Irreversible reinsurance: minimization of capital injections in presence of a fixed cost

Fede ico, Sal a o e; Fe a i, Gio gio; To en e, Ma ia-Lau a
A icle — Published Ve sion
I e e sible einsu ance: minimiza ion o capi al injec ions
in p esence o a ixed cos
Ma hema ics and Financial Economics
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Fede ico, Sal a o e; Fe a i, Gio gio; To en e, Ma ia-Lau a (2024) : I e e sible
einsu ance: minimiza ion o capi al injec ions in p esence o a ixed cos , Ma hema ics and
Financial Economics, ISSN 1862-9660, Sp inge , Be lin, Heidelbe g, Vol. 18, Iss. 4, pp. 707-733,
h ps://doi.o g/10.1007/s11579-024-00373-z
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/315458
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Ma hema ics and Financial Economics (2024) 18:707–733
h ps://doi.o g/10.1007/s11579-024-00373-z
I e e sible einsu ance: minimiza ion o capi al injec ions in
p esence o a ixed cos
Sal a o e Fede ico1·Gio gio Fe a i2·Ma ia-Lau a To en e3
Recei ed: 29 Janua y 2024 / Accep ed: 24 July 2024 / Published online: 4 Augus 2024
© The Au ho (s) 2024
Abs ac
We p opose a model in which, in exchange o he paymen o a ixed ansac ion cos , an
insu ance company can choose he e en ion le el as well as he ime a which subsc ibing
a pe pe ual einsu ance con ac . The su plus p ocess o he insu ance company e ol es
acco ding o he di usi e app oxima ion o he C amé -Lundbe g model, claims a i e a
a ixed cons an a e, and he dis ibu ion o hei sizes is gene al. Fu he mo e, we do no
speci y any pa icula unc ional o m o he e en ion le el. The aim o he company is
o ake ac ions in o de o minimize he sum o he expec ed alue o he o al discoun ed
low o capi al injec ions needed o a oid bank up cy and o he ixed ac i a ion cos o he
einsu ance con ac . We p o ide an explici solu ion o his p oblem, which in ol es he
esolu ion o a s a ic nonlinea op imiza ion p oblem and o an op imal s opping p oblem o
a e lec ed di usion. We hen illus a e he heo e ical esul s in he case o p opo ional and
excess-o -loss einsu ance, by p o iding a nume ical s udy o he dependency o he op imal
solu ion wi h espec o he model’s pa ame e s.
Keywo ds Reinsu ance ·Fixed cos ·Capi al injec ions ·Di usi e isk model ·Op imal
s opping
Ma hema ics subjec classi ica ion 97M30 ·91B30 ·60G40 ·49L20
Jel classi ica ion C61 ·G22 ·C41
BMa ia-Lau a To en e
[email p o ec ed]
Sal a o e Fede ico
[email p o ec ed]
Gio gio Fe a i
[email p o ec ed]
1Dipa imen o di Ma ema ica, Uni e si à di Bologna, Piazza di Po a S. Dona o 5, 40126 Bologna,
I aly
2Cen e o Ma hema ical Economics (IMW), Biele eld Uni e si y, Uni e si ä ss asse 25, 33615
Biele eld, Ge many
3Dipa imen o di Economia, Uni e si à di Geno a, Via Vi aldi 5, 16126 Geno a, I aly
123
708 Ma hema ics and Financial Economics (2024) 18:707–733
1 In oduc ion
Reinsu ance con ac s usually un o long ime pe iods (a leas o longe han he ypical
ma u i y o inancial con ac s) and a e exposed o high ic ional cos s. As a esul , einsu -
ance nego ia ions a e cos ly, leng hy, and can be hough o as i e e sible, c . [5]. As no iced
by [6], i is indeed he case ha “al hough einsu ance, in p inciple, is e e sible, in p ac ice
e e sing a einsu ance ansac ion exposes he insu e o ela i ely high ansac ion cos s as
well as addi ional cha ges o p o ec he einsu e agains ad e se selec ion.” Fu he mo e,
many ex e nal ac o s can in e e e wi h changes in he einsu ance con ac s. I is ecen news
ha “2023s enego ia ion o einsu ance policies has been he mos challenging in yea s as
einsu e s espond o p essu e om spi alling in la ion and la ge losses om na u al ca as-
ophes, as well as he allou om Russia’s in asion o Uk aine” (c . Ian Smi h, Insu ance
Co esponden o he “Financial Times”, Janua y 3 20231).
Op imal einsu ance decisions a e ypically o mula ed in e ms o egula con ol p ob-
lems, hus neglec ing he a o emen ioned i e e sibili y ea u e. Gi en he as i y o he
ela ed li e a u e, we e ain he e o m p o iding a lis o e e ences ( ha would necessa ily
esul in being no exhaus i e) and we simply e e o he discussion in Chap e 2 o [22]o in
Chap e 11 o [1] o models and solu ions. Howe e , in he las decade he ac ua ial li e a u e
has s a ed expe iencing models o op imal i e e sible einsu ance. In [2] i is in es iga ed
an op imal einsu ance p oblem unde ixed cos , o an insu ance company aiming a max-
imizing exponen ial expec ed u ili y a e minal ime. The p oblem o op imal einsu ance
nego ia ions wi h implemen a ion delay and ixed cos is conside ed in [9], he op imal iming
o he ac i a ion o an excess-o -loss einsu ance wi h ixed cos s is s udied in [19], while
he p esence o addi ional p opo ional ansac ion cos s o a company minimizing he uin
p obabili y is ea ed in [18]. Finally, a singula s ochas ic con ol model o he op imal
sequen ial adjus men o einsu ance con ac s has been ecen ly o mula ed in [27].
Ou pape con ibu es o ha bunch o li e a u e by p oposing a model in which, in
exchange o he paymen o a ixed ansac ion cos , an insu ance company can choose
he e en ion le el as well as he ime a which subsc ibing a (pe pe ual) einsu ance con ac .
Assuming ha in es o s injec capi al o a oid bank up cy o he company, he insu ance
company aims a minimizing he sum o he expec ed alue o he discoun ed ixed ac i a ion
cos and o he cumula i e discoun ed low o capi al injec ions. Capi al injec ion models
ha e been in oduced by Dickson and Wa e s in [8]. The ein, s a ing om he obse a ion
ha uin occu s almos su ely when he company pays di idends by ollowing he op imal
s a egy o he de Fine i’s p oblem, a model has been sugges ed in which he sha eholde s
a e obliged o injec capi al in o de o a oid bank up cy. We also e e o [14], [15], [20],
[21], [23], [29] and e e ence he ein o wo ks ela ed o he op imal di idends’ dis ibu ion
in p esence o capi al injec ions (see also [7]and[28] o he case o impulsi e injec ions o
capi al). In he con ex o op imal einsu ance p oblems, he employmen o he cumula i e
discoun ed low o capi al injec ions as a isk measu e al e na i e o he uin p obabili y has
i s ly been p oposed in [11], and la e also used in [10]and[13]. As a ma e o ac , he use
o he uin p obabili y as measu e o isk p esen s d awbacks: i s o all, i is no a cohe en
isk measu e, his po en ially leading o decisions ha a e no economically sounded; second
o all, i does no p o ide in o ma ion abou nei he he ime o uin no he se e i y o uin.
In o de o de ine a uni ied amewo k o he e alua ion o a a ie y o isk quan i ies, and
in pa icula o gi e indica ions abou he de ici a uin and he ime o uin, Ge be and
Shiu p oposed in [16] he so-called expec ed discoun ed penal y unc ion – also known as
1h ps://www. .com/con en / 5 9d450-c539-47a7-bc5c-44a8db57e74e
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Ma hema ics and Financial Economics (2024) 18:707–733 709
Ge be -Shiu unc ion – o which he capi al injec ion c i e ion ep esen s an example (see,
e.g., Sec ion 2.4.3 in [17]o Chap e 4in[26]).
In his pape , we assume ha he su plus p ocess o an insu ance company e ol es acco d-
ing o he di usi e app oxima ion o he C amé -Lundbe g model. Claims a i e a a ixed
cons an a e and he dis ibu ion o hei sizes is gene al. Fu he mo e, we do no speci y
any o m o he e en ion le el, which is simply assumed o be a con inuous unc ion, non
dec easing wi h espec o he einsu ance pa ame e . The company can choose he ime τa
which buying einsu ance and he desi ed e en ion le el, which will hen be kep om ime τ
on. Those once- o -all ac ions in ol e a ixed cos , which is immedia ely wi hd awn om he
company’s su plus a ime τ; u he , om ime τon, he company pays a pe pe ual p emium
a e compu ed acco ding o he Expec ed Value P inciple. The aim o he company is o ake
ac ions in such a way ha he o al discoun ed cos s o capi al injec ion and o he einsu ance
con ac a e minimized. We p o ide an explici solu ion o his p oblem which we show can
be sol ed ia a wo-s ep p ocedu e (see also [2]and[18], among o he s). We i s sol e o
he op imal e en ion le el, which is uniquely iden i ied h ough he solu ion o a nonlinea
algeb aic equa ion. Then, gi en he op imal e en ion le el, we look o he op imal ime a
which i is wo h ac i a ing he i e e sible einsu ance con ac . This u ns ou o be gi en as
he solu ion o a one-dimensional op imal s opping p oblem o a e lec ed d i ed B ownian
mo ion. We use he classical guess-and- e i y app oach by de e mining a smoo h solu ion
o he co esponding a ia ional inequali y wi h Neumann bounda y condi ion and hen by
e i ying he ac ual op imali y o he candida e policy. I is wo h no icing ha , gi en he
e lec ing condi ion o he su plus p ocess a ze o, he e i ica ion a gumen equi es qui e
some echnical wo k in o de o check ha he a ia ional inequali y is indeed sa is ied by he
candida e alue unc ion (see he p oo s o P oposi ion 3.5 and o Theo em 3.6 below). We
ind ha a ba ie -s a egy is op imal and ha einsu ance should be bough when he insu -
ance company’s su plus p ocess is su icien ly la ge, in pa icula la ge han an endogenously
de e mined igge le el ( ee bounda y) ha depends on he model’s pa ame e s. In e es -
ingly, we obse e ha he solu ion o ou p oblem is consis en wi h ha o [11], whe e,
gi en he absence o a ixed ansac ion cos , einsu ance is bough immedia ely. Namely,
he op imal e en ion le el in ou model is he same as ha in [11], and, when he ixed cos
K↓0, he ee bounda y con e ge o ze o as well, implying ha immedia e einsu ance is
in ac op imal.
We inally illus a e ou esul s in he wo ele an cases o p opo ional and excess-
o -loss einsu ance, when he dis ibu ion o he claims’ sizes a e Exponen ial o Pa e o
wi h pa ame e s (ζ, α), o someα>2andζ>0. We sol e nume ically he equa ions
ha uniquely de e mine he op imal e en ion le el and he ee bounda y and we s udy he
dependency o hose quan i ies wi h espec o ele an model’s pa ame e s. We obse e
ha bo h he op imal e en ion le el and he ee bounda y exhibi a mono onic beha io
wi h espec o he conside ed pa ame e s and we p o ide explana ions o hese indings.
Fu he mo e, we show ( o ixed alues o he model’s pa ame e s) ha , when he claim’s size
is exponen ially dis ibu ed, he alue unc ion one has in he case o p opo ional einsu ance
is smalle han he one ela ed o an excess-o -loss einsu ance, while no uni o m compa ison
can be made in he case o Pa e o-dis ibu ed claim’s size.
The es o he pape is o ganized as ollows. Sec ion 2p esen s he p oblem, which is
hen sol ed in Sec .3. Sec ion4illus a es nume ically he heo e ical indings in he case
o p opo ional and excess-o -loss einsu ance, while a inal appendix collec s mos o he
echnical p oo s o he pape .
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710 Ma hema ics and Financial Economics (2024) 18:707–733
2 P oblem o mula ion
Le (, F,F:= (F ) ≥0,P)be a comple e p obabili y space, ich enough o accommoda e a
one-dimensional F-B ownian mo ion (W ) ≥0and an independen squa e-in eg able andom
a iable Z, aking alues in Z⊂R+, and wi h law νZunde P. Wi hin his p obabilis ic
se ing, we conside he una ec ed su plus p ocess (
X ) ≥0o an insu ance company, wi h
ini ial alue 
X0=x>0, e ol ing h ough he di usion app oxima ion o he classical
C amé -Lundbe g model (see, e.g., Appendix D in [22] o Sec ion 8 in Chap e IV o [1])

Xx
=x+λημ +σ√λW , ≥0.(2.1)
He e, μ:= ZzνZ(dz)>0andσ2:= Zz2νZ(dz)>0 a e, espec i ely, he mean and he
second momen o he gene ic claim size Z,λis he a i al ime pa ame e o he claims, η
is he sa e y loading.
In o de o a oid bank up cy, in es o s a e asked by he insu ance company o injec capi al
whene e he su plus le el a emp s o become nega i e. Assuming ha in es o s a e impa ien
agen s, i is clea ha hose injec ions o capi al a e made only when s ic ly necessa y. The
cumula i e amoun o capi al injec ions (I ) ≥0will hen e lec (à la Sko okhod) he su plus
p ocess a x=0, so ha he esul ing dynamics a e
Xx
=x+λημ +σ√λW +I =
Xx
+I , ≥0,(2.2)
wi h
I =sup
0≤s≤ −
Xx
s+, ≥0.
Wi hin his model, we conside he possibili y o he insu e o adop ing a einsu ance
s a egy. Mo e p ecisely, we conside a con inuous unc ion
:Z×[0,1]→R+,
which ep esen s he e en ion le el o he insu e — ha is, he pa o isk emaining in
he cha ge — whose alue depends on he chosen le el b∈[0,1]. I is assumed (z,·)is
non dec easing and ha b=1 co esponds o no einsu ance and b=0 co esponds o ull
einsu ance; ha is,
(z,1)=z, (z,0)=0.(2.3)
Typical examples a e he case o p opo ional einsu ance, o which
(z,b)=bz,(2.4)
and ha o he excess-o -loss einsu ance, o which
(z,b)=z∧b
1−b.(2.5)
Rema k 2.1 I is wo h no icing ha choosing he einsu ance pa ame e b∈[0,1]allows
us o co e he ele an einsu ance models using a unique pa ame iza ion wi hin a uni ied
se ing. In he case o excess-o -loss einsu ance (c . (2.5) abo e), his leads o a de ia ion
om he classical o mula o he einsu ance e en ion le el which assumes b≥0(c .[22]).
123

Ma hema ics and Financial Economics (2024) 18:707–733 711
In ou model, we assume ha he einsu ance policy is i e e sible. This means ha a a
p ope ly picked F-s opping ime τ he insu e chooses he le el bτ, which will hen be kep
om ime τon. Fo mally, he einsu ance policy is hus a couple
a:= (τ, bτ)∈A:= T×MFτ,
whe e
T:= {τ:→R+∪{+∞}F-s opping ime},MFτ:= {b:(, Fτ)→[0,1]measu able}.
To implemen he einsu ance s a egy a, he insu e aces a ixed ansac ion cos K≥0a
he ime τ∈Ta which he einsu ance con ac is signed. F om ime τon, acco ding o he
Expec ed Value P inciple, he insu e pays o he einsu ance company a pe pe ual p emium
a e wi h alue
θλ(μ −M1(bτ)), whe e M1(b):= Z
(z,b)νZ(dz), (2.6)
whe e θ>ηdeno es he sa e y loading applied by he einsu ance company. On he o he
hand, he isk exposu e o he insu ance company is educed, leading o he di usion coe i-
cien
λM2(bτ), whe e M2(b):= Z
(z,b)2νZ(dz). (2.7)
In pa icula , om (2.3) i ollows ha M1(1)=μand M2(1)=σ2. Consequen ly, o ≥τ,
he insu e only aces he ou lows ela i e o he pa o isk, ep esen ed by he e en ion
le el . All in all, he su plus p ocess wi h capi al injec ion unde he einsu ance s a egy a
is
Xx,a
:= Xx
i <τ
Xx
τ−K+λ(θ M1(bτ)−(θ −η)μ)( −τ)+√λM2(bτ)(W −Wτ)+I ,i ≥τ,
(2.8)
whe e (I ) ≥0is now such ha
I =sup
τ≤s≤ −Xx
τ−K+λ(θ M1(bτ)−(θ −η)μ)( −τ)+λM2(bτ)(W −Wτ)+, ≥τ.
In he sequel, in o de o s ess he dependency o Ion he einsu ance policy and x,weshall
w i e Ix,a, when needed.
Following [10], [11], and [12], we assume ha he insu ance company employs he
expec ed o al amoun o discoun ed capi al injec ions as a measu e o isk and hus aims a
de e mining an admissible i e e sible einsu ance policy a∗∈Asuch ha
a∗∈a gmin
a∈A
E∞
0
e−ρ dIx,a
,
whe e ρ>0 is a subjec i e in e empo al discoun a e. Fo u u e equen use we also
de ine
U(x):= in
a∈A
E∞
0
e−ρ dIx,a
,x≥0.(2.9)
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712 Ma hema ics and Financial Economics (2024) 18:707–733
3 Solu ion o he p oblem
In his sec ion, we de e mine he explici solu ion o (2.9). To accomplish ha , we shall i s
e o mula e he p oblem in an handie way (c . Sec .3.1), hen we shall ob ain he op imal
le el (c . Sec .3.2) and, inally, he op imal ime o einsu ance (c . Sec .3.3).
3.1 Re o mula ion o he p oblem
In o de o ob ain an handy ep esen a ion o U, we compu e he injec ion cos s associa ed
o a ixed e en ion pa ame e b∈[0,1] aken a =0; ha is, gi en y∈Rand b∈[0,1],
we calcula e
Gb(y):= E∞
0
e−ρ dHy,b
,(3.1)
whe e
Hy,b
:= sup
0≤s≤ −Yy,b
s+, ≥0,(3.2)
wi h
Yy,b
:= y+λ(θ M1(b)−(θ −η)μ) +λM2(b)
W , ≥0,
o ano he F-B ownian mo ion (
W ) ≥0.
Following [25], we know ha , when y≥0, he unc ion Gbis he solu ion o he di e en ial
p oblem
⎧
⎪
⎪
⎨
⎪
⎪
⎩
1
2λM2(b)G
b(y)+λ(θ M1(b)−(θ −η)μ)G
b(y)−ρGb(y)=0,
G
b(0)=−1,lim
y→+∞Gb(y)=0.
(3.3)
I hen ollows om (3.3) ha
Gb(y)=− 1
γ−(b)eγ−(b)y,∀y≥0,(3.4)
whe e γ−(b)<0 is he nega i e solu ion o he equa ion (b,γ)=0, wi h
(b,γ):= 1
2λM2(b)γ 2+λ(θM1(b)−(θ −η)μ)γ−ρ, γ ∈R.(3.5)
On he o he hand, we ha e
Gb(y)=−y+Gb(0)=−y−1
γ−(b),∀y<0.(3.6)
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Ma hema ics and Financial Economics (2024) 18:707–733 713
Wi h he help o he p e iously de ined quan i ies (c . (3.1)and(3.2)), an applica ion o
he s ong Ma ko p ope y allows us o ew i e Uas ollows:
U(x)=in
a∈A
Eτ−
0
e−ρ dIx,a
+∞
τ−
e−ρ dIx,a

=in
a∈A
Eτ−
0
e−ρ dIx,a
+E∞
τ−
e−ρ dIx,a
Fτ−
=in
a∈A
E∞
0
e−ρ dHx,1
−∞
τ−
e−ρ dHx,1
+E∞
τ−
e−ρ dIx,a
Fτ−
=G1(x)+in
a∈A
Ee−ρτ Gbτ(Xx
τ−K)−G1(Xx
τ)
=: G1(x)+in
a∈A
J(x,a).
Le ing
V(x):= in
a∈A
J(x,a), (3.7)
whe e
J(x,a):= Ee−ρτ bτ(Xx
τ), b(y):= Gb(y−K)−G1(y), (3.8)
wi h he con en ion e−ρτ bτ(Xx
τ)=0on{τ=∞},weha e
U(x)=G1(x)+V(x). (3.9)
We now con inue ou analysis by de e mining he op imal a∗=(τ∗,b∗)s. . V(x)=
J(x,a∗).Clea ly,suchana∗will also be op imal o (2.9).
3.2 Op imal einsu ance
Recall ha he unc ion γ−(b)has been de ined as he nega i e solu ion o he equa ion
(b,·)=0, wi h as in (3.5). I s explici exp ession is
γ−(b)=−
(θ M1(b)−(θ −η)μ) +(θ M1(b)−(θ −η)μ)2+2ρM2(b)
λ
M2(b),b∈[0,1].
(3.10)
We deno e
B∗:=a gminb∈[0,1]γ−(b).
I is s aigh o wa d o no e ha B∗is no emp y since con inui y o he e en ion unc ion
(z,·)and M2(b)>0 o each b∈[0,1]imply con inui y o he unc ion γ−o e [0,1].A
ele an ac is ha
b∗∈B∗⇒ Gb∗(x)=min
b∈[0,1]Gb(x), ∀x∈R.(3.11)
Indeed
Gb(x)=H1(γ −(b), x)i x≥0
H2(γ −(b), x)i x<0,
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714 Ma hema ics and Financial Economics (2024) 18:707–733
whe e H1,H2:R−×R+→Ra e de ined by
H1(g,x):= −1
gegx
H2(g,x):= −x−1
g.
Since
∂H1
∂g(g,x)=1
g2egx(1−gx)>0,∀(g,x)∈R<0×R+
∂H2
∂g(g,x)=1
g2>0,∀(g,x)∈R<0×R<0,
we see ha H1and H2a e s ic ly inc easing wi h espec o he i s a iable. I ollows ha
b→ Gb(x)is minimized by he minimize s o γ−.
Rema k 3.1 I is wo h no icing ha he unc ion γ−(b),b∈[0,1],asde inedin(3.10)
coincides wi h he opposi e o he unc ion β(b),b∈[0,
b],de inedin[11](when
b=1).
In pa icula , up o a pa ame iza ion, any op imize b∗o γ−on [0,1]does also op imize
βin [11], and ice e sa. We shall see in he nex Theo em ha , as in [11], op imize s o
γ−ac ually gi e he op imal le el o be adop ed. The op imal iming o einsu ance is hen
de e mined gi en he op imal le el b∗(see Sec .3.3 below).
The nex esul shows how o educe he solu ion o (3.7) o a pu e op imal iming p oblem.
Theo em 3.2 Recall (3.7) and (3.8). Le b∗∈B∗and le τ∗(b∗)∈Tsuch ha
τ∗(b∗)∈a gminτ∈TEe−ρτ b∗(Xx
τ)=a gminτ∈TJ(x,(τ,b∗)),
wi h he con en ion e−ρτ b∗(Xx
τ)=0on {τ=∞}. Then, he couple a∗:= (τ∗(b∗), b∗)∈A
is an op imal einsu ance s a egy (wi h b∗ hough o as a cons an andom a iable).
P oo Since Gb∗(x)=minb∈[0,1]Gb(x)(see (3.11)) hen
U(x)=G1(x)+V(x)
≥G1(x)+in
τ∈T
Ee−ρτ Gb∗(Xx
τ−K)−G1(Xx
τ)
=G1(x)+in
τ∈T
J(x,(τ,b∗))
=G1(x)+J(x,(τ∗(b∗), b∗)).
On he o he hand
U(x)=G1(x)+V(x)
≤G1(x)+Ee−ρτ∗(b∗)Gb∗(Xx
τ∗(b∗)−K)−G1(Xx
τ∗(b∗))
=G1(x)+J(x,(τ∗(b∗), b∗)).
Consequen ly U(x)=G1(x)+J(x,(τ∗(b∗), b∗)) and (τ∗(b∗), b∗)is op imal. 
Theo em 3.2 p o ides su icien condi ions needed o iden i y an op imal einsu ance
pa ame e b∗.I b∗∈B∗, hen he le el co esponding o a andom a iable wi h cons an
alue belonging o he se B∗is he second componen o an op imal einsu ance s a egy.
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Ma hema ics and Financial Economics (2024) 18:707–733 721
Fig. 4 Dependency o b∗wi h espec o ρand μ
Fig. 5 Dependency o x∗
b∗wi h espec o ρ,μ,andK
alues o he pa ame e s as i ollows: θ=0.5, η=0.3, λ=0.05, ρ=0.04, μ=10,
K=10. Wi h such a choice, by P oposi ion 3.5 and om (4.7) (see P oposi ion 4.2), we
ob ain b∗=0.4627 and x∗
b∗=11.4785.
Figu e4shows how b∗depends on he pa ame e s ρand μ. Wi h espec o ρ,weobse e
abeha io o b∗which is line wi h ha we had in he case o p opo ional einsu ance, which
can be hen explained in he same way. No icing ha unde he Exponen ial dis ibu ion o
he claim’s size he pa ame e μmeasu es bo h he ampli ude o he B ownian luc ua ions
and he end o he a e age p o i s o he company, we obse e ha an inc ease in μleads
he company o einsu e less, i.e. o an inc ease o b∗. Hence, in he in e play o he wo oles
played by μ, he d i e ec appea s o be dominan and he beha io o μ→ b∗(μ) can be
hus explained as in he case o he p opo ional einsu ance (c . Fig. 2).
Figu e5plo s x∗
b∗as a unc ion o he pa ame e s ρ,μand K. While he beha io o x∗
b∗
wi h espec o ρand Kcan be explained by he same easoning ha we ollowed in he case
o p opo ional einsu ance, a di e en pa e n is obse ed o he dependency o x∗
b∗wi h
espec o μ. He e, we see ha x∗
b∗is dec easing wi h espec o μ. Recalling again ha μ
measu es bo h he ampli ude o he B ownian luc ua ions and he end o he a e age p o i s
o he company, we ind ha in he in e play o he wo oles played by μ, he ola ili y e ec
appea s o be dominan and he beha io o μ→ b∗(μ) can be hus explained as ha o
σ2→ b∗(σ2)in he case o he p opo ional einsu ance (c . Fig. 1).
123

722 Ma hema ics and Financial Economics (2024) 18:707–733
Fig. 6 Dependency o b∗wi h espec o ρand ζ
4.2.2 The case Z∼Pa e o(,˛)
We assume ha he claim sizes Z∼Pa e o(ζ, α),α>2, wi h densi y
p(z)=⎧
⎨
⎩
0i z<ζ
αζα
zα+1i z≥ζ.
By some compu a ions, we ind
M1(b)=⎧
⎪
⎪
⎨
⎪
⎪
⎩
b
1−bi b<ζ
1+ζ
ζ
α−1α−ζ1−b
bα−1i b≥ζ
1+ζ
M2(b)=⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
b
1−b2
i b<ζ
1+ζ
ζ2
α−2α−2ζ1−b
bα−2i b≥ζ
1+ζ.
Equa ion (3.5) now eads
(b,γ):= ⎧
⎪
⎨
⎪
⎩
1
2λb
1−b2γ2+λθb
1−b−ζ+ηζγ−ρi b<ζ
1+ζ
1
2λζ2
α−2α−2ζ1−b
bα−2γ2+λθζ
α−11−ζ1−b
bα−1+ηζγ−ρi b≥ζ
1+ζ,
wi h γ∈R.
We illus a e nume ically he sensi i i y o he op imal le el b∗and o he op imal ein-
su ance bounda y x∗
b∗wi h espec some ele an model’s pa ame e s. We choose benchma k
alues o he pa ame e s as i ollows: θ=0.5, η=0.3, λ=0.05, ρ=0.04, ζ=10,
K=10. Wi h such a choice, by P oposi ion 3.5 and om (4.7) (see P oposi ion 4.2), we
ob ain b∗=0.5195 and x∗
b∗=11.7572.
Figu e6shows how b∗depends on he pa ame e s ρand ζ, while Fig.7plo s x∗
b∗as a
unc ion o he pa ame e s ρ,ζand K. We obse e beha io s o b∗and x∗
b∗which a e simila
o hose obse ed wi h espec o ρ,μand Kin he case o an Exponen ial dis ibu ion o
he claim size, and which can be hen explained h ough he same a ionale. As a ma e o
ac , in he Pa e o dis ibu ion, he a e age and he a iance o he sizes o he claims a e
inc easing unc ions o he only pa ame e ζ, jus as hey a e unc ions o μin he case o an
Exponen ial dis ibu ion.
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Ma hema ics and Financial Economics (2024) 18:707–733 723
Fig. 7 Dependency o x∗
b∗wi h espec o ρ,ζand K
4.3 Compa ison o he alue unc ion in he case o p opo ional and excess-o -loss
einsu ance
In Fig.8we collec d awings o he alue unc ion in he case o p opo ional einsu ance
(solid line) and excess-o -loss einsu ance (dashed line), when he claim size Z∼Exp(1/μ)
(le panel) and Z∼Pa e o(ζ, α) ( igh panel). We obse e ha , when he claim’s size is
exponen ially dis ibu ed, he alue unc ion one ob ains in he case o p opo ional ein-
su ance is smalle han he one ela ed o an excess-o -loss einsu ance, while no uni o m
Fig. 8 Value unc ion (zoomed image in he bo om panels) in he case o p opo ional einsu ance (solid
line) and excess-o -loss einsu ance (dashed line), when he claim sizes Z∼Exp(1/μ) (le panel) and
Z∼Pa e o(ζ, α) ( igh panel)
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724 Ma hema ics and Financial Economics (2024) 18:707–733
compa ison can be made in he case o Pa e o-dis ibu ed claim’s size. We hus conclude ha
(a leas o he benchma k alues o he pa ame e s ha we ha e used) excess-o -loss ein-
su ance is no necessa ily a ou able o p opo ional einsu ance, a inding ha is in con as
o ha o Figu e 3 in [11] (see also he subsequen discussion a page 13 he ein).
Appendix
P oo o P oposi ion 3.3
P oo F om (3.4), (3.6)and(3.8), he explici exp ession o b∗is:
b∗(x)=⎧
⎪
⎨
⎪
⎩
−(x−K)−1
γ−(b∗)+1
γ−(1)eγ−(1)x0≤x≤K,
−1
γ−(b∗)eγ−(b∗)(x−K)+1
γ−(1)eγ−(1)xx>K,
(4.9)
om which b∗(0)=K+γ−(b∗)−γ−(1)
γ−(1)γ −(b∗)and limx→∞ b∗(x)=0. We compu e

b∗(x)=−1+eγ−(1)x0<x<K,
−eγ−(b∗)(x−K)+eγ−(1)xx>K.
I b∗=1, hen b∗is s ic ly dec easing in [0,+∞)and 0 < 1(x)≤ 1(0)=K,∀x≥0.
O he wise, i b∗= 1, hen b∗is s ic ly dec easing in [0,ˆxb∗]and s ic ly inc easing in
[ˆxb∗,∞),whe e ˆxb∗is de ined in (3.13). The poin ˆxb∗is he unique global minimum poin
o b∗, whose minimum alue is:
b∗(ˆxb∗)=−γ−(1)−γ−(b∗)
γ−(1)γ −(b∗)e−γ−(1)γ −(b∗)
γ−(1)−γ−(b∗).
I −Kγ−(1)γ −(b∗)≤γ−(b∗)−γ−(1)<0, hen b∗(0)≥0 and i em (i) ollows.
O he wise, i γ−(b∗)−γ−(1)<−Kγ−(1)γ −(b∗), hen b∗(0)<0 and i em (ii) ollows.

P oo o Theo em 3.4
P oo Le x≥0, T>0, τn:= in { ≥0Xx
≥n},n≥0, and τ∈T. Applying a change
o a iable o mula o semima ingales (see e.g. [4], Theo em 2.1) o {e−ρ w(Xx
), ∈
[0,τ
n∧τ∧T]}and hen aking expec a ions we ind:
Ee−ρ(τn∧τ∧T)w(Xx
τn∧τ∧T)=w(x)+Eτn∧τ∧T
0
e−ρs((L−ρ)w)(Xx
s)ds
+τn∧τ∧T
0
e−ρsw(Xx
s)dI
s,
whe e he B ownian-local ma ingale e m has anished in expec a ion since w∈
C1([0,∞);R)(by Sobole embedding) and because o he de ini ion o τn. Wi h ega ds
o he ac ha wsol es (3.15)and → I inc eases on { ≥0Xx
=0},weha e(a e
123
Ma hema ics and Financial Economics (2024) 18:707–733 725
ea anging e ms)
w(x)≤Ee−ρ(τn∧τ∧T) b∗(Xx
τn∧τ∧T).
As b∗is bounded (c . P oposi ion 3.3), by sending n↑∞and T↑∞, by he domina ed
con e gence heo em we ob ain
w(x)≤Ee−ρτ b∗(Xx
τ).
Gi en he a bi a iness o τ∈Tand x≥0, we ha e
w≤Fb∗on [0,∞).
Repea ing now he same a gumen s abo e, bu wi h τ eplaced by τ∗(b∗),we ind(byde i-
ni ion o τ∗(b∗))
w(x)=Ee−ρτ∗(b∗) b∗Xx
τ∗(b∗).
Hence,
w(x)≥in
τ∈T
Ee−ρτ b∗(Xx
τ)=Fb∗(x).
Gi en he a bi a iness o x≥0, we ha e w≥Fb∗, which, oge he wi h he p e iously
p o ed w≤Fb∗, implies ha w=Fb∗and ha τ∗(b∗)is op imal. 
P oo o P oposi ion 3.5
P oo S ep 1. We he e p o e exis ence and uniqueness o x∗
b∗.Using(3.17), we ew i e
p oblem (3.16) as ollows
⎧
⎨
⎩
C1γ−(1)+C2γ+(1)=0
C1eγ−(1)x+C2eγ+(1)x= b∗(x)
C1γ−(1)eγ−(1)x+C2γ+(1)eγ+(1)x= 
b∗(x),
(4.10)
whe e he explici exp ession o b∗is gi en in (4.9). The i s equa ion yields:
C2=−C1
γ−(1)
γ+(1),(4.11)
whe eas he second and hi d equa ion depend on he exp ession o b∗. We conside wo
dis inc cases.
(i) I 0 ≤x≤K, hen he second and he hi d equa ions o (4.10) become
⎧
⎨
⎩
C1γ−(1)eγ−(1)x−γ−(1)
γ+(1)eγ+(1)x=−γ−(1)(x−K)−γ−(1)
γ−(b∗)+eγ−(1)x
C1γ−(1)eγ−(1)x−eγ+(1)x=−1+eγ−(1)x.
(4.12)
Since γ−(1)= γ+(1), he p e ious sys em yields:
−γ−(1)(x−K)−γ−(1)
γ−(b∗)+eγ−(1)x
eγ−(1)x−γ−(1)
γ+(1)eγ+(1)x=−1+eγ−(1)x
eγ−(1)x−eγ+(1)x,
which can be ew i en as
F1(x)=F2(x), (4.13)
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726 Ma hema ics and Financial Economics (2024) 18:707–733
whe e
F1(x):= x−K+1
γ−(b∗)e−γ−(1)x−e−γ+(1)x
F2(x):= 1
γ−(1)1−e−γ+(1)x−1
γ+(1)1−e−γ−(1)x.
We ha e F1(0)=F2(0)=0and
F
1(x)=γ+(1)x−K+1
γ−(b∗)−1
γ+(1)e−γ+(1)x
−γ−(1)x−K+1
γ−(b∗)−1
γ−(1)e−γ−(1)x
F
2(x)=γ+(1)
γ−(1)e−γ+(1)x−γ−(1)
γ+(1)e−γ−(1)x.
F om (3.18) we no e ha
1
γ−(1)+1
γ+(1)=γ−(1)+γ+(1)
γ−(1)γ +(1)=λημ
ρ>0.
Consequen ly 1
γ−(b∗)<1
γ+(1)+1
γ−(1)and, o each x∈[0,K], i holds:
F
1(x)=γ+(1)x−K+1
γ−(b∗)−1
γ+(1)e−γ+(1)x
−γ−(1)x−K+1
γ−(b∗)−1
γ−(1)e−γ−(1)x
≤γ+(1)1
γ−(b∗)−1
γ+(1)e−γ+(1)x
−γ−(1)1
γ−(b∗)−1
γ−(1)e−γ−(1)x
<γ+(1)
γ−(1)e−γ+(1)x
−γ−(1)
γ+(1)e−γ−(1)x=F
2(x).
Hence, he unique solu ion o (4.13)in[0,K]is x=0, and (4.12)and(4.11) yield C1=
C2=0.
(ii) I x>K, hen he second and he hi d equa ion o (4.10) become
⎧
⎨
⎩
C1γ−(1)eγ−(1)x−γ−(1)
γ+(1)eγ+(1)x=−γ−(1)
γ−(b)eγ−(b∗)(x−K)+eγ−(1)x
C1γ−(1)eγ−(1)x−eγ+(1)x=−eγ−(b∗)(x−K)+eγ−(1)x.
(4.14)
Since γ−(1)= γ+(1), he p e ious sys em yields:
−γ−(1)
γ−(b∗)eγ−(b∗)(x−K)+eγ−(1)x
eγ−(1)x−γ−(1)
γ+(1)eγ+(1)x=−eγ−(b∗)(x−K)+eγ−(1)x
eγ−(1)x−eγ+(1)x,
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Ma hema ics and Financial Economics (2024) 18:707–733 727
which can be ew i en as (3.19); ha is,
(x)=D(K), (4.15)
whe e
(x):= γ+(1)(γ −(b∗)−γ−(1))e(γ −(b∗)−γ+(1))x−γ−(1)(γ −(b∗)−γ+(1))e(γ −(b∗)−γ−(1))x
and
D(K):= γ−(b∗)(γ +(1)−γ−(1))eγ−(b∗)K<0.
We ha e
(0)=γ−(b∗)(γ +(1)−γ−(1)) ≤γ−(b∗)(γ +(1)−γ−(1))eγ−(b∗)K=D(K).
and
(x)=(γ −(b∗)−γ−(1))(γ −(b∗)−γ+(1)) γ+(1)e−γ+(1)x−γ−(1)e−γ−(1)xeγ−(b∗)x>0.
Since is s ic ly inc easing and limx→+∞ (x)=0, (4.15)(and(3.19)) admi s in [0,+∞)
he unique posi i e solu ion x∗
b∗=x∗
b∗(K)=−1(D(K)) > 0. Fu he , since D(K)=
(γ −(b∗))2(γ +(1)−γ−(1))eγ−(b∗)K>0, hen x∗
b∗is s ic ly inc easing in K.
S ep 2. We show ha x∗
b∗≤ˆxb∗. Because
(ˆxb∗)=γ+(1)(γ −(b∗)−γ−(1))e
γ−(b∗)−γ+(1)
γ−(b∗)−γ−(1)γ−(b∗)K−γ−(1)(γ −(b∗)−γ+(1))eγ−(b∗)K
=γ+(1)(γ −(b∗)−γ−(1))e
γ−(1)−γ+(1)
γ−(b∗)−γ−(1)γ−(b∗)K−γ−(1)(γ −(b∗)−γ+(1))eγ−(b∗)K
≥γ+(1)(γ −(b∗)−γ−(1)) −γ−(1)(γ −(b∗)−γ+(1))eγ−(b∗)K
=γ−(b∗)(γ +(1)−γ−(1))eγ−(b∗)K=D(K),
hen he mono onici y o yields x∗
b∗≤ˆxb∗=γ−(b∗)
γ−(b∗)−γ−(1)K.
S ep 3. We now aim a p o ing ha x∗
b∗≥K.No ice ha
(K)=eγ−(b∗)Kγ+(1)(γ −(b∗)−γ−(1))e−γ+(1)K−γ−(1)(γ −(b∗)−γ+(1))e−γ−(1)K.
Le
S(K):= γ+(1)(γ −(b∗)−γ−(1))e−γ+(1)K−γ−(1)(γ −(b∗)−γ+(1))e−γ−(1)K,K≥0.
I holds S(0)=γ−(b∗)(γ +(1)−γ−(1)) < 0and
S(K)=−γ+(1)2(γ −(b∗)−γ−(1))e−γ+(1)K+γ−(1)2(γ −(b∗)−γ+(1))e−γ−(1)K
=−γ+(1)2(γ −(b∗)−γ−(1))e−γ−(1)Ke(γ −(1)−γ+(1))K−γ−(1)2γ−(b∗)−γ+(1)
γ+(1)2γ−(b∗)−γ−(1).
Since γ−(1)+γ+(1)<0, hen γ−(1)2(γ−(b∗)−γ+(1))
γ+(1)2(γ−(b∗)−γ−(1))>1. The e o e, S(K)<0 o each
K≥0, so ha
(K)=eγ−(b∗)KS(K)≤eγ−(b∗)KS(0)=γ−(b∗)(γ +(1)−γ−(1))eγ−(b∗)K=D(K)
and K≤x∗
b∗by mono onici y o .
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728 Ma hema ics and Financial Economics (2024) 18:707–733
S ep 4. Finally, using (4.14)and(4.11)wege C1=1
γ−(1)Band C2=1
γ+(1)B,whe e
B=eγ−(b∗)(x∗
b∗−K)−eγ−(1)x∗
b∗
eγ+(1)x∗
b∗−eγ−(1)x∗
b∗.(4.16)
We ew i e (3.19) (o equi alen ly (x)=D(K)) as ollows:
eγ−(b∗)(x∗
b∗−K)H(x∗
b∗)=γ−(b∗)γ+(1)−γ−(1)e(γ −(1)+γ+(1))x∗
b∗,(4.17)
whe e H(x∗
b∗)isgi enin(3.23). Using (4.17)in(4.16)wege
B=eγ−(b∗)(x∗
b∗−K)−eγ−(1)x∗
b∗
eγ+(1)x∗
b∗−eγ−(1)x∗
b∗
=γ−(b∗)γ+(1)−γ−(1)eγ+(1)x∗
b∗
H(x∗
b∗)−1eγ−(1)x∗
b∗
eγ+(1)x∗
b∗−eγ−(1)x∗
b∗
=γ+(1)(γ −(b)−γ−(1))(eγ+(1)x∗
b∗−eγ−(1)x∗
b∗)
H(x∗
b∗)
eγ−(1)x∗
b∗
eγ+(1)x∗
b∗−eγ−(1)x∗
b∗
=γ+(1)(γ −(b)−γ−(1))eγ−(1)x∗
b∗
H(x∗
b∗)=: B(x∗
b∗)
asgi enby(3.22). Since γ−(b)<γ
−(1)and H<0, hen B(x∗
b∗)>0; u he , H<
γ+(1)(γ −(b)−γ−(1))eγ−(1)x∗
b∗and he e o e B(x∗
b∗)<1. 
P oo o Theo em 3.6
P oo In o de o p o e ha w≡Fb∗, we need o show ha :
w(x)≤ b∗(x), ∀x>0
wis s. . 0 ≤(L−ρ)w(x), ∀x>0,(4.18)
which is implied by he ollowing wo condi ions:
(i) w(x)≤ b∗(x), ∀0<x≤x∗
b∗
(ii) 0 ≤(L−ρ)w(x), ∀x>x∗
b∗.
We s a p o ing (i). By P oposi ion 3.5
w(x)=B(x∗
b∗)1
γ−(1)eγ−(1)x−1
γ+(1)eγ+(1)x,
whe e B(x∗
b∗)isgi enby(3.22). We compu e w(0)=B(x∗
b∗)1
γ−(1)−1
γ+(1)and b∗(0)=
K−1
γ−(b∗)+1
γ−(1),so ha w(0)≤ b∗(0)i and only i
K≥B(x∗
b∗)1
γ−(1)−1
γ+(1)+1
γ−(b∗)−1
γ−(1)
=(γ −(b∗)−γ+(1))(γ −(b∗)−γ−(1))
γ−(b∗)
eγ+(1)x∗
b∗−eγ−(1)x∗
b∗
H(x∗
b∗)=Q(x∗
b∗),
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Ma hema ics and Financial Economics (2024) 18:707–733 729
whe e, o x>0,
Q(x):= (γ −(b∗)−γ+(1))(γ −(b∗)−γ−(1))
γ−(b∗)
eγ+(1)x−eγ−(1)
H(x),
and (c . (3.23))
H(x):= γ+(1)γ−(b∗)−γ−(1)eγ−(1)x−γ−(1)γ−(b∗)−γ+(1)eγ+(1)x<0.
By some compu a ions we ha e
Q(x)=(γ −(b∗)−γ+(1))(γ −(b∗)−γ−(1))(γ +(1)−γ−(1))2e(γ +(1)+γ−(1))x
(H(x))2>0.
Recalling ha x∗
b∗=x∗
b∗(K)=−1(D(K)) (see S ep 1 in he p oo o P oposi ion (3.5)), i
holds Q(x∗
b∗(0)) =Q(0)=0. Fu he ,
∂Q(x∗
b∗(K))
∂K=Q(x∗
b∗(K)) ·(x∗
b∗)(K).
Since x∗
b∗is s ic ly inc easing in Ki ollows ha Q(x∗
b∗(K)) is s ic ly inc easing in Kand
∂Q(x∗
b∗(0))
∂K=Q(0)·(x∗
b∗)(0)
=(γ −(b∗)−γ+(1))(γ −(b∗)−γ−(1))
γ−(b∗)2·γ−(b∗)2
(γ −(b∗)−γ−(1))(γ −(b∗)−γ+(1)) =1.
In o de o s udy he conca i y o K→ Q(x∗
b∗(K)), we compu e
∂2Q(x∗
b∗(K))
∂K2=Q(x∗
b∗(K)) ·(x∗
b∗)(K)2+Q(x∗
b∗(K)) ·(x∗
b∗)(K). (4.19)
We ind
Q(x)=(γ −(b∗)−γ+(1))(γ −(b∗)−γ−(1))(γ +(1)−γ−(1))3e(γ +(1)+γ−(1))x
(H(x))3A(x)
whe e
A(x):= γ+(1)(γ −(b∗)−γ−(1))eγ−(1)x+γ−(1)(γ −(b∗)−γ+(1))eγ+(1)x.
Since γ+(1)+γ−(1)=−2ημ
σ2<0, hen A(x∗
b∗(0)) =A(0)=γ−(b∗)(γ +(1)+γ−(1)) −
2γ−(1)γ +(1)>0; u he ,
A(x)=γ+(1)γ −(1)(γ −(b∗)−γ−(1))eγ−(1)x+(γ −(b∗)−γ+(1))eγ+(1)x>0,
he e o e A(x)≥0 o each x≥0, and consequen ly, since H(x)<0 o each x≥0, i
holds Q(x)≤0. In pa icula ,
Q(x∗
b∗(K)) ≤0, o each K≥0.(4.20)
Using ha x∗
b∗=x∗
b∗(K)is he unique solu ion o he equa ion (x,K)=(x)−D(K)=0
(see again s ep 1 o he p oo o P oposi ion 3.5), we ha e (see [3]) (x∗
b∗)(K)=
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730 Ma hema ics and Financial Economics (2024) 18:707–733
∂
∂K
∂2
∂K∂x−∂
∂x
∂2
∂K2/∂
∂x2
, so ha he sign o (x∗
b∗)(K)coincides wi h he sign
o he ollowing quan i y:
∂
∂K
∂2
∂K∂x−∂
∂x
∂2
∂K2=(x)·D(K)<0.(4.21)
Using (4.19), (4.20)and(4.21), we ge ∂2Q(x∗
b∗(K)
∂K2<0; consequen ly, K≥Q(x∗
b∗(K)),and
w(0)≤ b∗(0).Fu he ,wis nega i e and, because
w(x)=Beγ−(1)x−eγ+(1)x<0,
wis s ic ly dec easing. We ecall by P oposi ion 3.3 ha b∗is s ic ly dec easing in [0,ˆxb∗].
I x∈[0,K] hen
w(x)=Beγ−(1)x−eγ+(1)x<eγ−(1)x−eγ+(1)x<eγ−(1)x−1= 
b∗(x).
Consequen ly, i holds w(x)≤ b∗(x) o each x∈[0,K]. On he o he hand, since
x∗
b∗>K>γ−(b)K
γ−(b)−γ+(1), hen o each x∈[K,x∗
b∗]
w(x)=Beγ−(1)x−eγ+(1)x<eγ−(1)x−eγ+(1)x<eγ−(1)x−eγ−(b)(x−K)= 
b∗(x),
which, coupled wi h w(K)< b∗(K), yields w(x)≤ b∗(x) o each x∈[K,x∗
b∗].
We now p o e (ii). Fo each x>x∗
b∗>Kwe ha e
(L−ρ) b∗(x)=(L−ρ)−1
γ−(b∗)eγ−(b∗)(x−K)+1
γ−(1)eγ−(1)x
=− 1
γ−(b∗)eγ−(b∗)(x−K)1
2λσ2(γ −(b∗))2+λ(θμ −(θ −η)μ)γ −(b∗)−ρ
=− 1
γ−(b∗)eγ−(b∗)(x−K)(1,γ−(b∗)). (4.22)
Since γ−(1)is he unique nega i e solu ion o (1,γ) =0 and because γ−(b∗)<γ
−(1),
hen (1,γ−(b)) > 0 and, by (4.22), (L−ρ) b∗(x)>0 o each x>x∗
b∗.
P oo o P oposi ion 4.1
P oo F om (4.2), by some compu a ions, he de i a i e (γ −)o γ−is
(γ −)(b)=−γ−(b)φ(b)
D(b),(4.23)
whe e φis gi en by (4.6)and
D(b)=σ2b2γ−(b)+μ(θb−(θ −η)). (4.24)
The symme ic axis o he pa abola (b,γ)=0(see(4.2)) is γ(b)=−μ(θb−(θ −η))
σ2b2;as
a consequence, since γ−(b)is he nega i e solu ion o he equa ion (b,γ)=0, i ollows
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