Behavior of risk processes with random premiums after ruin and a multivariate ruin function
We establish relations for the distribution of functionals associated with the behavior of a risk process with random premiums after ruin and for a multivariate ruin function.
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Institute of Mathematics, NAS of Ukraine
2007
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509489819549696 |
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| author | Gusak, D. V. Гусак, Д. В. |
| author_facet | Gusak, D. V. Гусак, Д. В. |
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| description | We establish relations for the distribution of functionals associated with the behavior of a risk process with random premiums after ruin and for a multivariate ruin function. |
| first_indexed | 2026-03-24T02:41:55Z |
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UDK 519.21
D. V. Husak (In-t matematyky NAN Ukra]ny, Ky]v)
POVEDINKA PROCESIV RYZYKU
Z VYPADKOVYMY PREMIQMY PISLQ BANKRUTSTVA
TA BAHATOZNAÇNA FUNKCIQ BANKRUTSTVA
∗∗∗∗
For the distribution of functionals connected with the behavior of the risk process with stochastic
premiums after the ruin time and for the multivariate ruin function, relations are established.
Ustanovlen¥ sootnoßenyq dlq raspredelenyq funkcyonalov, svqzann¥x s povedenyem processa
ryska so sluçajn¥my premyqmy posle razorenyq, y dlq mnohoznaçnoj funkcyy razorenyq.
U roboti [1] vyvçalys\ pytannq pro bahatoznaçnu funkcig bankrutstva ta pro
povedinku klasyçnoho procesu ryzyku z linijnog funkci[g premij pislq bank-
rutstva. V danij roboti budemo rozhlqdaty taki Ω pytannq dlq nadlyßkovoho
sxidçastoho procesu ryzyku z dovil\no rozpodilenymy vymohamy { }′ ≥ξk k 1 ta z
vypadkovymy premiqmy { }′′ ≥ξk k 1.
Holovna meta ci[] roboty polqha[ u vstanovlenni spivvidnoßen\ dlq wil\-
nosti bahatoznaçno] funkci] bankrutstva ta dlq povedinky sxidçastoho procesu
ryzyku ζ ( t ) z pokaznykovo rozpodilenymy premiqmy pislq bankrutstva. Qk i v
[1], pry vstanovlenni spivvidnoßen\ dlq wil\nosti funkci] bankrutstva ta
inßyx xarakterystyk, wo opysugt\ povedinku rozhlqduvanoho procesu ryzyku
ζ ( t ) pislq bankrutstva, budemo vykorystovuvaty otrymani v [2 – 5] rezul\taty
dlq rozpodilu perestrybkovyx funkcionaliv.
Slid zaznaçyty, wo dlq procesu ryzyku z vypadkovymy premiqmy (oznaçennq
qkoho moΩna znajty v [6]) ryzykovi xarakterystyky, qk i dlq klasyçnoho proce-
su ryzyku, opysugt\sq rozpodilamy abo heneratrysamy vidpovidnyx hranyçnyx
funkcionaliv, poznaçennq qkyx podibni do poznaçen\ v [1] i navedeni nyΩçe. Za-
uvaΩymo takoΩ, wo rizni pytannq, pov’qzani z vyvçennqm povedinky procesiv ry-
zyku do i pislq bankrutstva, doslidΩuvalys\ u robotax [7 – 15].
Rozhlqnemo nadlyßkovyj proces ryzyku z vymohamy { }′ ≥ξk k 1 ta premiqmy
{ }′′ ≥ξk k 1:
ζ ( t ) = S ( t ) – C ( t ) , S ( t ) = ′
≤
∑ ξ
ν
k
k t1( )
, C ( t ) = ′′
≤
∑ ξ
ν
k
k t2 ( )
, (1)
de ν1 2, ( )t — nezaleΩni prosti puassonivs\ki procesy z intensyvnostqmy λ1 2, >
0,
P{ }′ <ξk x = F x1( ), x > 0, P{ }′′ >ξk x = e bx− , x > 0, b > 0.
{ }( ), , ( )ζ ζt t ≥ <0 0 0 moΩna rozhlqdaty qk skladnyj puassonivs\kyj proces iz
sumarnog intensyvnistg λ = λ1 + λ2 i strybkamy ξk dovil\noho znaku,
ξk =̇
− ′′ =
′ = −
ξ λ
λ
ξ
k
k
q
p q
z imovirnistg
z imovirnistg
2
1
,
,
F ( x ) = P{ }ξk x< = qe I x q pF x I xbx { } { }( ( ))< + + >0 01 , (2)
Eer tζ( ) = etk r( ) , k ( r ) = λ ϕq r
b r
p r
+
+ − −
( ( ))1 1 , ϕ1( )r = Ee r− ′ξ1 .
∗
Vykonano pry çastkovij pidtrymci Deutsche Forschungsgemeinschaft.
© D. V. HUSAK, 2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 11 1473
1474 D. V. HUSAK
Rezervnyj proces ryzyku ξu t( ) = R tu( ) = u t− ζ( ) [ analohom klasyçnoho
procesu ryzyku, ξu ( t ) = u + C ( t ) – S ( t ) — majΩe napivneperervnyj zverxu
sxidçastyj proces, ζ ( t ) — majΩe napivneperervnyj znyzu. Dlq porivnqnnq vve-
demo (qk i v [1]) poznaçennq osnovnyx doslidΩuvanyx funkcionaliv dlq ξu t( )
ta ζ ( t ) :
τ ( u ) = inf{ }: ( )t R tu < 0 , τ
+
( u ) = inf{ }: ( )t t uζ > ,
Y
+
( u ) = – R uu( ( ))τ , γ
+
( u ) = ζ τ( ( ))+ −u u ,
X
+
( u ) = R uu( ( ) )τ − 0 , γ + ( u ) = u u− −+ζ τ( ( ) )0 ,
X u Y u+ ++( ) ( ) = R u R uu u( ( ) ) ( ( ))τ τ− −0 , γ u
+ = γ γ+
+−( ) ( )u u .
Krim toho, poznaçymo ′τ ( )u = inf{ }( ), ( )t u tu> >τ ζ 0 ,
′T u( ) =
′ − < ∞
= ∞
τ τ τ
τ
( ) ( ), ( ) ,
, ( ) ,
u u u
u0
Z u+ ( ) = sup ( )
( )τ
ζ
+ ≤ <∞u t
t , Z u1
+ ( ) = sup ( )
( ) ( )τ τ
ζ
+ ≤ < ′u t u
t , ζ+ = sup ( )
0≤ <∞t
tζ .
′T u( ) nazyva[t\sq „çervonym periodom”, Z u+ ( ) — total\nyj maksymum deficy-
tu, Z u1
+ ( ) — maksymum deficytu za period ′T u( ), ζ± ( )t = sup (inf) ( )0≤ ≤ ′≤ ′t t t tζ
— ekstremumy ζ( )′t na intervali [ 0, t ] .
Spoçatku nahada[mo deqki rezul\taty j poznaçennq dlq heneratrys pere-
strybkovyx funkcionaliv { }( ), ( ), ( ),τ γ γ γ+ +
+
+u u u u ta par { }( ), ( )τ γ+ x xk , k =
= 1 3, , de γ + ( )u = γ1( )u , γ + ( )u = γ 2( )u , γ u
+ = γ 3( )u :
Φs
k u x( )( , ) = ∂
∂
>+
x
u u xs kP{ }( ) , ( ),ζ θ γ ,
V s u u u u( , , , , )1 2 3 = E e u
s u u uk kk− − +
+
=∑ < ∞
τ γ τ( ) ( )
, ( )1
3
=
=
0 0 0
1 2 3 1 2 3
1
3∞ ∞ ∞
−∫ ∫ ∫ =∑e u x x x dx dx dx
u x
s
k kk Φ ( , , , ) ,
Φs u x y z( , , , ) = ∂
∂ ∂ ∂
> < < <+
3
P
x y z
u u x u y u zs{ }( ) , ( ) , ( ) , ( )ζ θ γ γ γ1 2 3 ,
V s u uk k( , , ) = E e us u u uk k− − ++
< ∞[ ]τ γ τ( ) ( ), ( ) =
0
∞
−∫ e u x dxu x
s
kk Φ( )( , ) , k = 1 3, .
Zhidno z rezul\tatamy [5, 15] dlq sxidçastyx puassonivs\kyx procesiv osnovni
funkci] G s x u u u( , , , , )1 2 3 ta G s x uk k( , , ), çerez qki vidpovidno vyraΩagt\sq
heneratrysy V ta Vk , dewo uskladnggt\sq v porivnqnni z podibnymy funkciq-
my dlq napivneperervnyx puassonivs\kyx procesiv. Ce poqsng[t\sq tym, wo dlq
sxidçastyx procesiv obydvi atomarni jmovirnosti [ dodatnymy:
p s±( ) = P{ }( )ζ θ± =s 0 > 0, p s p s+ −( ) ( ) = s
s + λ
> 0, s > 0,
de θs — pokaznykovo rozpodilena vypadkova velyçyna P{ }θs t> = e st− , s > 0,
t ≥ 0. Dlq procesu (1) sam rozpodil ζ θ− ( )s vyznaça[t\sq vid’[mnym korenem
rivnqnnq Lundberha
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 11
POVEDINKA PROCESIV RYZYKU Z VYPADKOVYMY PREMIQMY … 1475
k rs( ) = s, rs = – ρ−( )s < 0, s > 0,
P s x−( , ) = P{ }( )ζ θ− <s x = q s e s x
−
−( ) ( )ρ , x < 0,
q s−( ) = 1 − −p s( ), ρ−( )s = b p s−( ) .
Tomu u zhadanyx vywe funkcij G ta Gk (qki [ zhortkamy pravyx çastyn inteh-
ral\nyx rivnqn\ dlq V ta Vk z rozpodilom P s x−( , ) ) vynykagt\ novi atomarni
dodanky. Zokrema,
G s x u u u( , , , , )1 2 3 =
−∞
− −∫
0
1 2 3A u u u dP s yx y ( , , ) ( , ) =
= p s A u u u q s A u u u dex x y
s y
− −
−∞
−
−+ ∫ −( ) ( , , ) ( ) ( , , ) ( )
1 2 3
0
1 2 3
ρ ,
A u u ux( , , )1 2 3 = λ
x
u u x u u ze dF z
∞
− − +∫ ( ) ( ) ( )1 2 1 3 , x > 0.
Vidpovidni atomarni dodanky vynykagt\ u G s x u u( , , , )1 2 = G s x u u( , , , , )1 2 0 ,
G s u ui i( , , ) = G s u u u u u r i ir
( , , , , ) , ,1 2 3 0 1 3= ∀ ≠ = .
Lema 1. Funkciq G s u u u( , , , )1 2 z vidpovidnym atomarnym dodankom zvo-
dyt\sq do vyhlqdu
G s u u u( , , , )1 2 = λ p s e e dF u zu u u z
−
−
∞
−∫ +( ) ( )2 1
0
+
+
λ ρ
ρ
ρq s s
s u u
e e e F u y dyu u u y s u y− −
−
−
∞
− − +
− +
−[ ] +∫ −( ) ( )
( )
( )( ( ) )
1 2 0
2 1 2 (3)
i dopuska[ obernennq (z poxidnog ′ <I y u{ } ≈ I y u{ }= v sensi Ívarca)
g s u x y( , , , ) = λ λ ρ ρp s F u x I y u q s s e F x y I y us u y
− − −
−′ + = + ′ + >−( ) ( ) ( ) ( ) ( ){ } { }( )( ) ,
(4)
funkci] zhortky G s u uk k( , , ) zvodqt\sq do vyhlqdu
G s u u1 1( , , ) = λ p s e F u y dyu y
−
∞
−∫ ′ +( ) ( )
0
1 +
+
λ ρ
ρ
ρq s s
s u
e e dF u yu y s y− −
−
∞
− −
−
−[ ] +∫ −( ) ( )
( )
( )( )
1 0
1 ,
G s u u2 2( , , ) = λ ρp s e F u bq s e F y dyu u
u
s u y u y
−
−
−
∞
− −+
∫ −( ) ( ) ( ) ( )( )( )2 2 , (5)
G s u u3 3( , , ) = λ λ ρ ρ
u
u z s u
u
u s ze dF z q s e e dF z
∞
−
−
∞
− +∫ ∫− − −3 3( ) ( ) ( )( ) ( ( ))
i dopuskagt\ obernennq po ui , i = 1 3, :
g s u x1( , , ) = λ λρ ρp s F u x q s s e dF u z
x
s x z
− − −
∞
−′ + + +∫ −( ) ( ) ( ) ( ) ( )( )( ) ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 11
1476 D. V. HUSAK
g s u y2( , , ) = λ λρ ρp s F u I y u q s s e F y I y us u y
− − −
−= + >−( ) ( ) ( ) ( ) ( ){ } { }( )( ) , (6)
g s u z3( , , ) = λ ρ′ −[ ] >−
−−F z q s e I z us u z( ) ( ) ( )( ) { }1 .
Qkwo m = Eζ( )1 = λ µ( )p qb− −1 < 0 ( )µ ξ= ′E k , to isnugt\ poxidni pry
s → 0
′g u x y( , , , )0 = λ λ
b m
F x u I y u
m
F x y I y u′ + = + ′ + >( ) ( ){ } { },
′g u x1 0( , , ) = λ λ
b m
F u x
m
F u x′ + + +( ) ( ),
(7)
′g u y2 0( , , ) = λ λ
b m
F u I y u
m
F y I y u( ) ( ){ } { }= + > ,
′g u z3 0( , , ) = λ
m
F z b z u I z u′ + − >−( )( ) { }1 .
Dovedennq. Funkciq G s u u u( , , , )1 2 v (3) z atomarnym dodankom oberta[t\sq
po u1 2, tak samo, qk i analohiçna funkciq bez atomarnoho dodanka v [1]. Analo-
hiçno obertagt\sq funkci] G s u ui i( , , ) , i = 1 2, , po ui . ZobraΩennq dlq
G s u u3 3( , , ) vyplyva[ iz spivvidnoßennq
G s u u3 3( , , ) =
−∞
− −∫
0
30 0A u dP s yx y( , , ) ( , ) =
= p s A u q s s A u e dyx x y
s y
− − −
−∞
−
−+ ∫ −( ) ( , , ) ( ) ( ) ( , , ) ( )0 0 0 03
0
3ρ ρ
pislq sprowennq dodanka z podvijnym intehralom
ρ ρ
−
−∞
−
−
∞
−∫ ∫ −( ) ( ) ( )s e dF y e dz
x z
u y s z
0
3 =
x
u y
x y
s ze dF y s e dz
∞
−
−
∞
−∫ ∫ −3 ( ) ( ) ( )ρ ρ =
=
x
u y s x ye e dF y
∞
− −∫ − −3 1( )( )( ) ( )ρ ,
G s u u3 3( , , ) tak samo lehko obernuty po u3. Poqva u spivvidnoßennqx dlq ober-
nen\ g s u x y( , , , ) ta g s u y2( , , ) dodanka z indykatorom I y u{ }= poqsng[t\sq
tym, wo pry umovi peretynu procesom krytyçnoho rivnq u (za dopomohog per-
ßoho strybka) nedostrybok γ 2( )u = γ +( )u = u nabuva[ fiksovanoho znaçennq z
dodatnog jmovirnistg. Tomu u nastupnyx tverdΩennqx (analohax teoremyJ1 ta
naslidkuJ1 v [1]) vynykagt\ novi spivvidnoßennq „atomarnoho” typu z indykato-
rom I y u{ }= .
Lemu dovedeno.
Teorema 1. Nexaj ζ ( t ) — majΩe napivneperervnyj znyzu proces ryzyku z vy-
padkovymy premiqmy. Todi spil\ni heneratrysy { }( ), ( ), ,τ γ+ =u u kk 1 3 vyzna-
çagt\sq spivvidnoßennqmy, podibnymy do (3), (11) v [1],
sV s u u u u( , , , , )1 2 3 =
0
1 2 3
u
G s u y u u u dP s y∫ − +( , , , , ) ( , ), (8)
de zhortka G s u u u u( , , , , )1 2 3 ma[ atomarnu skladovu
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 11
POVEDINKA PROCESIV RYZYKU Z VYPADKOVYMY PREMIQMY … 1477
G s u u u u( , , , , )1 2 3 = p s A u u u q s s A u u u e dyu u y
s y
− − −
−∞
−+ ∫ −( ) ( , , ) ( ) ( ) ( , , ) ( )
1 2 3
0
1 2 3ρ ρ .
Za lemogJ1 funkci] G s u u u( , , , )1 2 ta G s u uk k( , , ) dopuskagt\ obernennq po
uk , k = 1 3, . Za dopomohog obernennq g s u x y( , , , ) (dyv. (4)) vyznaça[t\sq
wil\nist\ bahatoznaçno] funkci] bankrutstva pry y ≠ u ta atomarne
spivvidnoßennq pry y = u :
s
x y
u u x u ys
∂
∂ ∂
> < <+
2
1 2P{ }( ) , ( ) , ( )ζ θ γ γ = p s g s u x y I y u+ >( ) ( , , , ) { } +
+
0
0
+
+∫ − < <
u
g s u z x y dP s z I y u( , , , ) ( , ) { }, (9)
∂
∂
> < = =+
x
u u x u y I y usP{ }( ) , ( ) , ( ) { }ζ θ γ γ1 2 =
λ
λ
p
s
F u x
+
′ +1( ) .
Çerez g s u xk k( , , ) (dyv. (6)) vyznaçagt\sq marhinal\ni wil\nosti funkci] bank-
rutstva
Φs
k u x( )( , ) = ∂
∂
< >+
x
u x uk sP{ }( ) , ( )γ ζ θ =
= p s g s u x g s u z x dP s zk
u
k+
+
++ −∫( ) ( , , ) ( , , ) ( , )
0
, k = 1 3, . (10)
Pry c\omu dlq γ 2( )u = γ +( )u vykonugt\sq atomarni spivvidnoßennq
P{ }( ) , ( )ζ θ γ+ > =s u u u2 =
λ
λ
p
s
F u
+ 1( ) , (11)
P{ }, ( )ζ γ+ > =u u u2 = pF u1( ), P u+ ( ) = P{ }ζ+ < u , u > 0, m < 0.
Dovedennq teoremy bazu[t\sq na obernenni vidpovidnyx spivvidnoßen\ dlq
V s u u u( , , , )1 2 = V s u u u( , , , , )1 2 0 ta V s u uk k( , , ), wo vyplyvagt\ iz (8). Naqvnist\
indykatorno] skladovo] z I y u{ }= u g s u x y( , , , ) ta g s u y2( , , ) obumovlg[
poqvu dodatkovyx atomarnyx spivvidnoßenn\ dlq par { }( ), ( )ζ θ γ+
s u2 ta
{ }, ( )ζ γ+
2 u pry m < 0.
Naslidok 1. Dlq sxidçastoho nadlyßkovoho procesu ryzyku (dyv. (1)) wil\-
nist\ skladno] bahatoznaçno] funkci] bankrutstva vyznaça[t\sq dvo]stym spiv-
vidnoßennqm pry y ≠ u ta atomarnym pry y = u, x > 0:
s u x ysΦ ( , , ) = s
x y
u u x u ys
∂
∂ ∂
> < <+ +
+
2
P{ }( ) , ( ) , ( )ζ θ γ γ =
=
λ ρ
λρ
ρ
ρ
q s s F x y e dP s z y u
s F x y q s e dP s z
b
P s u y y u
u
s u z y
u y
u
s u z y
− −
−
− −
+
− −
−
− −
+ +
′ + >
′ + + ′ −
< <
∫
∫
−
−
( ) ( ) ( ) ( , ), ,
( ) ( ) ( ) ( , ) ( , ) , ,
( )( )
( )( )
0
1 0
(12)
∂
∂
> < =+ +
+x
u u x u ysP{ }( ) , ( ) , ( )ζ θ γ γ = λ
λs
F u x
+
′ +( ), y = u
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 11
1478 D. V. HUSAK
Qkwo m = Eζ( )1 = λ µ( )p qb− −1 < 0, to
∂
∂ ∂
> < <+ +
+
2
x y
u u x u yP{ }, ( ) , ( )ζ γ γ =
=
λ ζ
λ ζ ζ
m
F x y u y u
m
F x y u y u
b u
u y y u
′ + < >
′ + − < < + ∂
∂
< −
< <
+
+ +
( ) , ,
( ) , ,
{ }
{ } { }
P
P P1 0
(13)
∂
∂
> < =+ +
+x
u u x u yP{ }, ( ) , ( )ζ γ γ = ′ +F u x( ), y = u
.
Intehruvannqm (9) poJx
Jvstanovlggt\sq spivvidnoßennq F u dF x u
u
( ) ( ),= >
∞
∫ 0
s
y
u u x u ys
∂
∂
> > <+ +
+P{ }( ) , ( ) , ( )ζ θ γ γ =
=
λ ρ
λρ
ρ
ρ
q s s F x y e dP s z y u
s F x y q s e dP s z
b
P s u y y u
u
s u z y
u y
u
s u y z
− −
−
− −
+
− −
−
− −
+ +
+ >
+ + ′ −
< <
∫
∫
−
−
( ) ( ) ( ) ( , ), ,
( ) ( ) ( ) ( , ) ( , ) , ,
( )( )
( )( )
0
1 0
P{ }( ) , ( ) , ( )ζ θ γ γ+ +
+> > =s u u x u u = λ
λs
F u x
+
+( ), y = u
, (14)
∂
∂
> > <+ +
+y
u u x u yP{ }, ( ) , ( )ζ γ γ =
=
λ ζ
λ ζ
m
F x y u y u F x pF x x
m
F x y u y u
b u
P u y y u
( ) , , ( ) ( ), ,
( ) ( ) , .
{ }
{ }
+ < > = >
+ − < < + ∂
∂
−
< <
+
+
+
P
P
1 0
1 0
Wil\nist\ rozpodilu perßo] marhinal\no] funkci] bankrutstva vyznaça[t\sq
spivvidnoßennqmy (pry s > 0 ta dlq s = 0 pry m = Eζ( )1 < 0 )
Φs u x( )( , )1 = : ∂
∂
> <+ +
x
u u xsP{ }( ) , ( )ζ θ γ =
= λ
λ
λ
λ
ρ
+
′ + +
+
′ +−
∞
−∫ −
s
F u x q s b
s
e F u z dz
x
s x z( ) ( ) ( )( )( ) +
+ s p s F u y x dP s y q s b e F u y z dzdP s y
u u
x
s x z−
− + −
∞
−
+∫ ∫ ∫′ − + + ′ − +
−1
0 0
λ ρ( ) ( ) ( , ) ( ) ( ) ( , )( )( ) ,
(15)
Φ0
1( )( , )u x = ∂
∂
> <+ +
x
u u xP{ }, ( )ζ γ = ′ + + +F u x bF u x( ) ( ) +
+ λ ′ ′ − + + + −
− + +∫ ∫p F u y x dP y b F u x y dP y
u u
( ) ( ) ( ) ( ) ( )0
0 0
, ′−p ( )0 = 1
b m
.
Wil\nist\ druho] marhinal\no] funkci] bankrutstva pry y ≠ u ma[ vyhlqd
(pry y = u (dyv. (11))
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 11
POVEDINKA PROCESIV RYZYKU Z VYPADKOVYMY PREMIQMY … 1479
s u ysΦ( )( , )2 = : ∂
∂
> <+
+y
u u ysP{ }( ) , ( )ζ θ γ =
=
λ ρ
λρ
ρ
ρ
q s s F y e dP s z y u
s F y q s e dP s z
b
P s u y y u
u
s u z y
u y
u
s u y z
− −
−
− −
+
− −
−
− −
+ +
∫
∫
−
−
>
+ ′ −
< <
( ) ( ) ( ) ( , ), ,
( ) ( ) ( ) ( , ) ( , ) , ,
( )( )
( )( )
0
1 0
(16)
Φ0
2( )( , )u y =
λ
λ ζ
m
F y P u y u
m
F y u y u
b u
P u y y u
( ) ( ), ,
( ) ( ) , .{ }
+
+
+
>
− < < + ∂
∂
−
< <
P 1 0
Wil\nist\ vymohy, wo spryçynyla bankrutstvo ( pry s > 0 ta s = 0 ), ma[
vyhlqd
s u zsΦ( )( , )3 =
λ
λ
ρ
ρ
′ −[ ] >
′ −[ ] < <
∫
∫
−
− −
+
−
−
−
− −
+
−
−
F z q s e dP s y z u
s F z q s e dP s y z u
u
s u z y
u z
u
s u y z
( ) ( ) ( , ), ,
( ) ( ) ( , ), ,
( )( )
( )( )
0
1
1
1 0
(17)
Φ0
3( )( , )u z = λ
m
F z z y u
b
dP y I z u
u
′ + − +
>∫ +( ) ( ) { }
0
1 +
+ λ
m
F z z y
b
dP y u I z u
z
′ + +
− < <∫ +( ) ( ) { }
0
1 0 .
Heneratrysy { }( ), ( )τ γ+ 0 0k , k = 1, 2, 3, vyznaçagt\sq spivvidnoßennqmy
E e xs− + ++
> < ∞[ ]τ γ τ( ), ( ) , ( )0 0 0 = λ
λ
ρ
s
F x bq s e F y dy
x
x y s
+
+
−
∞
−∫ −( ) ( ) ( )( ) ( ) ,
E e ys−
+
++
> < ∞[ ]τ γ τ( ), ( ) , ( )0 0 0 =
b
s
q s e F x dx
y
x sλ
λ
ρ
+ −
∞
−∫ −( ) ( )( ) , (18)
E e zs− + ++
> < ∞[ ]τ γ τ( ), , ( )0
0 0 = λ
λ
ρ
s
F z bq s e dF x
z
s x
+
+ −( )
−
∞
−∫ −( ) ( ) ( )( )1 ,
qki uzhodΩugt\sq z (15) – (17) pry u → 0.
Dovedennq. Pry dovedenni (12) slid vraxuvaty, wo
0
u
I y u z dP s z∫ ′ < − +{ } ( , ) = p s I y u I z u y dP s z
u
+
+
+= + ′ < −∫( ) ( , ){ } { }
0
=
= p s I y u P s u y I y u+ += + ′ − < <( ) ( , ){ } { }0 .
Pislq pidstanovky g ( s , u, x, y ) iz (4) v (9) oderΩymo spivvidnoßennq (14) pry
s > 0 ta pry s → 0 ( m < 0 ) . Z (14) lehko oderΩaty marhinal\ni funkci]
bankrutstva (15), (16) dlq γ1 2, ( )u . Heneratrysa dlq { }( ), ( )τ γ+ u u3
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 11
1480 D. V. HUSAK
E e us u u u− − ++
< ∞[ ]τ γ τ( ) ( ), ( )3 3 = p s G s u u G s u y u dP s y
u
+
+
++ −∫( ) ( , , ) ( , , ) ( , )3 3
0
3 3
dopuska[ obernennq po u3 pry s > 0 ta s = 0:
s u zsΦ( )( , )3 = p s g s u z g s u y z dP s y
u
+
+
++ −∫( ) ( , , ) ( , , ) ( , )3
0
3 ,
Φ0
3( )( , )u z = p g u z g s u y z dP s y
u
+
+
+′ + ′ −∫3
0
30( , , ) ( , , ) ( , ).
Pislq pidstanovky g s u z3( , , ) ta ′g u z3 0( , , ) iz (6) z ostannix spivvidnoßen\ vy-
plyva[ (17). Pry c\omu slid vraxuvaty dvo]stist\ zobraΩennq skladovo] çastyny
zhortky g3 z dP s y+ ( , )
0
3
u
g s u y z dP s y∫ − +( , , ) ( , ) =
=
0
u
I y u z dP s y∫ … > − +{ } ( , ) = 0
0
u
u z
u
dP s y z u
dP s y z u
∫
∫
… >
… < <
+
−
+
( , ), ,
( , ), .
Oçevydno, wo marhinal\ni funkci] bankrutstva vyznaçagt\sq spivvidnoßennqmy
P{ },ζ γ+ > >u zk k =
z
k
k
u z dz
∞
∫ Φ0
( )( , ) , k = 1 3, .
Spivvidnoßennq (18) vyplyvagt\ z (41) v [1], qkwo vraxuvaty, wo Π ( y ) =
= λ F y( ), y > 0. Z (18) pry x, y → 0 vyplyvagt\ spivvidnoßennq dlq γ + ( )0 ,
γ + ( )0 :
P{ }( ) , ( )γ ζ θ+ +> >0 0 0s = λ
λ
ρ
s
F bq s s
+
+[ ]− −( ) ( ) ˜ ( ( ))0 Π ,
(19)
P{ }( ) , ( )γ ζ θ+
+> >0 0 0s = b
s
q s s
+ − −λ
ρ( ) ˜ ( ( ))Π ,
qki, na perßyj pohlqd, moΩut\ vydatysq supereçlyvymy. Naspravdi livi çasty-
ny cyx spivvidnoßen\ vyznaçagt\ rizni jmovirnosti:
P{ }( ) , ( )γ ζ θ+ +> >0 0 0s = P{ }( ) , ( )γ ζ θ+ +≥ >0 0 0s = q s+( ) ,
(20)
q s+( ) > P{ }( ) , ( )γ ζ θ+
+> >0 0 0s = q s
s
F+ −
+
( ) ( )λ
λ
0 .
Naslidok dovedeno.
Dlq vyvçennq povedinky sxidçastoho procesu ryzyku pislq bankrutstva
sformulg[mo lemu pro rozpodil maksymumu procesu ζv( )t = v + ζ( )t (ζ0 =
= ζ( )t , v ≥ 0)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 11
POVEDINKA PROCESIV RYZYKU Z VYPADKOVYMY PREMIQMY … 1481
ζv
+ ( )t =
sup ( )
0≤ ′≤
′
t t
tζv , ζ θv
+ ( )s =
sup ( )
0≤ ≤t s
t
θ
ζv .
Lema 2. Heneratrysa ζ θv
+ ( )s vyznaça[t\sq spivvidnoßennqmy
Φ ( s , v, z ) = : Ee z s− +ζ θv ( ) = Φ( , , )s z e z0 − v , v ≥ 0; (21)
Φ ( s , 0, z ) = Ee z s− +ζ θ( ) =
p s
q s g zs
+
+−
( )
( ) ( )1
,
(22)
g zs( ) = E e z
s
− ++
>[ ]γ ζ θ( ) ( )0 0 =
G s z
G s
1
1
0
0 0
( , , )
( , , )
,
G s z1 0( , , ) = λ ϕ
ρ
ϕ ϕ ρpp s z
bq s
s z
z s−
−
−
−+
−
−
( ) ( )
( )
( )
( ) ( ( ))( )1 1 1 ,
G s1 0 0( , , ) = λ ϕ ρp q s s1 1−[ ]− −( ) ( ( )) , ϕ1( )z = Ee z− ′ξ1 .
Spivvidnoßennq (22) my nazyva[mo dohranyçnym uzahal\nennqm formuly Polq-
çeka – Xinçyna, hranycq qkoho pry s → 0 dlq m < 0 zvodyt\sq do formuly
Polqçeka – Xinçyna
Φ ( 0 , 0, z ) = Ee z− +ζ =
p
q g z
+
+−1 0( )
,
(23)
g z0( ) = E e z− ++
>[ ]γ ζ( )0 0 = 1
1
0
1 1+
+[ ]
∞
−∫b
e dF x bF x dxz x
µ
( ) ( ) .
Dovedennq. Formulu (21), qk i dlq klasyçnoho procesu ryzyku, otrymu[mo
za dopomohog stoxastyçnoho spivvidnoßennq
ζv
+ ( )t ⋅=
v, ( ), ,
( ( )), ( ),
{ }
( )
t t e
t t
t< > =
+ − ≥
+ −
+ + +
+
τ ζ
ζ ζ τ τ
λ
γ
0
0 0
0
P
z qkoho vyplyva[ perße spivvidnoßennq v (21). Pry v = 0 z c\oho stoxastyçno-
ho spivvidnoßennq vyplyva[ formula
Φ ( s , 0, z ) = p s T s z+
−−[ ]( ) ( , , )1 0 1, T s z( , , )0 = E e z
s
− ++
>[ ]γ ζ θ( ), ( )0 0 .
Na pidstavi (8) i toho, wo p s+( ) > 0, heneratrysa T s z( , , )0 = V s z( , , , , )0 0 0 vy-
znaça[t\sq spivvidnoßennqm
T s z( , , )0 = q s g zs+( ) ( ) = s p s G s z−
+
1
1 0( ) ( , , ) ,
z qkoho vyplyva[ formula (22) z vidpovidnym znaçennqm g zs( ) u terminax
G s z1 0( , , ). Hranyçnym perexodom s → 0 z (22) vstanovlg[t\sq (23).
Lemu dovedeno.
Analiz povedinky sxidçastoho nadlyßkovoho procesu pislq bankrutstva zvo-
dyt\sq do rozhlqdu vidpovidnoho sxidçastoho procesu z vypadkovym startovym
znaçennqm
ζ∗( )t = ζγ + ( )
( )
u
t , t ≥ 0, ζ∗( )0 = γ + ( )0 , ζ∗
+ = sup ( )
( )
0< <∞
+
t
u
tζγ , (24)
dlq qkoho ma[ misce analoh teoremyJ2 v [1].
Teorema 2. Nexaj ζ ( t ) — sxidçastyj nadlyßkovyj proces ryzyku z vypadko-
vymy premiqmy. Todi heneratrysa
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 11
1482 D. V. HUSAK
Φ∗( , , )s u z = E e u
z
s
u s− ++
+
>
ζ θ
γ ζ θ( )
( )
, ( )
vyznaça[t\sq spivvidnoßennqm
Φ∗( , , )s u z = Φ( , , ) ( , , )s z T s u z0 , (25)
sT s u z( , , ) = sV s u z( , , , , )0 0 = p s G s u z G s u y z dP s y
u
+
+
++ −∫( ) ( , , ) ( , , ) ( , )1
0
1 ,
sT s z( , , )0 = p s G s z+( ) ( , , )1 0 ,
Φ( , , )s z0 — dohranyçnym uzahal\nennqm formuly Polqçeka – Xinçyna (22). Qk-
wo m = Eζ( )1 = λ µ( )p qb− −1 < 0
λ p
b m
+ =
1 , to heneratrysa ta rozpodil
Ωorstkosti bankrutstva vyznaçagt\sq spivvidnoßennqmy
T u z( , , )0 = E e uz u− ++
< ∞[ ]γ τ( ), ( ) = p s G u z G u y z dP y
u
+
+
+′ + ′ −∫( ) ( , , ) ( , , ) ( )1
0
10 0 ,
P{ }( ) ,γ ζ+ +> >u x u = F u x bF u x( ) ( )+ + + +
+ λ
b m
F u x y bF u x y dP y
u
0
∫ + − + + −[ ] +( ) ( ) ( ),
(26)
P{ }( ) ,γ ζ+ +> >0 0x = p F x bF x1 1( ) ( )+( ), F x1( ) =
x
F y dy
∞
∫ 1( ) , x > 0.
Heneratrysa Z u+( ) ⋅= u + ∗
+ζ vyznaça[t\sq spivvidnoßennqm
E e uzZ u− ++
>[ ]( ), ζ = e z T u zuz Φ( , , ) ( , , )0 0 0 , (27)
Φ( , , )0 0 z — formulog Polqçeka – Xinçyna (23), T u z( , , )0 — perßym spivvid-
noßennqm v (26) z
′G u z1 0( , , ) = λ
b m
e F u y bF u y dyzy
0
∞
−∫ ′ + + +[ ]( ) ( ) . (28)
Dovedennq. Vraxovugçy (24), pislq userednennq heneratrysy dlq ζγ +
+ ⋅
( )
( )
u
po
E e u ds u− ++
∈[ ]τ γ( ), ( ) v oderΩu[mo spivvidnoßennq (25). Druhe spivvidnoßen-
nq v (25) vyplyva[ z (8) pry u u1 = , u u2 3 0= = z funkci[g G s u u1 1( , , ) (dyv.
vJ(5)), a z n\oho pry s → 0 — formula dlq heneratrysy Ωorstkosti bankrut-
stva, pislq obernennq qko] oderΩugt\ funkcig rozpodilu Ωorstkosti (dyv.
(26)). Rozpodil { }( ),γ ζ+ +u moΩna oderΩaty takoΩ intehruvannqm ostann\oho
spivvidnoßennq v (15). Iz (25) pry s → 0 vstanovlg[t\sq (27), oskil\ky
Z u+( ) = u + ∗
+ζ .
Teoremu dovedeno.
Qk dlq klasyçnyx procesiv ryzyku, tak i dlq sxidçastyx majΩe napivne-
perervnyx procesiv „çervonyj period” ′T u( ) stoxastyçno ekvivalentnyj
τ γ− +−( ( ))u . Na osnovi spivvidnoßennq
′T u( ) ⋅= τ γ− +−( ( ))u , u ≥ 0,
vstanovlg[t\sq analoh teoremyJ3 z [1].
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 11
POVEDINKA PROCESIV RYZYKU Z VYPADKOVYMY PREMIQMY … 1483
Teorema 3. Qkwo m = λ µb pb q− −1( ) < 0, to heneratrysa ′T u( ) vyzna-
ça[t\sq spivvidnoßennqm
E e T usT u− ′ ′ < ∞[ ]( ), ( ) = p e F u x bF u x dxx s
0
1
∞
−∫ − ′ + + +( )ρ ( ) ( ) ( ) +
+
λ ρp
m
e F u x y dxdP y
u
x s
0 0
1
+
∞
−
+∫ ∫ − + −( ) ( ) ( ),
(29)
P{ }( )′ < ∞T u = p F u bF u
p
m
F u y dP y
u
1 1
0
1( ) ( ) ( ) ( )+( ) + −
+
+∫λ
.
Wil\nist\ rozpodilu ′T u( ) (u dyferencialax) ma[ vyhlqd
P{ }( )′ ∈T u dt = p x dt F u x bF u x dx
0
1
∞
−∫ − ∈ ′ + + +( )P{ }( ) ( ) ( )τ +
+ λ τ
m
x dt F u x y dxdP y
u
0 0
1
+
∞
−
+∫ ∫ − ∈ + −P{ }( ) ( ) ( ) . (30)
Dovedennq. Pislq userednennq za wil\nistg Φ0
1( )( , )u x heneratrysy
E e xs x− − −−
− < ∞[ ]τ τ( ), ( ) = e x s− −ρ ( )
iz spivvidnoßennq
E e T usT u− ′ ′ < ∞[ ]( ), ( ) = E e us u− − − +− +
− < ∞[ ]τ γ τ γ( ( )), ( ( )) =
=
0
∞
− + +∫ − ∂
∂
< >e
x
u x u dxs xρ γ ζ( ) { }( ) ,P
vyplyva[ (29), pislq obernennq qkoho oderΩugt\ wil\nist\ (30). Pry s → 0 iz
perßoho spivvidnoßennq v (29) vyznaça[t\sq jmovirnist\
P{ }( )′ < ∞T u
u→
→
0
P{ }( )′ < ∞T 0 = p b( )1 + µ < p + q = 1.
Teoremu dovedeno.
U vypadku majΩe napivneperervnosti znyzu heneratrysa çysla vymoh N u∗( ) =
= n T u( ( ))′ za „çervonyj period” ′T u( )
n u z∗( , ) = E z T uN u∗
′ < ∞[ ]( ), ( )
vyznaça[t\sq v nastupnomu tverdΩenni (analoh teoremyJ4 v [1]).
Teorema 4. Heneratrysa çysla vymoh za period ′T u( ) pry m < 0 vyznaça-
[t\sq çerez sz = λ1 1( )− z , 0 < z < 1, spivvidnoßennqm
n u z∗( , ) = p e F u x dx b e F u x dxx s s xz z
0
1
0
1
∞
−
∞
−∫ ∫− −′ + + +
ρ ρ( ) ( )( ) ( ) +
+
λ ρp
m
e F u x y dP y dx
u
x sz
0 0
1
∞
+
−
+∫ ∫ − + −( ) ( ) ( ) , (31)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 11
1484 D. V. HUSAK
n z∗( , )0 = p s b s sz z zϕ ρ ρ ϕ ρ1
1
11( ( )) ( )( ( ( )))− −
−
−+ −[ ]. (32)
Pry z → 1 ( sz → 0 ) magt\ misce spivvidnoßennq, wo uzhodΩugt\sq z ostan-
nim spivvidnoßennqm v (29) pry s → 0. Heneratrysa çysla vymoh do bankrut-
stva n u( ( ))τ+
vyznaça[t\sq spivvidnoßennqm
E z un u( ( )), ( )τ τ
+ + < ∞[ ] = E z us uz− ++
< ∞[ ]τ τ( ), ( ) =
= P{ }( )ζ θ+ >sz
u
z→
→
1
P{ }ζ+ > u . (33)
Dovedennq. Formulu (31) moΩna otrymaty userednennqm heneratrysy
n ( t ) = ν1 ( t ) E zn t( ) = et zλ1 1( )−
za rozpodilom ′T u( ) (dyv.J(30)):
n u z∗( , ) =
0
11
∞
−∫ ′ ∈ ′ < ∞e T u dt T ut zλ ( ) { }( ) , ( )P .
Pry u = 0 z (31) vyplyva[ (32). Spivvidnoßennq (33) oderΩugt\ userednennqm
heneratrysy n ( t ) = ν1 ( t ) za rozpodilom τ+ ( )u
0
11
∞
− +∫ <e d u tt zλ τ( ) { }( )P = E z us uz− ++
< ∞[ ]τ τ( ), ( ) , sz = λ1 1( )z − .
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OderΩano 30.06.06
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 11
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| id | umjimathkievua-article-3405 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:41:55Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a6/1352d97a1883f9894396314ae2fff9a6.pdf |
| spelling | umjimathkievua-article-34052020-03-18T19:53:28Z Behavior of risk processes with random premiums after ruin and a multivariate ruin function Поведінка процесів ризику з випадковими преміями після банкрутства та багатозначна функція банкрутства Gusak, D. V. Гусак, Д. В. We establish relations for the distribution of functionals associated with the behavior of a risk process with random premiums after ruin and for a multivariate ruin function. Установлены соотношения для распределения функционалов, связанных с поведением процесса риска со случайными премиями после разорения, и для многозначной функции разорения. Institute of Mathematics, NAS of Ukraine 2007-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3405 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 11 (2007); 1473–1484 Український математичний журнал; Том 59 № 11 (2007); 1473–1484 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3405/3553 https://umj.imath.kiev.ua/index.php/umj/article/view/3405/3554 Copyright (c) 2007 Gusak D. V. |
| spellingShingle | Gusak, D. V. Гусак, Д. В. Behavior of risk processes with random premiums after ruin and a multivariate ruin function |
| title | Behavior of risk processes with random premiums after ruin and a multivariate ruin function |
| title_alt | Поведінка процесів ризику з випадковими преміями після банкрутства та багатозначна функція банкрутства |
| title_full | Behavior of risk processes with random premiums after ruin and a multivariate ruin function |
| title_fullStr | Behavior of risk processes with random premiums after ruin and a multivariate ruin function |
| title_full_unstemmed | Behavior of risk processes with random premiums after ruin and a multivariate ruin function |
| title_short | Behavior of risk processes with random premiums after ruin and a multivariate ruin function |
| title_sort | behavior of risk processes with random premiums after ruin and a multivariate ruin function |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3405 |
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