , Ch is iansen [6]a -
gues ha a decomposi ion should be no malised, i.e., i a isk ac o emains cons an ,
i s con ibu ion o he P&L should be ze o. I is also desi able o a decomposi ion
o conside he ull pa h o each isk ac o , i.e., o use all a ailable in o ma ion; see
Mai [18] and Flaig and Junike [9].
A common me hod o c ea ing decomposi ions is o sequen ially upda e he isk
ac o s one by one while “ eezing” all o he isk ac o s. This idea da es back a
leas o Oaxaca [20] and Blinde [3], who de eloped a sequen ial upda ing (SU)
decomposi ion echnique in a one-pe iod se ing. The SU decomposi ion is gi en by
DX=X1Y0−X0Y0,D
Y=X1Y1−X1Y0,
when we upda e he isk ac o X i s . Al e na i ely, one may upda e Y i s o ob ain
DX=X1Y1−X0Y1,D
Y=X0Y1−X0Y0.
Each SU decomposi ion is exac , bu i he e a e d isk ac o s, he e a e d!di e en
upda ing o de s and he e o e d!di e en SU decomposi ions. Candland and Lo z [4]
call he one-pe iod SU decomposi ion wa e all and apply i o P&L a ibu ion. See
Fo in e al. [10] o an o e iew on how he SU decomposi ion is used in a ious
ields o economics.
The SU decomposi ion can also be de ined in a mul i-pe iod se ing by di iding
he ime ho izon in o subin e als and applying he SU decomposi ion ecu si ely
on each subin e al. Je ses and Ch is iansen [16] and Ch is iansen [6] analysed he
limi o he SU decomposi ion when he mesh size o he ime g id con e ges o ze o.
In he limi , he decomposi ion akes he whole pa h in o accoun , and he limi ing
SU decomposi ion is called he in ini esimal sequen ial upda ing (ISU) decomposi-
ion. The ISU decomposi ion is independen o any ime g id, which is help ul “ o
p e en inconsis encies when using con lic ing subin e als o di e en pu poses”;
see Flaig and Junike [9, Sec . 1].
P&L decomposi ion in con inuous ime and app oxima ions 1077
The a e aged sequen ial upda ing (ASU) decomposi ion, also known as he Shap-
ley alue, is simply he a i hme ic a e age o he d!possible SU decomposi ions. I
has many desi able p ope ies; in pa icula , i is exac and symme ic. Shapley [27]
in oduces he ASU decomposi ion o coope a i e games. Shubik [28] de ines he
ASU decomposi ion o cos unc ions. Sp umon [29] and F iedman and Moulin [12]
p o ide an axioma isa ion o he ASU decomposi ion o cos unc ions. Je ses
and Ch is iansen [16] de ine he in ini esimal a e aged sequen ial upda ing (IASU)
decomposi ion as he a e age o he d!possible ISU decomposi ions.
We now summa ise ou main con ibu ions. In his pape , we s a di ec ly in a
ime-con inuous se ing. I he po olio is a C2- unc ion o he isk ac o s and he
la e ha e con inuous pa hs, I ô’s o mula p o ides a na u al addi i e decomposi ion
o he po olio. Ou main con ibu ions a e as ollows. In o de o ea isk ac o s
wi h jumps, we p o ide a ea anged e sion o I ô’s o mula and use i o de ine
a la ge class o easonable decomposi ions, which we call I ô decomposi ions and
which include all d!ISU and he IASU decomposi ions as special cases. We p o e
ha he e is a unique I ô decomposi ion (up o indis inguishabili y) ha sa is ies he
h ee axioms o exac ness, symme y and no malisa ion. We show ha i is indis in-
guishable om he IASU decomposi ion. We u he show ha he IASU decompo-
si ion can be in e p e ed as he limi ing case o he ASU decomposi ion. Compa ed
o Je ses and Ch is iansen [16], who assume ha he co a ia ions be ween he isk
ac o s a e ze o, we use much weake assump ions o p o e he con e gence o he
SU/ASU decomposi ions o he ISU/IASU decomposi ions.
In summa y, we p opose o use he IASU decomposi ion o ob ain a P&L a ibu-
ion because i conside s he whole pa hs o he isk ac o s and sa is ies he axioms
o exac ness, symme y and no malisa ion. Howe e , in p ac ical applica ions, he
IASU decomposi ion has wo d awbacks: a) simila ly o he ASU decomposi ion, i
su e s om he cu se o dimensionali y; b) he IASU decomposi ion is de ined by
s ochas ic in eg als, which somehow mus be app oxima ed in p ac ice. Nai ely ap-
p oxima ing hese in eg als can lead o decomposi ions ha a e no longe exac . As
ano he impo an con ibu ion o his pape , we show ha he IASU decomposi ion
does no su e om he cu se o dimensionali y i he isk ac o s do no ha e simul-
aneous jumps. In his case, he IASU decomposi ion is indis inguishable om he
a e age o wo (sui ably selec ed) ISU decomposi ions. To a oid poin b), we sugges
app oxima ing ISU/IASU by SU/ASU.
Up o now, mos p ac i ione s ha e applied an a bi a y SU decomposi ion in a
one-pe iod se ing o ob ain an annual P&L a ibu ion; see Candland and Lo z [4].
Wo king wi h eal ma ke da a, Flaig and Junike [9] empi ically show ha he SU de-
composi ion depends signi ican ly on he o de o labelling o he isk ac o s, and ha
some SU decomposi ions may e en igno e ele an isk ac o s, which may “lead o
w ong ading and hedging decisions”; see Flaig and Junike [9, Sec . 1].
Ou heo e ical analysis sugges s using he a e age o only wo SU decomposi ions
wi h a su icien ly ine ime g id o ob ain a P&L a ibu ion, since such a decomposi-
ion is a bi a ily close o he IASU decomposi ion when he isk ac o s do no ha e
simul aneous jumps. To ob ain hese wo SU decomposi ions, de ine one SU decom-
posi ion in any o de , e.g. alphabe ically ascending, and ano he SU decomposi ion
by he e e se o de , e.g. alphabe ically descending; see Theo em 3.10 o de ails.
1078 G. Junike e al.
Thus ou analysis is highly ele an o p ac i ione s: we ecommend compu ing wo
SU decomposi ions ins ead o one and using a ine g id han jus annual da a o
ob ain a decomposi ion ha is much close o he IASU decomposi ion han a single
SU decomposi ion. While he choice o he decomposi ion ( he a e age o wo SU de-
composi ions) is heo e ically jus i ied, we ha e only nume ical expe imen s a ailable
o es ima e he ime g id, and we ecommend using mon hly o weekly da a.
Is he e any o he way o b eak he cu se o dimensionali y? Ch is iansen [6]
p o es ha he ISU decomposi ion is symme ic i i is s able wi h espec o small
pe u ba ions in he empi ical obse a ion o he isk ac o s. In Appendix A.3,we
show ha he ISU decomposi ion o a simple p oduc o wo co ela ed B ownian
mo ions is no s able. This shows ha s abili y is a a he s ong assump ion.
The e a e o he decomposi ion p inciples as well. The e is he so-called one-a -
a- ime (OAT) decomposi ion, which is also known as bump and ese ; see Cand-
land and Lo z [4]. The OAT decomposi ion is closely ela ed o he SU decompo-
si ion. I is symme ic, bu in gene al no exac . F ei [11] analyses he limi o he
OAT decomposi ion when he mesh size o he ime g id con e ges o ze o.
The e a e also comple ely di e en app oaches. Fische [8] uses a condi ional ex-
pec a ions app oach. Rosen and Saunde s [24] use he Hoe ding me hod o a de-
composi ion o c edi isk po olios. F ei [11] and Bielecki e al. [1] use he Eule
p inciple o isk a ibu ion. Ramlau-Hansen [23] and No be g [19] decompose su -
plus in li e insu ance by heu is ic in eg al ep esen a ions, whe e he in eg a o s a e
in e p e ed as he d i ing o ces o change and de e mine he a ibu ion. A simila
idea is used in Schilling e al. [25] based on he ma ingale ep esen a ion heo em.
This a icle is s uc u ed as ollows. In Sec . 2, we es ablish some no a ion. In
Sec . 3, we de elop a ea anged e sion o I ô’s o mula and in oduce he am-
ily o I ô decomposi ions. We show ha he IASU decomposi ion is he only exac
and symme ic I ô decomposi ion, and we b eak he cu se o dimensionali y o he
IASU decomposi ion in Theo em 3.10. In Sec . 4, we p o e ha he IASU decompo-
si ion can be app oxima ed by he ASU decomposi ion. In Sec . 5, we p o ide some
nume ical applica ions. Sec ion 6concludes.
2No a ion
Le (Ω, ℱ,𝔽=(ℱ ) ≥0,P)be a il e ed p obabili y space sa is ying he usual con-
di ions, i.e., ℱ0con ains all nullse s and 𝔽is igh -con inuous. Le 𝒳be he se o
all eal- alued 𝔽-semima ingales. A so-called isk basis o in o ma ion basis is a
d-dimensional semima ingale X∈𝒳d, and i s dcomponen s a e called isk ac o s
o sou ces o isk. We deno e he s opped semima ingale by Xσ=(X1,σ ,...,Xd,σ)
o a s opping ime σ. Equali y o andom a iables is unde s ood in he almos su e
sense, and equali y o s ochas ic p ocesses is unde s ood up o indis inguishabili y.
Le C2be he se o wice con inuously di e en iable unc ions om ℝd o ℝ.Fo
∈C2and i, j =1,...,d, we w i e iand ij o he pa ial de i a i es ∂i and
∂i∂j .Byx∧y, we deno e he minimum o wo eal numbe s xand y. We call a
map F:𝒳d→𝒳non-an icipa i e i o any s opping ime σ, i holds ha o all
P&L decomposi ion in con inuous ime and app oxima ions 1079
X∈𝒳d,
F (Xσ)=F ∧σ(X), ≥0.(2.1)
Such a non-an icipa i e mapping depends only on he in o ma ion up o ime
, i.e., on X .Byℳ, we deno e some subspace o all non-an icipa i e map-
pings. By ℳ(C2), we deno e he space o unc ionals F:𝒳d→𝒳such ha
F(X) =( (X )) ≥0 o X∈𝒳dand some ∈C2, which a e clea ly non-
an icipa i e. By σd, we deno e he se o all d!pe mu a ions o {1,...,d}.Le id∈σd
be he iden i y. In a sligh abuse o no a ion, we de ine o π∈σd,
π(x) =(xπ(1),...,xπ(d)), x ∈ℝd,
π(X) =(Xπ(1),...,Xπ(d)), X ∈𝒳d.
Fo wo one-dimensional semima ingales Zand Yand a càglàd p ocess H, we deno e
by ∫︁
0HsdZs:= ∫︁(0, ]HsdZs he s ochas ic in eg al. In pa icula , ∫︁0
0HsdZs=0
by con en ion. We u he se Z0−=0,
Z −=lim
ε↘0Z −ε, >0,
ΔZ =Z −Z −, ≥0,
[Z,Y]=ZY −Z0Y0−∫︂·
0Zu−dYu−∫︂·
0Yu−dZu,
[Z,Y]c=[Z,Y]− ∑︂
0<s≤·
ΔZsΔYs.
We w i e p
→ o he con e gence in p obabili y o a sequence o andom a iables.
Fo A⊆{1,...,d}, we de ine he p ojec ion
pA:ℝd→ℝd,x↦→ (︁x11A(1), . . . , xd1A(d))︁,
whe e he unc ion 1A(h) is 1 i h∈Aand 0 o he wise.
3 Family o I ô decomposi ions
Simila ly o Sho ocks [26], Ch is iansen [6], we de ine a decomposi ion as ollows.
Defini ion 3.1 Amap
δ:ℳ×𝒳d→𝒳d,(F,X)↦→ (︁δ1(F,X),...,δd(F, X))︁
is called a decomposi ion.
1080 G. Junike e al.
We in e p e δi
(F, X) as he con ibu iono Xi o he p o i and loss F (X)−F0(X)
in [0, ]. We ecall he ollowing h ee axioms om he li e a u e:
i) A decomposi ion is called exac i o all F∈ℳand X∈𝒳d, i holds ha
F(X)−F0(X) =δ1(F, X) +···+δd(F, X).
ii) A decomposi ion is called symme ic i o all π∈σd,F∈ℳand X∈𝒳d,i
holds ha
F(X) =F(︁π(X))︁=⇒ δi(F, X) =δπ−1(i)(︁F,π(X))︁.
iii) A decomposi ion is called no malised i o all 0 ≤ <s<∞,i=1,...,d,
F∈ℳand X∈𝒳d, i holds ha
Xiis indis inguishable om a cons an p ocess on ( , s]
=⇒ δi(F, X) is indis inguishable om a cons an p ocess on ( , s].
Axiom i) ensu es ha a decomposi ion is able o ully explain he P&L; see
Sho ocks [26] and Ch is iansen [6]. Axiom ii) conside s symme ic maps Fand
s a es ha i Fdoes no depend on he o de o labelling o he isk ac o s, hen
nei he does he decomposi ion. The symme y axiom is mo i a ed by he ac ha
δi(F, X) ep esen s he con ibu ion o Xiand ha he e m δπ−1(i)(F, π(X)) also
desc ibes he con ibu ion o
(︁π(X))︁π−1(i) =(︁Xπ(1),...,Xπ(d))︁π−1(i) =Xi.
The symme y axiom has al eady been men ioned in simila o m in F iedman and
Moulin [12] and Sho ocks [26]. Finally, o axiom iii), i he isk ac o Xi emains
cons an du ing ( , s], i does no con ibu e o Fs(X)−F (X), and so he con ibu ion
o Xiin ( , s]should also be ze o. This is exac ly e lec ed by he no malisa ion
axiom, aken om Ch is iansen [6].
Nex , we indica e how I ô’s o mula helps o de ine decomposi ion p inciples. Le
:ℝd→ℝbe in C2.Fo i, j =1,...,d,le
Ii:= ∫︂·
0 i(Xs−)dXi
s,I
ij := ∫︂·
0 ij (Xs−)d[Xi,Xj]c
s,(3.1)
S:= ∑︂
0<s≤·(︃ (X
s)− (X
s−)−
d
∑︂
i=1
i(Xs−)ΔXi
s)︃.(3.2)
I ô’s o mula s a es ha o ≥0, we ha e o any semima ingale X∈𝒳d ha
(X
)− (X
0)=
d
∑︂
i=1
Ii
+1
2
d
∑︂
i=1
Iii
+1
2
d
∑︂
i,j=1
i=j
Iij
+S .(3.3)
P&L decomposi ion in con inuous ime and app oxima ions 1081
I we assume ha Xhas con inuous pa hs wi hou in e ac ion e ec s, i.e., Iij =0,
i=j, and S=0, hen (3.3) p o ides a na u al way o addi i ely decompose he P&L
(X
)− (X
0). Indeed, by he no malisa ion axiom, Iiand Iii should be assigned
o δi, which is in e p e ed as he con ibu ion o Xi. To see his, assume ha some δj
depends on Iio Iii o i= j. Assume ha Xjis cons an e e ywhe e. Acco ding o
he no malisa ion axiom, we should hen ha e δj=0. So δjmus no depend on Ii
o on Iii.
Howe e , how o handle he in e ac ion e ec s Iij ,i= j, and he jump pa S
is no so ob ious. The e o e, we p o ide in P oposi ion 3.3 a ea anged e sion o
I ô’s o mula. Based on ha esul , we de ine he la ge amily o I ô decomposi ions
in De ini ion 3.4 and show in Sec . 3 ha his amily con ains many well-known
decomposi ion p inciples as special cases. Wi hin he amily o I ô decomposi ions,
we iden i y a single decomposi ion ha sa is ies he axioms o exac ness, symme y
and no malisa ion. Fo A⊆{1,...,d},i∈{1,...,d}and s>0, de ine
Yi,A
s:= (︁Xs−+pA(ΔXs))︁− (︁Xs−+pA {i}(ΔXs))︁− i(Xs−)ΔXi
s
and
Si,A(X) := ∑︂
0<s≤·
Yi,A
s.
Fo π∈σd, de ine
Si,π (X) := Si,{j:π(j)≤π(i)}(X). (3.4)
To ob ain Si,π (X), all ime poin s swhe e Xijumps a e conside ed. All isk ac o s
excep Xia e ixed a so s−, depending on he choice o π, and only Xi a ies
be ween s−and s.
Lemma 3.2 Fix i∈{1,...,d},X∈𝒳dand A⊆{1,...,d}.I i∈A, hen Si,A(X)
is a semima ingale wi h a.s.pa hs o ini e a ia ion on compac s.
P oo Fix X∈𝒳d.Le Nbe a nullse such ha u↦→ |Xi
u(ω)|,i=1,...,d,is
càdlàg o ω∈Ω Nand
d
∑︂
h,j=1∑︂
0<s≤ |ΔXh
s(ω)ΔXj
s(ω)|<∞,ω∈Ω N, ≥0.(3.5)
Such an Nexis s as Xis a semima ingale. Le ω∈Ω Nand ≥0. Le Mω⊆ℝd
be he closu e o he se {Xu(ω) :u∈[0, ]}, which is compac . The unc ion and
i s de i a i es a e con inuous and each a maximum and minimum on he con ex hull
o Mω, which is compac by Ca a héodo y’s heo em; see G ünbaum [14, Sec . 2.3].
Hence and i s de i a i es a e bounded on he con ex hull o Mω.Fo s∈(0, ],we
de elop a ound Xs−(ω) using a Taylo expansion. We ha e ha
(︂Xs−(ω) +pA(︁ΔXs(ω))︁)︂= (︁Xs−(ω))︁+∑︂
h∈A
h(︁Xs−(ω))︁ΔXh
s(ω) +R(ω),
1082 G. Junike e al.
whe e R(ω) is he emainde e m o he Taylo expansion, i.e., o some
θ(ω) ∈[0,1], i holds ha
R(ω) =1
2∑︂
h,j∈A
hj (︂Xs−(ω) +θ(ω)pA(︁ΔXs(ω))︁)︂ΔXh
s(ω)ΔXj
s(ω).
The e m (X
s−(ω)+pA {i}(ΔXs(ω))) can be ea ed simila ly. Since i∈A, i holds
o some C(ω) > 0, which does no depend on so θ(ω), ha
Yi,A
s≤C(ω) ∑︂
h,j∈A|ΔXh
s(ω)ΔXj
s(ω)|.
I ollows by (3.5) ha
∑︂
0<s≤ |Yi,A
s(ω)|<∞,ω∈Ω N. (3.6)
Since was a bi a y, (3.6) implies ha u↦→ Si,A
u(X)(ω),ω∈Ω N, is càdlàg and
o ini e a ia ion on compac s. The e o e Si,A(X) is a semima ingale. □
P oposi ion 3.3 Le π∈σd, ∈C2and X∈𝒳d.Fo all ≥0, i holds ha
(X
)− (X
0)=
d
∑︂
i=1(︃Ii
+1
2Iii
+1
2
d
∑︂
j=1
j=i
Iij
+Si,π
)︃,
whe e Iiand Iij a e de ined in (3.1)and Si,π is de ined in (3.4).
P oo Since he se ies elescopes, we ha e ha
(X
s)− (X
s−)
=
d
∑︂
i=1
(︁Xs−+p{j:π(j)≤π(i)}(ΔXs))︁− (︁Xs−+p{j:π(j)<π(i)}(ΔXs))︁.
By (3.6), i holds o any ≥0 ha
d
∑︂
i=1
Si,π
(X) =∑︂
0<s≤
d
∑︂
i=1
Yi,{j:π(j)≤π(i)}
s=S ,(3.7)
whe e Sis de ined in (3.2). The claim hen ollows by I ô’s o mula. □
Defini ion 3.4 Le λij ∈[0,1] o i, j =1,...,d.Le μπ∈[0,1] o π∈σd.The
decomposi ion
δI ô :ℳ(C2)×𝒳d→𝒳d,(F,X)↦→ (︁δI ô,1(F,X),...,δI ô,d (F, X))︁,
P&L decomposi ion in con inuous ime and app oxima ions 1089
semima ingales and hence a semima ingale. By P o e [21, Sec . II.2] and since
σn→∞a.s. o n→∞, he p ocess δSU,i,π,γ (F, X) is a semima ingale and he
decomposi ion δSU,π,γ is he e o e well de ined. The ac ha σn→∞a.s. implies
ha he sum in (4.1) e alua ed a is a.s. ini e.
We show exac ness. Le x∈ℝd. Since
p{j:π(j)≤π(π−1(d))}(x) =xand p{j:π(j)<π(π−1(1))}(x) =0,
we ha e o any ∈[0,∞)and n∈ℕby (4.3) ha
d
∑︂
i=1
δSU,i,π,γ
∧σn(F, X)
=
d
∑︂
i=1
i=π−1(d)
n−1
∑︂
ℓ=0
F ∧σn(︁Xσℓ+p{j:π(j)≤π(i)}(Xσℓ+1−Xσℓ))︁+
n−1
∑︂
ℓ=0
F ∧σn(Xσℓ+1)
−
d
∑︂
i=1
i=π−1(1)
n−1
∑︂
ℓ=0
F ∧σn(︁Xσℓ+p{j:π(j)<π(i)}(Xσℓ+1−Xσℓ))︁−
n−1
∑︂
ℓ=0
F ∧σn(Xσℓ).
Fo each i∈{1,...,d} {π−1(d)}, he e is exac ly one k∈{1,...,d} {π−1(1)}
such ha
p{j:π(j)≤π(i)}(x) =p{j:π(j)<π(k)}(x),
since π(k) =π(i) +1 i and only i k=π−1(π(i) +1). Thus we ge
d
∑︂
i=1
δSU,i,π,γ
∧σn(F, X) =
n−1
∑︂
ℓ=0
F ∧σn(Xσℓ+1)−
n−1
∑︂
ℓ=0
F ∧σn(Xσℓ)
=F ∧σn(Xσn)−F ∧σn(Xσ0)
=F ∧σn(X) −F0(X).
Since and nwe e a bi a y and σn→∞a.s., he decomposi ion δSU,π,γ is exac .
To see he las poin , no e ha wo p ocesses wi h càdlàg pa hs a e indis inguishable
i hey a e modi ica ions. □
Example 4.4 Assume d=2. The SU decomposi ion wi h espec o γde ines
d!=2 decomposi ions, namely δSU,id,γ (F, X) and δSU,ϱ,γ (F, X) wi h ϱ(1)=2
and ϱ(2)=1, by
δSU,1,id,γ (F, X) =∞
∑︂
ℓ=0(︁F(X1,σℓ+1,X2,σℓ)−F(X1,σℓ,X2,σℓ))︁,
δSU,2,id,γ (F, X) =∞
∑︂
ℓ=0(︁F(X1,σℓ+1,X2,σℓ+1)−F(X1,σℓ+1,X2,σℓ))︁
1090 G. Junike e al.
and
δSU,1,ϱ,γ (F, X) =∞
∑︂
ℓ=0(︁F(X1,σℓ+1,X2,σℓ+1)−F(X1,σℓ,X2,σℓ+1))︁,
δSU,2,ϱ,γ (F, X) =∞
∑︂
ℓ=0(︁F(X1,σℓ,X2,σℓ+1)−F(X1,σℓ,X2,σℓ))︁.
Defini ion 4.5 Le γ={0=σ0<σ
1<···}be an unbounded andom pa i ion.
The ASU (a e aged sequen ial upda ing) decomposi ion δASU,γ :ℳ×𝒳d→𝒳dis
de ined by
δASU,i,γ (F, X) =1
d!∑︂
π∈σd
δSU,i,π,γ (F, X), i =1,...,d.
Rema k 4.6 As in Sho ocks [26], we obse e ha
δASU,i,γ (F, X) =1
d!∑︂
π∈σd
δSU,i,π,γ (F, X) =∑︂
A⊆{1,...,d}
i∈A
δSU,i,A,γ (F, X)ξi,A
o ξi,A de ined in (3.8) and
δSU,i,A,γ (F, X)
:= ∞
∑︂
ℓ=0(︂F(︁Xσℓ+pA(Xσℓ+1−Xσℓ))︁−F(︁Xσℓ+pA {i}(Xσℓ+1−Xσℓ))︁)︂.
The eby, he compu a ional cos o ob ain δASU,i,γ can be educed om 𝒪(d!)
o 𝒪(2d−1).
Theo em 4.7 Fix π∈σdand le (γn)n∈ℕbe a sequence o unbounded andom pa -
i ions ending o he iden i y.Le F∈ℳ(C2),X∈𝒳d, ≥0and i∈{1,...,d}.
Then i holds o n→∞ ha
δSU,i,π,γn
(F, X) p
−→ δISU,i,π
(F, X),
δASU,i,γn
(F, X) p
−→ δIASU,i
(F, X).
P oo See Appendix A.2.□
The nex example shows ha he assump ion F∈ℳ(C2)in Theo em 4.7 is
impo an o ensu e con e gence.
Example 4.8 Le Zbe a s ochas ic p ocess wi h independen inc emen s and Z0=0.
Suppose he jumps o Zonly occu a ixed imes J={2−ℓ−1:ℓ∈ℕ}, and o
P&L decomposi ion in con inuous ime and app oxima ions 1091
each ℓ∈ℕ, he p ocess jumps by ±ℓ−1wi h equal p obabili y. The p ocess Zs ays
cons an be ween jumps. Then Zis a semima ingale; see ˇ
Ce ný and Ru [5]. Le
(x1,x2)=|x1−x2|
so ha /∈C2.Le (γn={0=σn
0<σ
n
1<···})n∈ℕbe a de e minis ic se-
quence o unbounded pa i ions ending o he iden i y such ha γncon ains he
i s nsmalles elemen s o J, bu he in e sec ion wi h (2−n−1,2]is emp y. Assume
ha X=(Z, Z). Then o ≥2, i ollows ha
∞
∑︂
ℓ=0(︁ (X1,σn
ℓ+1
,X2,σn
ℓ
)− (X1,σn
ℓ
,X2,σn
ℓ
))︁=
n
∑︂
ℓ=1
ℓ−1,
which is di e gen o n→∞; so he SU decomposi ion does no con e ge o he
map F (X) := (X
), ≥0.
How can he IASU decomposi ion be compu ed e icien ly in p ac ice? I we
nai ely app oxima e he in eg als in De ini ion 3.6 nume ically, we may lose exac -
ness o he decomposi ion, which is undesi able in many applica ions. Theo em 4.7
sugges s using he ASU decomposi ion as an app oxima ion o he IASU decom-
posi ion. Howe e , his becomes compu a ionally in easible o mode a ely la ge d
since he compu a ional cos o ob ain δASU,i,γ scales like 𝒪(2d−1). The nex esul
p o ides an elegan solu ion when he e a e no simul aneous jumps.
Defini ion4.9 Le γ={0=σ0<σ
1<···}be an unbounded andom pa i ion. The
2SU (a e age o wo sequen ial upda ing) decomposi ion δ2SU,π,γ :ℳ×𝒳d→𝒳d
wi h upda ing o de π∈σdis de ined by
δ2SU,i,π,γ (F, X) =1
2(︁δSU,i,π,γ (F, X) +δSU,i,π′,γ (F, X))︁,i=1,...,d,
whe e π′=d+1−π.
Co olla y 4.10 Fix π∈σdand le (γn)n∈ℕbe a sequence o unbounded andom
pa i ions ending o he iden i y.Le F∈ℳ(C2),X∈𝒳d,i∈{1,...,d}and
≥0.
i) I ΔXhΔXj=0 o all h, j ∈{1,...,d}wi h h= j, hen
δ2SU,i,π,γn
(F, X) p
−→ δIASU,i
(F, X), n →∞.
ii) I [Xh,Xj]=0 o all h, j ∈{1,...,d}wi h h= j, hen
δSU,i,π,γn
(F, X) p
−→ δIASU,i
(F, X), n →∞.
P oo I ΔXhΔXj=0, h=j, Theo em 3.10 implies ha
δIASU,i(F, X) =1
2(︁δISU,i,π (F, X) +δISU,i,π′(F, X))︁,
1092 G. Junike e al.
Fig. 1 O e iew o disc e e app oxima ions o he IASU decomposi ion
which is he limi o δ2SU,i,π,γn(F, X) by Theo em 4.7.I [Xh,Xj]=0, h= j,
apply Co olla y 3.12 and Theo em 4.7.□
In pa icula , he 2SU decomposi ion wi h a bi a y upda ing o de πis exac and
app oxima es he IASU decomposi ion when he isk ac o s do no ha e simul a-
neous jumps. In his case, he compu a ionally expensi e a e aging o ob ain he
ASU decomposi ion can be omi ed and he compu a ional complexi y o app oxi-
ma e δIASU,i dec eases om 𝒪(2d−1) o 𝒪(1). Theo em 4.7 and Co olla y 4.10 a e
also illus a ed in Fig. 1.
Finally, we de ine he OAT decomposi ion. To ob ain he con ibu ion o Xi,all
isk ac o s a e ixed a he o igin and only Xiis allowed o change om he beginning
o a subin e al o he end o ha subin e al.
Defini ion 4.11 Le γ={0=σ0<σ
1<···}be an unbounded andom pa i ion.
The OAT (one-a -a- ime) decomposi ion δOAT,γ :ℳ×𝒳d→𝒳dis de ined by
δOAT,i,γ (F, X)
=∞
∑︂
ℓ=0(︁F(X1,σℓ,...,Xi−1,σℓ,Xi,σℓ+1,Xi+1,σℓ,...,Xd,σℓ)−F(Xσℓ))︁.
Rema k 4.12 The OAT decomposi ion is symme ic, bu in gene al no exac . Le
(γn)n∈ℕbe a sequence o unbounded andom pa i ions ending o he iden i y. Fo
each i∈{1,...,d}, choose a pe mu a ion πi∈σdsuch ha πi(i) =1. Then
δOAT,i,γnis indis inguishable om δSU,i,πi,γn.I F∈ℳ(C2), Theo em 4.7 gi es
o ≥0 ha
δOAT,i,γn
(F, X) p
−→ δISU,i,πi
(F, X), i =1,...,d,
o n→∞. Thus by Co olla y 3.12, he h ee decomposi ions p inciples OAT, SU
(wi h a bi a y o de π∈σd) and ASU a e asymp o ically indis inguishable i he e
a e no in e ac ion e ec s.
5 Applica ions
In es men po olios o inancial ins i u ions o insu ance companies may include in-
s umen s such as s ocks, plain anilla o callable bonds, con e ible bonds, in la ion-
P&L decomposi ion in con inuous ime and app oxima ions 1093
linked bonds, con ingen con e ible bonds (CoCos), baske op ions, o eign ex-
change op ions and s uc u ed p oduc s. These ins umen s o en depend on mul i-
ple isk ac o s such as di e en o eign exchange a es, in e es a es o di e en
ma u i ies, c edi sp eads, in la ion a e, some igge ac i a ions o CoCos, mul iple
equi ies and ime decay. Candland and Lo z [4] also conside ed de aul s and a ing
changes as isk ac o s.
In o de o ob ain a P&L a ibu ion o such ins umen s, we p opose he IASU de-
composi ion because i is exac , symme ic and no malised, and i akes in o accoun
he whole pa hs o he isk ac o s, i.e., uses all a ailable in o ma ion. The las poin
also a oids inconsis encies when epo ing a P&L a ibu ion o di e en ime g ids,
e.g. on an annual, qua e ly, mon hly and weekly basis. The IASU decomposi ion
in ol es a s ochas ic in eg al. To app oxima e he IASU decomposi ion, we p opose
he ASU o 2SU decomposi ion wi h a su icien ly ine ime g id, as such an ap-
p oxima ion is always an exac decomposi ion. The use o he 2SU decomposi ion is
heo e ically jus i ied when he isk ac o s do no ha e simul aneous jumps.
In Sec . 5.1, we p o ide an exempla y decomposi ion o a plain anilla call op ion
wi h s ochas ic in e es a es on a o eign s ock. A change in he P&L o his op ion
can be explained by mo emen s in he s ock, he yield cu e, he o eign exchange
a e and ime decay. Thus he e a e d=4 isk ac o s. We analyse he unexplained
P&L o he OAT decomposi ion, he ange o he SU and 2SU decomposi ions o e
all possible upda ing o de s π∈σd o di e en ime g ids, and he con e gence o
he ASU decomposi ion o he IASU decomposi ion.
Compu ing he ASU decomposi ion o app oxima e he IASU decomposi ion be-
comes in easible when he numbe o isk ac o s dis mode a ely la ge; o example,
a plain anilla bond paying coupons may depend on dyield cu es. A baske op ion
may depend on ds ocks. In p ac ice, d=30 is a common case o baske op ions;
see G zelak e al. [15]. In Sec . 5.2, we decompose a digi al cash-o -no hing baske
pu op ion. We illus a e ha i is impossible o ob ain he ASU decomposi ion in
easonable ime when d=30, and we show how he 2SU decomposi ion is able o
b eak he cu se o dimensionali y.
5.1 Decomposing a call op ion wi h s ochas ic in e es a es
In his sec ion, we alloca e he P&L o he p ice o a plain anilla Eu opean call op-
ion wi h s ike Kand ma u i y T=10 wi h s ochas ic in e es a es and o eign
exchange exposu e. The s ock p ice Sis gi en by a Black–Scholes model wi h con-
s an ola ili y σS>0 and wi h s ochas ic in e es a es . The dynamics unde he
isk-neu al measu e a e gi en by
dS = S d +σSS dBS
,
d =κ(η − )d +σ dB
wi h cons an ola ili y σ >0, long- e m mean η∈ℝand speed κ>0 o mean-
e e sion. Unde he physical measu e, he s ock has d i μS∈ℝ, and he o eign
exchange a e Yis assumed o ollow a geome ic B ownian mo ion wi h d i μY∈ℝ
1094 G. Junike e al.
and ola ili y σY>0 d i en by he B ownian mo ion BY. The B ownian mo ions a e
assumed o ha e co ela ions
d⟨BS,B ⟩ =ρS d , d⟨BS,BY⟩ =ρSY d , d⟨BY,B ⟩ =ρY d .
The ime o ma u i y is deno ed by τ( ) =T− . The p ice pcall( ) a ime o he
plain anilla call op ion is gi en by a C2- unc ion :ℝd→ℝ, see Rabino i ch [22],
i.e.,
pcall( ) = (︁S ,
,Y
,τ( )
)︁=:F (S, ,Y,τ), >0,
wi h
(s, ,y,τ)=ysΦ(︁d+(s, ,τ)
)︁−yKP( ,τ)Φ(︁d−(s, ,τ)
)︁,
whe e Φdeno es he dis ibu ion unc ion o a s anda d no mal dis ibu ion and
d±(s, ,τ)=1
√ (τ)(︃log s
KP( ,τ) ±1
2 (τ))︃,
(τ) =σ2
Sτ+σ2
τ−2gκ(τ) +g2κ(τ)
κ2−2ρS σSσ
τ−gκ(τ)
κ,
gκ(τ) =1−e−κτ
κ.
The bond p ice P( ,τ)is gi en by
P( ,τ)=A(τ)e−gκ(τ ) ,
whe e
A(τ) =exp (︃(︂η+σ2
λ
κ−σ2
2κ2)︂(︁gκ(τ) −τ)︁−1
κ(︂σ gκ(τ)
2)︂2)︃
and λdeno es he ma ke p ice o isk. Fo simplici y, we se he ma ke p ice o
isk o ze o and hence assume ha he dynamics o unde he physical and he
isk-neu al measu e a e iden ical. Bjö k [2, Sec . 24.2] desc ibes how o es ima e he
pa ame e s o om ma ke da a. We simula e 1000 pa hs o he s ock, in e es a e
and o eign exchange a e unde he physical measu e o e one yea . Fo each pa h,
we decompose he p ice o he call op ion a ime =1 wi h espec o he d=4
isk ac o s X:= (S, ,Y,τ). We use he ollowing pa ame e s: K=S0=100,
μS=0.05, σS=0.4, Y0=1.1, μY=0, σY=0.05, 0=0.08, κ=0.1, η=0.05,
σ =0.01 and ρS =−0.7, ρSY =−0.4, ρY =0.7.
Figu e 2shows he ela i e unexplained P&L o he OAT decomposi ion, i.e.,
|(F1(X) −F0(X)) −∑︁d
i=1δOAT,i,γ
1(F, X)|
|F1(X) −F0(X)|.
P&L decomposi ion in con inuous ime and app oxima ions 1095
Fig. 2 Rela i e unexplained
P&L o he OAT decomposi ion
o a plain anilla call op ion in a
o eign cu ency a ime =1
o di e en ime g ids
Fig. 3 Rela i e ange o all SU
and 2SU decomposi ions o he
isk ac o S
Weuseas imeg idsγannual, qua e ly, mon hly, weekly and daily ime s eps.
As obse ed in Flaig and Junike [9], we also see ha he unexplained P&L o he
OAT decomposi ion is signi ican o all ime g ids.
Figu e 3shows he ela i e ange o he d!SU decomposi ions o he isk ac o S,
i.e.,
max
π∈σd
δSU,1,π,γ
1(F, X)
δIASU,1
1(F, X) −min
π∈σd
δSU,1,π,γ
1(F, X)
δIASU,1
1(F, X) ,
and he ela i e ange o he d!
22SU decomposi ions o he isk ac o S. The limi ing
IASU decomposi ion is app oxima ed by an ASU decomposi ion wi h 10’000 ime
s eps pe yea . We obse e ha he ange is signi ican o he SU decomposi ions and
insigni ican o he 2SU decomposi ions.
The speed o con e gence o he ASU o he IASU decomposi ion is illus a ed
in Fig. 4 o he isk ac o S, i.e., we show he con e gence
δASU,1,γ
1(F, X)
δIASU,1
1(F, X) −→ 1asγ ends o he iden i y.
Figu es 3and 4look simila o o he isk ac o s.
1096 G. Junike e al.
Fig. 4 Con e gence o he ASU
decomposi ion o he IASU
decomposi ion o he isk ac o
S
In u he nume ical expe imen s, we calcula e he ela i e di e ence be ween he
ASU decomposi ion and he 2SU decomposi ions,
δ2SU,i,π,γ
1(F, X) −δASU,i,γ
1(F, X)
δIASU,i,γ
1(F, X) ,
o e all isk ac o s i∈{1,...,d}, ime g ids γand upda ing o de s π∈σd, and
obse e alues o less han 0.6% in 95% o he simula ions. In conclusion, we ind
ha he ASU decomposi ion and he 2SU decomposi ions a e s ongly dependen
on he ime g id, bu using mon hly o weekly ins ead o annual ime s eps signi -
ican ly educes he de ia ion o he ASU and 2SU decomposi ions om he IASU
decomposi ion.
5.2 Decomposing a baske op ion
In his sec ion, we compa e he compu a ional cos o ob aining a one-yea P&L a -
ibu ion o a baske op ion using a nai e SU decomposi ion wi h annual ime g id
o he compu a ional cos o ob aining an ASU and a 2SU decomposi ion based on a
mon hly ime g id, espec i ely. We conside d isk ac o s, namely ime decay and
d−1 di e en s ocks. A digi al cash-o -no hing baske pu op ion pays $1 a ma u-
i y Ti S1
T≤K,...,Sd−1
T≤Kand ze o o he wise. The s ock p ices a e gi en by
a Black–Scholes model. We se he in e es a e o ze o. We se Si
0=K=100,
i=1,...,d −1 and T=2. The p ice o he op ion a ime ∈[0,T) is equal o
Φ(log K,...,log K), whe e Φis he dis ibu ion unc ion o a (d −1)-dimensional
no mal dis ibu ion wi h loca ion
(︃logS1
−(︂ −1
2σ2)︂(T − ),...,logSd−1
−(︂ −1
2σ2)︂(T − ))︃∈ℝd−1
and co a iance ma ix Σ(T − ), whe e we se σ=0.2, ρ=0.5 and
Σij ={︄σ2,i=j,
ρσ2,i= j.
P&L decomposi ion in con inuous ime and app oxima ions 1097
Table 1 CPU ime o compu e he dcon ibu ions o he SU, ASU and 2SU decomposi ions o a baske
op ion o e one yea using di e en ime g ids. The CPU ime o Φis ob ained om a Mon e Ca lo
simula ion. The CPU imes in b acke s a e es ima ed using he CPU ime o Φand he known complexi ies
o he h ee decomposi ions
Numbe o
e alua ions o Φ
d=4d=15 d=30
E alua ion o Φ1 0.018 s 0.15 s 0.54 s
SU wi h annual g id d+1 0.09 s 2.4 s 16.7 s
2SU wi h mon hly g id (12d+1)2 1.76 s 54.3 s 390 s
ASU wi h mon hly g id (12d+1)2(d−1)7.06 s 123.6 h 3318.7 y s
Baske op ions a e o en p iced by using Mon e Ca lo echniques; see Glasse -
man [13, Sec . 3.2.3]. Fo mode a e dimensions, many baske op ions can also be
p iced by using as e Fou ie echniques; see Ebe lein e al. [7] and Junike and
S ie [17]. We compu e Φby using a simple Mon e Ca lo simula ion implemen ed in
C++ wi h 100’000 simula ions. The expe imen s a e pe o med on a lap op wi h In el
i7-11850H p ocesso and 32 GB RAM.
Table 1shows he CPU ime needed o ob ain Φ o d∈{4,15,30}. We mea-
su e CPU imes by a e aging o e 100 uns. Since in some cases, he a gumen s o
Φ o ob ain an SU decomposi ion wi h a ce ain upda e o de πa e he same o
di e en con ibu ions, we need o e alua e Φonly dL +1 imes, whe e Lis he
numbe o subin e als o [0,T], o ob ain he dindi idual con ibu ions. Fo exam-
ple, (12d+1)2 and (12d+1)2d−1e alua ions o Φa e equi ed o he 2SU and
ASU decomposi ions wi h a mon hly ime g id.
Table 1also shows he CPU ime o compu e he SU, ASU and 2SU decomposi-
ions. A nai e SU decomposi ion based on an annual ime g id is a mos 24 imes
as e han a 2SU decomposi ion wi h a mon hly ime g id. The compu a ional cos
o he 2SU decomposi ion o each con ibu ion is dimension-independen , excep
o he longe ime equi ed o e alua e Φ. Compa ed o he ASU decomposi ion,
he 2SU decomposi ion is 2d−2 imes as e . The ASU decomposi ion canno be
compu ed in easonable ime o d≥30.
Rema k 5.1 To educe he compu a ional ime, i is possible o compu e he dcon i-
bu ions o he SU, 2SU and ASU decomposi ions in pa allel, which would educe
he nume ical e o by a ac o o d. Fu he mo e, he sums o he SU, 2SU and
ASU decomposi ions can also be pa allelised. Fo example, o he 2SU decomposi-
ion, we need o pe o m 2(dL+1) unc ion e alua ions o ob ain all dcon ibu ions.
I a unc ion e alua ion akes 0.54 s in d=30 dimensions as in Table 1, he com-
pu a ion ime o he 2SU decomposi ion wi h mon hly ime g id could be educed
om 390 s o abou 0.54 s using 722 co es o pa allelisa ion.
6Conclusions
We showed ha he IASU decomposi ion is he only (up o indis inguishabili y) exac
and symme ic decomposi ion in he amily o I ô decomposi ions, which is a la ge
1098 G. Junike e al.
class o no malised decomposi ions based on a ea anged e sion o I ô’s o mula.
This axioma ic esul , oge he wi h he ac ha he IASU decomposi ion is g id-
independen and conside s he ull pa hs o he isk basis, makes i a decomposi ion
o choice om a heo e ical pe spec i e. In p ac ice, he calcula ion o he IASU de-
composi ion comes wi h wo challenges: i in ol es s ochas ic in eg als ha mus be
app oxima ed, and he compu a ional e o explodes as he numbe o isk ac o s
inc eases.
We ha e shown ha he IASU decomposi ion can be app oxima ed by he ASU de-
composi ion (which is always exac and symme ic) i we use a su icien ly ine ime
g id, bu he ASU decomposi ion also su e s om he cu se o dimensionali y as
he numbe o isk ac o s inc eases. Fo applica ions whe e di e en isk ac o s
may ha e in e ac ions, bu almos su ely do no ha e simul aneous jumps, we ha e
shown ha he IASU decomposi ion is indis inguishable om he a e age o wo ISU
decomposi ions, hus b eaking he cu se o dimensionali y. The e o e, om a heo-
e ical poin o iew, he 2SU decomposi ion wi h a su icien ly ine ime g id is an
app op ia e app oxima ion o he IASU decomposi ion.
Based on ou own nume ical expe imen s and he empi ical analysis o Flaig and
Junike [9], we ecommend using mon hly o e en weekly ins ead o annual ime
s eps.
The addi ional compu a ional cos o ou wo ecommenda ions is mode a e, bu
he heo e ical p ope ies o he decomposi ion a e d ama ically imp o ed.
Appendix
A.1 Auxilia y esul s
Lemma A.1 Le i, j ∈{1,...,d}.Le π,η ∈σdand x∈ℝd.Then i holds ha
η−1(︂p{j:π(j)≤π(η−1(i))}(︁η(x))︁)︂=p{j:π(η−1(j))≤π(η−1(i))}(x). (A.1)
P oo Le k∈{j:π(j) ≤π(η−1(i))}, which is equi alen o
η(k) ∈{︁j:π(︁η−1(j))︁≤π(︁η−1(i))︁}︁.
Since (η−1(x))η(k) =xkand (η(x))k=xη(k), we ob ain ha
(︃η−1(︂p{j:π(j)≤π(η−1(i))}(︁η(x))︁)︂)︃η(k) =(︂p{j:π(j)≤π(η−1(i))}(︁η(x))︁)︂k
=(︁p{j:π(η−1(j))≤π(η−1(i))}(x))︁η(k),
which leads o (A.1). □
P&L decomposi ion in con inuous ime and app oxima ions 1105
le 𝒯also con ain (˜τn)n∈ℕ=((τn
2,τn
1))n∈ℕ. Since τn
2(an
ℓ,1)=τn
2(bn
ℓ,1)=τn
1(bn
ℓ,1)
and by he mul idimensional Taylo heo em,
δISU,1
(X ⋄τn)=∑︂
ℓ(︂ϱ(︁(X ⋄τn)bn
ℓ,1∧ )︁−ϱ(︁(X ⋄τn)an
ℓ,1∧ )︁)︂
=∑︂
ℓ
ϱ1(︁(X ⋄τn)an
ℓ,1∧ )︁(︁X1
τn
1(bn
ℓ,1∧ ) −X1
τn
1(an
ℓ,1∧ ))︁.
By he de ini ions o X1,X2and ρ,
δISU,1
(X ⋄τn)=∑︂
ℓ
Bτn
2(an
ℓ,1∧ )(Bτn
1(bn
ℓ,1∧ ) −Bτn
1(an
ℓ,1∧ ))
=∑︂
ℓ
Bτn
1(bn
ℓ,1∧ )(Bτn
1(bn
ℓ,1∧ ) −Bτn
1(an
ℓ,1∧ ))
=∑︂
ℓ
B ℓ(B ℓ∧ −B ℓ−1∧ )
=2∑︂
ℓ
(B ℓ+B ℓ−1)
2(B ℓ∧ −B ℓ−1∧ )−∑︂
ℓ
B ℓ−1(B ℓ∧ −B ℓ−1∧ )
o n
ℓ:= τn
1(bn
ℓ,1)=τn
1(an
ℓ+1,1)=τn
2(bn
ℓ−1,2)=τn
2(an
ℓ,2).Le ∫︁
0Bs◦dBsdeno e
he S a ono ich in eg al and ∫︁
0BsdBs he I ô in eg al. I holds ha
δISU,1
(X ⋄τn)p
−→ 2∫︂
0Bs◦dBs−∫︂
0BsdBs=1
2B2
+1
2
o n→∞. By he same a gumen s,
δISU,1
(X ⋄˜τn)=∑︂
ℓ(︂ϱ(︁(X ⋄˜τn)bn
ℓ,2∧ )︁−ϱ(︁(X ⋄˜τn)an
ℓ,2∧ )︁)︂
=∑︂
ℓ
ϱ1(︁(X ⋄˜τn)an
ℓ,2∧ )︁(X2
τn
2(bn
ℓ,2∧ ) −X2
τn
2(an
ℓ,2∧ ))
=∑︂
ℓ
Bτn
1(an
ℓ,2∧ )(Bτn
2(bn
ℓ,2∧ ) −Bτn
2(an
ℓ,2∧ ))
=∑︂
ℓ
Bτn
2(an
ℓ,2∧ )(Bτn
2(bn
ℓ,2∧ ) −Bτn
2(an
ℓ,2∧ ))
=∑︂
ℓ
B ℓ(B ℓ+1∧ −B ℓ∧ )
p
−→ ∫︂
0BsdBs
=1
2B2
−1
2
1106 G. Junike e al.
o n→∞. The e o e,
p-limn→∞δISU,1
(X ⋄τn)= p-limn→∞δISU,i
(X ⋄˜τn), i =1,2,
o >0, and hence he ISU decomposi ion o ϱ(X) canno be s able a X.□
Acknowledgemen s We hank Be nd Buchwald and And eas Mä ke om Hanno e Re, Sol eig Flaig
om Deu sche Rück, Julien Hambucke s om Uni e si y o Liège, Jan-F ede ik Mai om XAIA In-
es men GmbH, Tho s en Schmid om Uni e si ä F eibu g and an anonymous e e ee o e y help ul
commen s ha imp o ed his pape .
Funding in o ma ion Open Access unding enabled and o ganized by P ojek DEAL.
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and ins i u ional a ilia ions.
