Behavior of classical risk processes after ruin and a multivariate ruin function
We establish relations for the distribution of functionals associated with the behavior of a classical risk process after ruin and a multivariate ruin function.
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| Date: | 2007 |
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Institute of Mathematics, NAS of Ukraine
2007
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509477094031360 |
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| author | Gusak, D. V. Гусак, Д. В. |
| author_facet | Gusak, D. V. Гусак, Д. В. |
| author_sort | Gusak, D. V. |
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| description | We establish relations for the distribution of functionals associated with the behavior of a classical risk process after ruin and a multivariate ruin function. |
| first_indexed | 2026-03-24T02:41:43Z |
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UDK 519.21
D. V. Husak (In-t matematyky NAN Ukra]ny, Ky]v)
POVEDINKA KLASYÇNYX PROCESIV RYZYKU
PISLQ BANKRUTSTVA
TA BAHATOZNAÇNA FUNKCIQ BANKRUTSTVA
*
We establish relations between distributions of functionals that depend on the behavior of the classical
risk process after the ruin time and the multivariate ruin function.
Ustanovlen¥ sootnoßenyq dlq raspredelenyq funkcyonalov, svqzann¥x s povedenyem klassy-
çeskoho processa ryska posle momenta razorenyq, y mnohoznaçnoj funkcyy razorenyq.
Ostannim çasom zris interes do vyvçennq riznyx xarakterystyk procesiv ryzyku,
pov’qzanyx z ]x povedinkog pislq bankrutstva. Ce poqsng[t\sq tym, wo pislq
bankrutstva straxova kompaniq moΩe prodovΩyty funkcionuvannq, vzqvßy v
kredyt deqkyj kapital. Dlq vyznaçennq peredbaçuvanoho kredytu vaΩlyvo zna-
ty rozpodil xarakterystyk klasyçnoho (rezervnoho) procesu ryzyku
ξu ( t ) = Ru ( t ) = u + c t – S ( t ) , S ( t ) = ξ
ν
k
k t≤ ( )
∑ , P { ν ( t ) = k } =
( ) −λ λt
k
e
k
t
!
,
abo nadlyßkovoho procesu ryzyku ζ ( t ) = S ( t ) – c t ( c, λ, u > 0 ) z kumulqntog
k ( r ) = t–
1
ln E er
ζ
(
t
) = c r + λ ( E e–
r
ξ1 – 1 ) . Ci xarakterystyky vyznaçagt\sq pere-
strybkovymy funkcionalamy { τ+
( u ), γ+
( u ), γ+ ( u ) } (]x poznaçennq dyv. nyΩçe),
rozpodily qkyx vyvçalys\ v bahat\ox robotax, zokrema v [1 – 4]. Rozpodily zha-
danyx funkcionaliv vyznaçagt\ takoΩ bahatoznaçnu funkcig bankrutstva
(dyv. (5.1.18) v [5])
φ ( u, x, y ) = P { γ+
( u ) > k, γ+ ( u ) > y, τ+
( u ) < ∞ }. (1)
Vyvçenng wil\nosti dewo uzahal\neno] bahatoznaçno] funkci] bankrutstva
ta rozpodilu funkcionaliv, wo opysugt\ povedinku procesiv ryzyku pislq ban-
krutstva, prysvqçeno danu robotu. Zaznaçymo, wo bahato riznyx pytan\, pov’q-
zanyx z povedinkog procesiv ryzyku pislq bankrutstva, doslidΩuvalys\ u robo-
tax [5 – 14].
Spoçatku navedemo dlq porivnqnnq vykorystani namy poznaçennq dlq do-
slidΩuvanyx funkcionaliv z poznaçennqmy v [5, 6] (]x hrafiçni zobraΩennq na-
vedeno na rysunku):
*
Vykonano pry çastkovij pidtrymci Deutsche Forschungsgemeinschaft.
© D. V. HUSAK, 2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10 1339
1340 D. V. HUSAK
τ ( u ) = inf { t : Ru ( t ) < 0 }, τ+
( u ) = inf { t : Ru ( t ) > u },
Y+
( u ) = – Ru ( τ ( u ) ), γ+
( u ) = ζ ( τ+
( u ) ) – u,
X
+
( u ) = Ru ( τ ( u ) – 0 ), γ+ ( u ) = u – ζ ( τ+
( u ) – 0 ),
X
+
( u ) + Y
+
( u ) = Ru ( τ ( u ) – 0 ) – Ru ( τ ( u ) ), γ u
+ = γ+
( u ) + γ+ ( u ),
τ ( u ) =̇ τ+
( u ) vyznaça[ moment bankrutstva, Y
+
( u ) =̇ γ+
( u ) — Ωorstkist\ bank-
rutstva, X
+
( u ) =̇ γ+ ( u ) — znaçennq Ru ( t ) pered nastannqm bankrutstva, γ u
+
—
rozmir vymohy, wo spryçynyla bankrutstvo, ζ±
( t ) = sup 0 ≤ t ′ ≤ t ( inf ) ζ ( t ′ ) — ek-
stremumy ζ ( t ′ ) na intervali [ 0, t ], τ′ ( u ) = inf { t > τ ( u ), Ru ( t ) > 0 } — moment
povernennq Ru ( t ) pislq bankrutstva u pivplowynu Π+ = { y > 0 }. Poznaçymo
we
T ′ ( u ) =
′( ) − ( ) ( ) < ∞
( ) = ∞
τ τ τ
τ
u u u
u
, ,
, .0
T ′ ( u ) nazyva[t\sq „çervonym periodom”, qkyj vyznaça[ tryvalist\ perebuvannq
Ru ( t ) u pivplowyni Π
– = { x < 0 },
Z
+
( u ) = sup sup
τ τ
ζ
+ ( )≤ <∞ ( )≤ <∞
( ) = − ( ){ }
u t u t
ut R t ,
Z u t R t
u t u u t u
u1
+
( )≤ < ′( ) ( )≤ ≤ ′( )
( ) = ( ) = − ( )
+ +
{ }sup sup
τ τ τ τ
ζ .
Z
+
( u ) vyznaça[ total\nyj maksymum deficytu, Z u1
+( ) — maksymum deficytu za
period T ′ ( u ).
Rozhlqnemo wil\nist\ uzahal\neno] bahatoznaçno] (skladno]) funkci] ban-
krutstva
ϕs ( u, dx, dy ) = P { γ
+
( u ) ∈ dx, γ+ ( u ) ∈ dy, ζ
+
( θs ) > u } =
= E [ e–
s
τ
+
(
u
), γ
+
( u ) ∈ dx, γ+ ( u ) ∈ dy, τ+
( u ) < ∞ ], (2)
de θs — pokaznykovo rozpodilena vypadkova velyçyna P { θs > t } = e–
s
t, s > 0,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
POVEDINKA KLASYÇNYX PROCESIV RYZYKU PISLQ BANKRUTSTVA … 1341
t ≥ 0. Pislq intehruvannq (2) pry s → 0 znaxodymo samu skladnu funkcig ban-
krutstva
φ ( u, x, y ) = ϕ0( ′ ′)
∞∞
∫∫ u dx dy
yx
, , , x > 0, y > 0.
Intehral\ne peretvorennq wil\nosti (2) [ heneratrysog trijky { τ+
( u ), γ+
( u ),
γ+ ( u ) }:
˜ , , , ,ϕ ϕs
u x u y
su u u e u dx dy( ) = ( )− −
∞∞
∫∫1 2
00
1 2 =
= E ,[ ]− ( )− ( )− ( ) ++ +
+ ( ) < ∞e us u u u u uτ γ γ τ1 2 = V ( s, u, u1 , u2 ).
Spil\na heneratrysa { τ+
( u ), γ+
( u ), γ+ ( u ), γ u
+
} ( γ+
( u ) = γ1 ( u ), γ+ ( u ) = γ2 ( u ),
γ u
+ = γ3 ( u ) )
V ( s, u, u1 , u2 , u3 ) = E ,[ ]− ( )− ( ) +
+
=∑ ( ) < ∞e u
s u u uk kkτ γ
τ1
3
vyznaça[t\sq spivvidnoßennqm (dyv. (13) v [4])
V ( s, u, u1 , u2 , u3 ) = s G s u y u u u dP s y
u
−
+( − ) ( )∫1
1 2 3
0
, , , , , , (3)
G ( s, x, u1 , u2 , u3 ) = A u u u dP s yx y− −
−∞
( ) ( )∫ 1 2 3
0
, , , ,
P± ( s, y ) = P { ζ+
( θs ) < y }, ± y > 0,
Ax ( u1 , u2 , u3 ) = λ e dF zu u x u u z
x
( − ) −( + )
∞
( )∫ 1 2 1 3 , x > 0,
Ax ( u1 , u2 ) = Ax ( u1 , u2 , 0 ),
F ( x ) = P { ξk < x }, F x( ) = 1 – F ( x ), F x F z dz
x
( ) = ( )
∞
∫ , x > 0.
Dlq neperervnoho znyzu procesu P– ( s, y ) = eρ–
(
s
)
y, y ≤ 0, P+ ( s, + 0 ) = p+ ( s ) > 0,
tomu
G ( s, u, u1 , u2 ) = G ( s, u, u1 , u2 , 0 ) =
ρ
ρ
−
−
−
( )
( ) − +
s e
s u u
u u2
1 2
×
× [( ) ]−
−( ( )+ ) −
∞
( ) + − ( + )−∫ ρ ρs u e u e u z dzs u z u z
2 1
0
2 1 Π , u > 0, (4)
Π ( z ) = λF z( ), F z( ) = 1 – F ( z ), z > 0; – ρ– ( s ) — vid’[mnyj korin\ rivnqnnq
Lundberha k ( – ρ– ( s ) ) = s. Pislq intehruvannq çastynamy z (4) vyplyva[
G ( s, x, u1 , u2 ) =
λρ
ρ
ρ−
−
−
− −( ( )+ )
∞( )
( ) − +
− ( + )[ ]−∫s e
s u u
e e dF u y
u u
u y s u y
2
1 2
1 2 0
. (5)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
1342 D. V. HUSAK
Poznaçymo G i ( s, u , u i ) = G s u u u u u r i ir
( ) = ∀ ≠ =, , , , , ,1 2 3 0 1 3 i zauvaΩymo, wo
p+ ( s ) ρ– ( s ) = s c–
1
.
Ma[ misce take tverdΩennq.
Lema 1. Funkciq G ( s, u, u1 , u2 ) dopuska[ obernennq po u1 , u2 , a funkci]
Gi ( s, u, ui ), i = 1 3, , dopuskagt\ obernennq po ui z poxidnog F ′ ( y ) (isnuvannq
qko] abo prypuska[t\sq, abo F ′ ( y ) vΩyva[t\sq v sensi Ívarca):
g ( s, u, x, y ) = λρ ρ
−
− ( )( − )( ) ′( + ) { > }−s e F x y I y us u y , x > 0; (6)
G1 ( s, u, u1 ) =
λρ
ρ
ρ−
−
− − ( )
∞( )
( ) −
− ( + )( )−∫s
s u
e e dF u yu y s y
1 0
1
,
(7)
g1 ( s, u, x ) = λρ ρ
−
( )( − )
∞
( ) ( + )−∫s e dF u ys x y
x
;
G2 ( s, u, u2 ) = λρ ρ
−
− ( + )− ( )
∞
( ) ( + )−∫s e F u z dzu u z s z2
0
,
(8)
g2 ( s, u, y ) = λρ ρ
−
( )( − )( ) ( ) { > }−s e F y I y us u y
;
G3 ( s, u, u3 ) = λ λ ρ ρe dF z e e e F z dzu z
u
s u u z s z
u
−
∞
( ) − − ( )
∞
∫ ∫( ) − ′( )− −3 3
,
(9)
g3 ( s, u, z ) = λ ρ′( ) − { > }[ ]−( )( − )F z e I z us u z1 .
Qkwo m = E ζ ( 1 ) = µ λ – c < 0, to isnugt\ poxidni pry s → 0
g′ ( 0, u, x, y ) =
λ
m
F x y I y u′( + ) { > }, x > 0,
′( ) = ( + )g u x
m
F u x1 0, ,
λ
, x > 0,
(10)
′ ( ) = ( ) { > }g u y
m
F y I y u2 0, ,
λ
,
′ ( ) = ′( )( − ) { > }g u z
m
F z z u I z u3 0, ,
λ
.
Dovedennq. Lehko pereviryty, wo zhortka
J ( s, u, x, u2 ) = λρ ρ
−
− ( ( )+ )( − )
∞
( ) ( + )−∫s e e dF x yu u s u x y
x
2 2 , x > 0,
[ obernennqm G ( s, u, u1 , u2 ) po u1 , qka pislq zaminy zminnyx ma[ vyhlqd
J ( s, u, x, u2 ) = λρ ρ
−
− ( )( − )
∞
( ) ( + )−∫s e e dF x yu y s u y
u
2
,
zruçnyj dlq obertannq po u2 , v rezul\tati çoho vstanovlg[t\sq (6). Analohiç-
no obertagt\sq po ui funkci] Gi ( s, u, ui ), i = 1 3, , i vstanovlggt\sq spivvid-
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
POVEDINKA KLASYÇNYX PROCESIV RYZYKU PISLQ BANKRUTSTVA … 1343
noßennq (7) – (9). Qkwo m < 0, to s–
1
ρ– ( s ) → ( | m | )
–
1
pry s → 0, tomu
pislq dyferencigvannq po s ( s = 0 ) vstanovlggt\sq spivvidnoßennq (10).
Teorema 1. Dlq neperervnoho znyzu procesu ryzyku heneratrysy V ( s, u, u1 ,
u2 , u3 ) ta ˜ , ,ϕs u u u( )1 2 vyznaçagt\sq spivvidnoßennqmy
s V ( s, u, u1 , u2 , u3 ) = p+ ( s ) G ( s, u, u1 , u2 , u3 ) + G s u y u u u dP s y
u
( − ) ( )
+
+∫ , , , , ,1 2 3
0
,
(11)
s u u u p s G s u u u G s u y u u dP s ys
u
˜ , , , , , , , , ,ϕ ( ) = ( ) ( ) + ( − ) ( )+
+
+∫1 2 1 2 1 2
0
.
Qkwo m < 0, to heneratrysa wil\nosti bahatoznaçno] funkci] bankrut-
stva ma[ vyhlqd
˜ , , lim ˜ , , E ,ϕ ϕ τγ γ
0 1 2
0
1 2
1 2( ) = ( ) = ( ) < ∞
→
− ( )− ( ) +[ ]
+
+u u u u u u e u
s
s
u u u u =
= p G u u G u y u u d y
u
+
+
+′( ) + ′( − ) { < }∫0 01 1 2
0
, , , , , P ζ , (12)
de p+ = P { ζ+ = 0 }, ζ+ = sup
0≤ <∞
( )
t
tζ ,
G′ ( 0, u, u1 , u2 ) =
λ
m
e
u u
u e u e F u z dz
u u
u z u z
−
− −
∞
−
− ( + )[ ]∫
2
2 1
2 1
2 1
0
.
Pislq obernennq druhoho spivvidnoßennq v (11) ta (12) po u1 t a u2 v y -
znaçagt\sq wil\nosti (v dyferencialax) skladno] funkci] bankrutstva (pry
s > 0 ta s → 0)
s u dx dy p s g s u x y g s u z x y dP s z dx dys
u
ϕ ( ) = ( ) ( ) + ( − ) ( )
+
+
+∫, , , , , , , , ,
0
,
(13)
ϕ ζ0
0
0 0( ) = ′( ) + ′( − ) { < }
+
+
+∫u dx dy p g u x y g u z x y dP z dx dy
u
, , , , , , , , .
Dlq marhinal\nyx wil\nostej { γi ( x ), ζ+
( θs ) }
Φs
i
i su x
d
dx
u x u( ) +( ) = ( ) < ( ) >{ }, P ,γ ζ θ , x ≠ u, i = 1 3, ,
vykonugt\sq spivvidnoßennq
s u x p s g s u x g s u z x dP s zs
i
i i
u
Φ( )
+
+
+( ) = ( ) ( ) + ( − ) ( )∫, , , , , ,
0
,
(14)
Φ0
0
0 0( )
+
+
+( ) = ′( ) + ′( − ) { < }∫i
i i
u
u x p g u x g u z x d z, , , , , P ζ .
Dovedennq. Spivvidnoßennq (11) vyplyvagt\ z (3), oskil\ky p+ ( s ) > 0. Pry
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
1344 D. V. HUSAK
m < 0 iz (11) pry s → 0 vyplyva[ (12). Pislq obernennq (12) oderΩu[mo (13).
Analohiçno vstanovlggt\sq spivvidnoßennq (14) dlq marhinal\nyx wil\nostej.
Naslidok 1. Dlq nadlyßkovoho procesu ryzyku ζ ( t ) z linijnog funkci[g
premij c ( t ) = c t i poçatkovym kapitalom u > 0 wil\nosti (dohranyçna pry
s > 0 ta hranyçna pry s → 0) skladno] funkci] bankrutstva vyznaçagt\sq pry
y > 0 ( y ≠ u ) spivvidnoßennqmy
s
x y
u u x u ys
∂
∂ ∂
( ) > ( ) < ( ) <{ }+ +
+
2
P , ,ζ θ γ γ =
=
λρ
λρ
ρ
ρ
−
( )( − − )
+
−
( )( − − )
+
−
( ) ′( + ) ( ) >
( ) ′( + ) ( ) < <
−
−
∫
∫
s F x y e dP s z y u
s F x y e dP s z y u
s u y z
u
s u z y
u y
u
, , ,
, , ,
0
0
(15)
∂
∂ ∂
> ( ) < ( ) <{ }+ +
+
2
x y
u u x u yP , ,ζ γ γ =
=
λ ζ λµ
λ ζ
m F x y u y u m c
m F x y u y u y u
− +
− +
′( + ) < > = − <
′( + ) − < < < <
{ }
{ }
1
1
0
0
P , , ,
P , .
Dlq wil\nosti rozpodilu Ωorstkosti bankrutstva magt\ misce spivvidno-
ßennq pry s > 0 ta s → 0 ( x > 0 )
s
x
u u x c s e dF u ys
s x y
x
∂
∂
( ) > ( ) < = ( + ){ }+ + − ( )( − )
∞
−∫P ,ζ θ γ λ ρ1 +
+ λρ ρ
−
( )( − )
∞
+( ) ( + − ) ( )−∫∫s e dF u y z dP s zs x y
x
u
0
, , ρ– ( s ) p+ ( s ) =
s
c
,
(16)
∂
∂
> ( ) < = ( ) + ( + − ) <{ } { }+ + +∫x
u u x
c
F u x
m
F u x z d z
u
P , , Pζ γ λ λ ζ
0
, m < 0.
Wil\nist\ rozpodilu nadlyßku pered nastannqm bankrutstva γ+ ( u ) vyzna-
ça[t\sq spivvidnoßennqmy (dohranyçnym pry s > 0 ta hranyçnym pry s → 0)
pry y ≠ u
s
y
u u ys
∂
∂
( ) > ( ) <{ }+
+P ,ζ θ γ =
=
λρ
λρ
ρ
ρ
−
( )( − − )
+
−
−
( )( − − )
+
−
( ) ( ) ( ) >
( ) ( ) ( ) < <
−
−
∫
∫
s F y e dP s z y u
s F y e dP s z y u
s u y z
u
s u y z
u y
u
, , ,
, , ,
0
0
(17)
∂
∂
> ( ) <{ }+
+y
u u yP ,ζ γ =
λ ζ
λ ζ
m F y u y u m
m F y u y u y u
− +
− +
( ) < > <
( ) − < < < <
{ }
{ }
1
1
0
0
P , , ,
P , .
Rozpodil vymohy γ u
+
, wo spryçynyla bankrutstvo, vyznaça[t\sq spivvidno-
ßennqmy ( z ≠ u )
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
POVEDINKA KLASYÇNYX PROCESIV RYZYKU PISLQ BANKRUTSTVA … 1345
s
z
u z p s g s u z g s u y z dP s ys u
u
∂
∂
( ) > < = ( ) ( ) + ( − ) ( ){ }+ +
+ +
+
∫P , , , , , ,ζ θ γ 3 3
0
,
∂
∂
< > = ′( ) ( − + ) < { > }{ } { }+ + +∫z
z u
m
F z z u y d y I z uu
u
P , Pγ ζ λ ζ
0
+
+
λ ζ′( ) ( + ) < − { < < }{ }+∫F z
m
z d u I z u
z
v vP
0
0 , (18)
P ,{ }+ +< >γ ζu z u0 =
=
λ ζ ζ
λ ζ ζ
λ ζ
m
yd y z u u dF z z u m
m
F u yd y u F u
m
z y d y u dF z
u
z
u
z
z
P P , , ,
P P
P ,
{ } { }
{ } { }
{ }
+ +
∞
+ +
+
< + ( − ) <
( ) > <
( ) < + < ( )
+
+ ( + ) < − ( )
∫∫
∫
∫
0
0
0
0
0
0
00
0 0
u
z u∫ < <
.
Intehruvannqm (15) po x ∈ [ x0
, ∞ ) vstanovlggt\sq spivvidnoßennq ( y ≠
≠ u )
s
y
u u x u ys
∂
∂
( ) > ( ) < ( ) <{ }+ +
+P , ,ζ θ γ γ0 =
=
λρ
λρ
ρ
ρ
−
( )( − − )
+
−
( )( − − )
+
−
( ) ( + ) ( ) >
( ) ( + ) ( ) < <
−
−
∫
∫
s F x y e dP s z y u
s F x y e dP s y y u
s u y z
u
s u y z
u y
u
0
0
0 0
, , ,
, , ,
(19)
∂
∂
> ( ) > ( ) <{ }+ +
+y
u u x u yP , ,ζ γ γ0 =
=
λ ζ
λ ζ
m
F x y u y u m x
m
F x y u y u y u
( + ) < > < >
( + ) − < < < <
{ }
{ }
+
+
0 0
0
0 0
0
P , , , ,
P , ,
P , , , ,{ }+ +
+> ( ) > ( ) > = ( )ζ γ γ φu u x u y u x y0 0 0 0 =
=
λ ζ
λ ζ
λ ζ
m
F x y u y u m x
m
F x u u
m
F x y u y u dy y u
y
u
( + ) < > < >
( + ) < +
+ ( + ) − < < < <
{ }
{ }
{ }
+
+
+∫
0 0 0 0
0
0 0
0 0
0
0
P , , , ,
P
P , .
Dovedennq. Dovedennq naslidku vyplyva[ iz spivvidnoßen\ teoremy 1 pislq
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
1346 D. V. HUSAK
pidstanovky znaçen\ funkcij G ( s, u, u1 , u2 ), G i ( s, u, ui ), i = 1 3, , ta ]x ober-
nen\ g ( s, u, ui ) po ui , a takoΩ znaçen\ poxidnyx po s pry s = 0. Pry dovedenni
(15) slid vraxuvaty dvo]stist\ zobraΩennq zhortky
g s u z dP s z I z u y dP s z
u u
( − …) ( ) = … { > − } ( )+ +∫ ∫, , , ,
0 0
=
=
… ( ) >
… ( ) < <
+
+
−
∫
∫
dP s z y u
dP s z y u
u
u y
y
, , ,
, , .
0
0
Na pidstavi (7) – (9) z (14) pry i = 1 3, vidpovidno vyplyvagt\ spivvidnoßennq
(16) – (18). V rezul\tati intehruvannq (15) po x ∈ [ x0
, ∞ ) ta y ∈ [ y0
, ∞ ) otry-
mu[mo spivvidnoßennq (19). Ostann[ spivvidnoßennq v (18) oderΩu[mo z druhoho
v (18) intehruvannqm po z ∈ [ z0
, ∞ ). ZauvaΩymo, wo ostann[ spivvidnoßennq v
(19), oderΩane z poperedn\oho intehruvannqm po y ∈ [ y0
, ∞ ), vyznaça[ bahato-
znaçnu funkcig bankrutstva φ ( u, x0 , y0 ) (dyv. (1)). Tak samo pislq intehruvan-
nq (16) po x ∈ [ 0, ∞ ) ta x ∈ [ x0
, ∞ ) oderΩymo vidpovidno spivvidnoßennq dlq
φ ( u, 0 , 0 ) ta φ ( u, x0 , 0 ) :
P { ζ+ > u } =
λ λ
c
F u
m
F u z dP z
u
( ) + ( − ) ( )∫ +
0
, P+ ( z ) = P { ζ+ < z },
P ,{ }+ +> ( ) >ζ γu u x0 =
=
λ λ λ
c
F u x
m
F u x z dP z
c
F u
u
x
( + ) + ( + − ) ( ) → ( )∫ + →0 0
0
00
.
Z ostann\oho spivvidnoßennq v (19) pry x0 = 0 znaxodymo marhinal\nu funkcig
bankrutstva
φ ζ γ( ) = > ( ) >{ }+
+u y u u y, , P ,0 0 0 =
=
λ
λ λ
m
F y P u y u
m
F u P u
m
F y P u P u y dy y u
y
u
( ) ( ) >
( ) ( ) + ( ) ( ) − ( − ) < <
+
+ + +∫ [ ]
0 0
0
0
0
, ,
, .
Zvidsy pry y0 = 0 vyplyva[, wo P { γ+ ( u ) > 0 } = 1, oskil\ky
P ,{ }+
+
+ +( ) > > = ( ) ( ) + ( − ) ( )
∫γ ζ λ
u u
m
F P u P u y dF y
u
0 0
0
=
=
λ ζ γ
m
p F u F u z dP z u u P u
u
+ +
+ +
+( ) + ( − ) ( )
= > ( ) > = ( )∫ { }
0
0P , .
Naslidok dovedeno.
Perß niΩ vyvçaty rozpodily funkcionaliv, pov’qzanyx z povedinkog proce-
su ζ ( t ) pry t > τ+
( u ), dovedemo dopomiΩne tverdΩennq dlq procesu ζv ( t ) = v +
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POVEDINKA KLASYÇNYX PROCESIV RYZYKU PISLQ BANKRUTSTVA … 1347
+ ζ ( t ) ( ζ0 ( t ) = ζ ( t ), v, t ≥ 0 ) ta joho maksymumu
ζ ζv v
+
≤ ′≤
( ) = ( )t t
t t
sup
0
,
ζ θ ζ
θ
v v
+
≤ ′≤
( ) = ( ′)s
t s
tsup
0
.
Vvedemo poznaçennq heneratrys dlq ζ θv
+( )s ta dlq pary { τ+
( v ), γ+
( v ) }:
Φ ( s, v, z ) = Ee z s− ( )+ζ θv
,
(20)
T ( s, v, z ) = E , E ,[ ] [ ]− ( )− ( ) + − ( ) ++ + +
( ) < ∞ = ( ) >e es z z
s
τ γ γτ ζ θv v vv v .
Lema 2. Heneratrysa Φ ( s, v, z ) vyznaça[t\sq spivvidnoßennqmy
Φ ( s, v, z ) = Φ ( s, 0, z ) e–
z
v, v ≥ 0,
(21)
Φ ( s, 0, z ) = p+ ( s ) [ 1 – T ( s, 0, z ) ]
–
1, p+ ( s ) = P { ζ+
( θs ) = 0 } > 0.
Pry m < 0, v = 0 z (21) vyplyva[ spivvidnoßennq, qke my nazvemo dohranyç-
nym uzahal\nennqm formuly Polqçeka – Xinçyna
Φ ( s, 0, z ) = Ee
p s
q s g z
p
q g z
z
s
s
s− ( ) +
+
→
+
+
+
= ( )
− ( ) ( )
→
− ( )
ζ θ
1 10
0
, p+ = P { ζ+ = 0 },
(22)
gs ( z ) = E
, ,
, ,
˜ / ˜
[ / ] ( )− ( ) + − −
−
+
( ) > = ( )
( )
= ( ) − ( ) ( )
( ) −
e
G s z
G s
s z z s
s z
z
s
γ ζ θ ρ ρ
ρ
0 1
1
0
0
0 0
Π Π
.
Qkwo m < 0, to pry s → 0 (22) zvodyt\sq do zvyçajno] formuly Polqçe-
ka – Xinçyna z
g0 ( z ) = lim ˜ ˜
s
sg z z
→
( ) = ( ) ( )/
0
0Π Π , Π̃( ) = ( )−
∞
∫z e F x dxzxλ
0
.
Dovedennq. Ma[ misce stoxastyçne spivvidnoßennq
ζ
τ
ζ τ τγ
v
v
v
+
+
( )
+ + +( ) =
< ( )
+ − ( ) ≥ ( )
+ ( )t
t
t t
˙
, ,
, ,
0
0 0
0
z qkoho vyplyva[ intehral\ne spivvidnoßennq
E P E , ,e e t e e dx dyz z z z t x
t
s y− ( ) − + − − ( − ) + +
∞
+ +
= ( ) > + ( ) ∈ ( ) ∈{ } [ ]∫∫ζ θ ζτ τ γv v vv 0 0
00
.
Pry dovil\nomu fiksovanomu y > 0 proces ζy ( t ) dlq t ≥ τ+
( 0 ) i joho maksymum
ne zaleΩat\ vid τ+
( 0 ) < t, tomu z oderΩanoho intehral\noho spivvidnoßennq
pislq peretvorennq Laplasa – Karsona po t vyplyva[ intehral\ne rivnqnnq dlq
Φ ( s, v, z ) :
Φ ( s, v, z ) =
e T s e s y z e dyz z s− − − ( ) +
∞
( ) [ ]− ( ) + ( ) ( ) ∈
+
∫v v1 0 0 00
0
, , , , E ,Φ τ γ . (23)
Qkwo poznaçyty
Φ* ( s, z ) = Φ( ) ( ) ∈[ ]− ( ) +
∞
+
∫ s y z e dys, , E ,τ γ0
0
0 ,
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1348 D. V. HUSAK
to pislq userednennq (23) po E ,[ ]− ( ) ++
( ) ∈e dsτ γ0 0 v oderΩymo rivnqnnq dlq
vyznaçennq Φ* ( s, z ) :
Φ* ( s, z ) = T ( s, 0, z ) ( 1 – T ( s, 0, 0 ) ) + Φ* ( s, z ) T ( s, 0, z ).
Vraxovugçy, wo
T ( s, 0, 0 ) = P { ζ+
( θs ) > 0 } = q+ ( s ), p+ ( s ) = 1 – q+ ( s ),
znaxodymo znaçennq
Φ* ( s, z ) = p s
T s z
T s z+( ) ( )
− ( )
, ,
, ,
0
1 0
,
pislq pidstanovky qkoho v (21) oderΩu[mo spivvidnoßennq
Φ ( s, v, z ) =
p s e
T s z
z
+
−( )
− ( )
v
1 0, ,
, T ( s, 0, z ) = E ,[ ]− ( ) ++
( ) >e z
s
γ ζ θ0 0 ,
z qkoho vyplyva[ (21) ta (22).
Lemu dovedeno.
Pislq bankrutstva nadlyßkovyj proces ryzyku nahadu[ proces çekannq
ζ* ( t ) z vypadkovym startovym znaçennqm v = γ+
( u ). Pry c\omu slid vraxuvaty,
wo γ+
( u ) vyznaça[t\sq na podi]
{ ω : ζ
+
( t ) > u } = { ω : τ+
( u ) < t }.
Tomu dlq doslidΩuvanoho procesu ζ* ( t ) = ζγ + ( )( )
u
t budemo rozhlqdaty spil\ni
rozpodily par { }+ ( )
+ +( ) ( )ζ τγ u
t u, , { }+ ( )
+ +( ) ( ) >ζ θ ζ θγ u s s u, abo vidpovidnu henerat-
rysu
Φ*
( s, u, z ) = E ,[ ]
− ( ) ++ ( )
+
( ) >e u
z
s
u sζ θ
γ ζ θ =
= E ,[ ]
− ( )− ( ) +
+
+ ( )
+
( ) < ∞e u
s u z
u sτ ζ θ
γ τ , (24)
dlq qko] ma[ misce take tverdΩennq.
Teorema 2. Heneratrysy ζ*
+ = ζγ + ( )
+
u
ta Z +
( u ) vyznaçagt\sq spivvidno-
ßennqmy
Φ*
( s, u, z ) = Φ ( s, 0, z ) T ( s, u, z ), E ,[ ]− ( ) ++
>e uzZ u ζ = eu
z
Φ*
( s, u, z ), (25)
de heneratrysa T ( s, u, z ) v (20) i vidpovidna wil\nist\ vyznaçagt\sq spivvid-
noßennqmy
s T ( s, u, z ) = p s G s u z G s u y z
y
P s y dy
u
+
+
+( ) ( ) + ( − ) ∂
∂
( )∫1 1
0
, , , , , ,
(26)
s
x
u x us
∂
∂
( ) < ( ) >{ }+ +P ,γ ζ θ = p s g s u x g s u y x
y
P s y dy
x
+
+
+( ) ( ) + ( − ) ∂
∂
( )∫1 1
0
, , , , , ,
a funkci] G1 ( s, u, z ) ta g1 ( s, u, x ) navedeno v (7).
Qkwo m = E ζ ( 1 ) = λ µ – c < 0 ( µ = E ξ1 ), to heneratrysa total\noho mak-
symumu deficytu Z
+
( u ) vyznaça[t\sq spivvidnoßennqm
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POVEDINKA KLASYÇNYX PROCESIV RYZYKU PISLQ BANKRUTSTVA … 1349
E ,[ ]− ( ) ++
>e uzZ u ζ = e uz− Φ ( 0, 0, z ) T ( 0, u, z ), Φ ( 0, 0, z ) =
p
T z
+
− ( )1 0 0, ,
, (27)
a heneratrysy
T ( 0, u, z ) = E ,[ ]− ( ) ++
>e uz uγ ζ , T ( 0, 0, z ) = E ,[ ]− ( ) ++
>e zγ ζ0 0
vyznaçagt\sq spivvidnoßennqmy
T ( 0, u, z ) = p G u z G u y z d y
u
+
+
+′( ) + ′( − ) { < }∫1 1
0
0 0, , , , P ζ ,
′( ) = ( + )−
∞
∫G u z
m
e F u y dyzy
1
0
0, ,
λ
, m < 0, (28)
T ( 0, 0, z ) =
p
m
e F y dy
c
e F y dyzy zy+ −
∞
−
∞
( ) = ( )∫ ∫λ λ
0 0
p
m
c
q
c+ += =
,
λµ
.
Dovedennq. Pislq userednennq po E ,[ ]− ( ) ++
( ) ∈e u ds uτ γ v z (21) vyplyva[
spivvidnoßennq (25). Perße spivvidnoßennq v (26) lehko obernuty po z i oder-
Ωaty dohranyçnu ( s > 0 ) wil\nist\ Ωorstkosti bankrutstva (dyv. druhu for-
mulu v (26)). Qkwo m < 0, to iz (25) ta (26) pry s → 0 vyplyvagt\ spivvidno-
ßennq (27), (28). Slid vidmityty, wo pry m < 0 perße spivvidnoßennq v (28)
takoΩ lehko oberta[t\sq po z, v rezul\tati obernennq vstanovlg[t\sq spivvid-
noßennq dlq hranyçno] ( s → 0 ) wil\nosti Ωorstkosti bankrutstva (dyv. os-
tanni spivvidnoßennq v (16) ta (26)).
Teoremu dovedeno.
Perß niΩ rozhlqdaty „çervonyj period” T ′ ( u ) ta çyslo vymoh za cej pe-
riod, vidmitymo podibnist\ ]x vidpovidno do periodu zajnqtosti θ̃1 ta çysla vy-
moh n( )θ̃1 u teori] SMO. Ostanni [ prostißymy xoça b tomu, wo vony ne zale-
Ωat\ vid parametra u. Funkcional T ′ ( u ) moΩna interpretuvaty qk moment
perßoho dosqhnennq nulq procesom ζγ + ( )
+ ( )
u
t abo moment perßoho dosqhnennq
rivnq x = – γ+
( u ) procesom ζ ( t ). Na pidstavi spivvidnoßennq
T ′ ( u ) =̇ τ–
( – γ+
( u ) ), E , E ,[ ] [ ]− ′( ) − (− ( ))… = …
− +
e esT u s uτ γ
vstanovlg[t\sq take tverdΩennq.
Teorema 3. Qkwo m < 0, to heneratrysa tryvalosti „çervonoho periodu”
T ′ ( u ) vyznaça[t\sq spivvidnoßennqm pry u ≥ 0
E ,[ ]− ′( ) − ( )
∞
′( ) < ∞ = ( + )−∫e T u
c
e F u dsT u sλ ρv v v
0
+
+
λ ζρ
m
e F u y d y ds
u
− ( )
+
∞
+− ( + − ) { < }∫∫ v v v
00
P . (29)
Qkwo s → 0, to
P { T ′ ( u ) < ∞ } =
λ λ ζ
c
F u
m
F u y d y
u
( ) + ( − ) ( < )
+
+∫
0
P . (30)
Wil\nist\ rozpodilu T ′ ( u ) (u dyferencialax) ma[ vyhlqd
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
1350 D. V. HUSAK
P { T ′ ( u ) ∈ dt } =
λ τ
c
F u dt d( + ) (− ) ∈{ }−
∞
∫ v v vP
0
+
+
λ τ ζ
m
dt F u y d y d
u
P P{ }−
+
∞
+(− ) ∈ ( + − ) { < }∫∫ v v v
00
. (31)
Dovedennq. Proces ζ ( t ) [ napivneperervnym znyzu, tomu pry v > 0
P E ,{ } [ ]− − (− ) − − ( )(− ) ∈ ⇔ (− ) < ∞ =
−
−τ ττ ρv vv vdt e es s
. (32)
Pislq userednennq heneratrysy E ,[ ]− (− ( ))− +
…e s uτ γ
po hranyçnij wil\nosti v
(16) pry m < 0 zhidno z (32) vstanovlg[t\sq spivvidnoßennq (29), z qkoho pry
s → 0 vyplyva[ (30). Na pidstavi (32) heneratrysu (29) lehko obernuty po s i
oderΩaty wil\nist\ (31).
Teoremu dovedeno.
Dlq vyznaçennq heneratrysy çysla vymoh za period T ′ ( u ) N*
( u ) = n ( T ′ ( u ) )
vykorysta[mo formulu dlq çysla vymoh n ( t ) = ν ( t ) na intervali [ 0, t ]
nt ( z ) = E z en t t z( ) − ( − )= λ 1 , 0 < z < 1, t > 0, (33)
i poznaçymo ßukanu heneratrysu
n*
( u, z ) = E ,
*[ ]( ) ′( ) < ∞z T uN u
.
Teorema 4. Qkwo m < 0, to heneratrysa n*
( u, z ) vyznaça[t\sq spivvidno-
ßennqm
n*
( u, z ) =
λ ρ
c
e F u dsz− ( )
∞
− ( + )∫ v v v
0
+
+
λ ζρ
m
e F u y d y ds
u
z− ( )
+
∞
+− ( + − ) { < }∫∫ v v v
00
P , sz = λ ( 1 – z ),
(34)
n*
( 0, z ) =
λ ρ
c
e F dsz− ( )
∞
− ( )∫ v v v
0
p
m c
+ =
1
.
Pry z → 1 ma[ misce spivvidnoßennq (podibne do (30) pry s → 0 )
P { T ′ ( u ) < ∞ } =
λ λ ζ
c
F u
m
F u y d y
u
( ) + ( − ) ( < )
+
+∫
0
P , (35)
oskil\ky m = λ µ – c < 0, ρ– ( sz )
z→ →
1
0, to pry u = 0 P { T ′ ( 0 ) < ∞ } =
λµ
c
<
< 1.
Analohiçno, heneratrysa çysla vymoh do bankrutstva n ( τ+
( u ) ) vyznaça-
[t\sq spivvidnoßennqm
E , E ,[ ] [ ]( ( )) + − ( ) ++ +
( ) < ∞ = ( ) < ∞z u z un u s uzτ ττ τ =
= P P{ } { }+
→
+( ) > → >ζ θ ζs zz
u u
1
. (36)
Dovedennq. Pislq userednennq (32) za wil\nistg (31) oderΩymo spivvidno-
ßennq
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POVEDINKA KLASYÇNYX PROCESIV RYZYKU PISLQ BANKRUTSTVA … 1351
n*
( u, z ) = e T u dtt z− ( − )
∞
∫ { }′( ) ∈λ 1
0
P =
=
λ τ λ
c
dz e F u dt zP{ }−
∞∞
− ( − )(− ) ∈ ( + )∫∫ v v v
00
1 +
+
λ τ ζλ
m
e dt F u y d y dt z
u
− ( − ) −
∞∞
+{ }(− ) ∈ ( + − ) { < }∫ ∫∫ 1
0 00
P Pv v v ,
v qkomu pidkreslenyj intehral, zhidno z (32), zbiha[t\sq z eksponentog
e s
s z
z
z
− ( )
= ( − )
−vρ
λ 1
( sz → 0 pry z → 1 ). Tomu z ostann\oho spivvidnoßennq dlq
n*
( u, z ) vyplyva[ (34), z qkoho pry u = 0 vyznaça[t\sq heneratrysa n*
( 0, z ). Z
(34) pry z → 1 vyplyva[ spivvidnoßennq (35), wo zbiha[t\sq z (30) pry s → 0.
Spivvidnoßennq (36) oderΩu[mo userednennqm heneratrysy n ( t ) = ν ( t ) za roz-
podilom τ+
( u )
e d u t z ut z s uzλ ττ τ( − ) +
∞
− ( ) +{ } [ ]( ) < = ( ) < ∞∫
+1
0
P E , , sz = λ ( z – 1 ).
Qkwo vymohy ξk magt\ pokaznykovyj rozpodil, to
Π ( x ) = λ λF x e bx( ) = − , x > 0, b > 0,
P s y y q s es
s y
+
+
+
− ( )( ) = ( ) > = ( ){ } +, P ζ θ ρ , y > 0,
ρ+ ( s ) = b p+ ( s ), c p+ ( s ) ρ– ( s ) = s,
spivvidnoßennq dlq G ( s, u, u1 , u2 ) ta Gi ( s, u, ui ), i = 1 3, , sprowugt\sq i ]x
obernennq po ui magt\ vyhlqd
g ( s, u, x , y ) = λ ρ ρb s e I y us y u
−
− ( )( − )( ) { > }−
,
g1 ( s, u, x ) =
λ ρ
ρ
b s
b s
e b x u−
−
− ( + )( )
+ ( )
,
g2 ( s, u, y ) = λρ ρ
−
( )( − )−( ) { > }−s e I y us u y by
,
g3 ( s, u, z ) = λ ρbe e I z ubz s u z− ( )( − )[ ]− { > }−1 ,
a ]x poxidni po s pry s = 0 sprowugt\sq (dyv. (10))
g′ ( 0, u, x, y ) =
λb
m
e I y ub u x− ( + ) { > },
′( ) = − ( + )g u x
m
e b u x
1 0, ,
λ
, m < 0,
′ ( ) = { > }−g u y
m
e I y uby
2 0, ,
λ
,
′ ( ) = ( − ) { > }−g u z
b
m
e z u I z ubz
3 0, ,
λ
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
1352 D. V. HUSAK
Znaçno sprowugt\sq teoremy 3 ta 4, qkwo vraxuvaty, wo
d
dy
y q e yP{ < } =+
+ +
− +ζ ρ ρ , ρ+ = b p+ , y > 0, m < 0.
1. Husak D. V., Korolgk V. S. O momente proxoΩdenyq zadannoho urovnq dlq processov s ne-
zavysym¥my pryrawenyqmy // Teoryq veroqtnostej y ee prymenenyq. – 1968. – 13 , # 3. –
S.S471 – 478.
2. Husak D. V. O sovmestnom raspredelenyy vremeny y velyçyn¥ pervoho pereskoka dlq odno-
rodn¥x processov s nezavysym¥my pryrawenyqmy // Tam Ωe. – 1969. – 14, # 1. – S. 15 – 23.
3. Husak D. V., Korolgk V. S. Raspredelenye funkcyonalov ot odnorodn¥x processov s neza-
vysym¥my pryrawenyqmy // Teoryq veroqtnostej y mat. statystyka. – 1970. – V¥p. 1. – S. 55
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OderΩano 30.06.2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
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| id | umjimathkievua-article-3393 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:41:43Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ff/06163b19e1a38f93b5db2842c28c5dff.pdf |
| spelling | umjimathkievua-article-33932020-03-18T19:53:10Z Behavior of classical risk processes after ruin and a multivariate ruin function Поведінка класичних процесів ризику 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 classical risk process after ruin and a multivariate ruin function. Установлены соотношения для распределения функционалов, связанных с поведением классического процесса риска после момента разорения, и многозначной функции разорения. Institute of Mathematics, NAS of Ukraine 2007-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3393 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 10 (2007); 1339–1352 Український математичний журнал; Том 59 № 10 (2007); 1339–1352 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3393/3529 https://umj.imath.kiev.ua/index.php/umj/article/view/3393/3530 Copyright (c) 2007 Gusak D. V. |
| spellingShingle | Gusak, D. V. Гусак, Д. В. Behavior of classical risk processes after ruin and a multivariate ruin function |
| title | Behavior of classical risk processes after ruin and a multivariate ruin function |
| title_alt | Поведінка класичних процесів ризику після банкрутства та багатозначна функція банкрутства |
| title_full | Behavior of classical risk processes after ruin and a multivariate ruin function |
| title_fullStr | Behavior of classical risk processes after ruin and a multivariate ruin function |
| title_full_unstemmed | Behavior of classical risk processes after ruin and a multivariate ruin function |
| title_short | Behavior of classical risk processes after ruin and a multivariate ruin function |
| title_sort | behavior of classical risk processes after ruin and a multivariate ruin function |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3393 |
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