1

.

2009

2

, .

, , .

, , , .

3

1.

2.

1:

1.

1.1

1.2

1.3

1.3.1 VaR

1.3.2 (Expected Shortfall)

1.4

1.4.1

1.4.2 VaR

1.4.3

2:

2.1 (Market Risk)

2.1.1 -

2.1.2

2.1.3 Monte Carlo

2.2 (Operational Risk)

2.2.1

2.2.1.1 (BI)

2.2.1.2

2.2.2

3: (Credit Risk)

3.1

3.1.1 Merton

3.1.2 Merton

3.1.3

3.1.4

3.1.5

3.1.6 KMW

3.1.7

3.1.8

4

3.2

3.2.1

3.2.2

3.3

3.3.1 Bernoulli

3.3.2 Poisson

3.3.3

3.3.4

3.4 Monte-Carlo

3.4.1

3.4.2 Bernoulli

3.4.3 Bernoulli

4:

4.1

4.1.1 VaR

4.2

4.2.1

4.2.2 Euler

4.2.3 Euler

5:

5.1 Fsn

5.2 Poisson

5.3 -

5.4 Panjer

5.5 Poisson

5.6 Poisson

5.7 Poisson

5

1.

20 . 1950

.

Hanry Markowitz 1952

. ,

-

.

1970.

:

i)

.

ii)

.

2.

1970 1988

(Bassel I).

:

.

,

.

. 2001 ,

(Bassel I), .

, ,

:

6

i) : .

ii) : .

iii) : .

7

1 :

1.

1.1

(, F, P)

.

1.1: ,

S

V(s).

1.2:

[S, S+] :

SSss VVL , ,

.

1.3: ssL ,

.

,

.

,tY t tYY1 .

ttttt VVLL 11,1

tZttFV , (1.1),

t d- dtt ZZ ,1, ,..., ,

F dx .

(1.1) .

,tX t

1ttt ZZX ,

ttt ZtXZttFFL ,),1(1 1

8

1.4: t d

tt ZtXZttFFx ,),1( , dx

)( 11 ttt XL .

F , -

d

iittZttt XZtfZtFL i

1

,11 ),(,

L - .

1.5: :

d

i

itZttt XZtfZtFx i1

),(,

.

9

1.2

.

.

1tL t

F F

tF - Boole. :t sx s tF , -

Boole

(). 1tx t

F F 1tx

tF .

1tL t

F F :

11 1( ) ( )

t tt t t tL F tF P x P L F

1tL

F

xF d . ,tX t

.

1 1 1( ) ( )

tL t ttF P x P L

,

.

10

1.3

.

:

)

)

)

) .

. -- (VR)

(Expected Shortfall).

1.3.1. VR

--

.

, :

( ) ( )LF P L (1.3.1)

LF

.

:

: ( ) 1LinF F

LF

.

1.3.1 (--).

0,1 . -- ,

,

L (1-).

: ( ) 1 : ( )LV R inF P L inF F (1.3.2)

11

0,95 0,99 1 10

.

VR

(1-).

VR .

1.3.2 ( )

i) :T ,

T :

( ) : ( )T y inF x T x y

.

ii) F , F

F .

0,1 - F

: ( )q F F inF x F x

1.3.1 0x - F

:

i) 0( )F x ii) ( )F x

1.3.2. (Expected Shortfall)

1.3.2 ( ). L E L

LF ,

0,1 :

11

1 u LES q F du (1.3.2)

( )u L Lq F F u LF .

:

11

1 uES V R L du

12

, VR

u .

ES L . ES V R .

1.3.2 L

LF 0,1 , :

( ; ( ))( )

1

E L L q LES E L L V R (1.3.3)

( ; ) ( )AE X A E XI

X .

1: LF (1.3.3)

:

1( ; ) (1 )

1ES E L L q q P L q (1.3.4)

1.3.3 i iL

LF :

1

,

1lim[ (1 )]

n

i ni

n

L

ESn

(1.3.5)

1, ,n n nL L 1, , nL L [ (1 )]n

(1 )n .

ES

VR. ,

VR.

.

13

1.4

(coherent risk)

1.4.1

(, F, P) .

L0 (, F, P) (, F)

.

L0 (, F, P)

.

.

) L1 L2 , L1+L2

) L1 L1 >0

: R.

(L)

L,

.

(L) 0 .

: R

, :

1 ( ) L

(L+) = (L)+

2 () L1, L2 (L1+L2) (L1) + (L2)

3 ( ) L >0 (L) = (L)

4 () L1, L2 L1L2, (L1) (L2)

14

1.4.1 ( )

,

, ( ) .

1.4.2 VR

d=100 .

.

(2%).

100 .

, 1t 105,

.

Li i-, 100 -5

i i-.

i 1 [ , 1]t t

.

Li = 100Yi -5(1 - Yi) = 105 Yi 5

Li

P(Li = -5) = 0.98 P(Li = 100) =0.02.

. 10.000.

100 100 .

.

VR.

VR 95%

.

: LA = 100L1 VR0.95(LA) = 100VR0.95(L1)

P(L1-5) = 0.98 0.95 P(L11) = 0 < 0.95 l

15

VR0.95(LA) = -500

500

VR

95%.

. 100 100

1 1

105 500B i iIi Ii

L L Y

100

1

( ) 105 500a B a iIi

VaR L q Y

100

1

B (100, 0.02)ii

M Y

P(M5) 0.984 0.95 P(M4) = 0.949 < 0.95

0.95( ) 5q M

0.95( ) 25BVaR L .

125

VR 95%.

.

.

VR .

, :

100 100

1 11 1

( ) 100 ( ) (100 )i ii i

P L P L P L P L

, ,

.

--

.

1.4.2 ( VR

)

16

X ~ Ed (,,)

01

: , d

i i ii

M L L X R

L1, L2 M 0.5 < 1

1 2 1 2( ) ( ) ( )V R L L V R L V R L

1.4.3

1.4.3. .

: ,

11

( )1

ES V R L duu

.

:

L1, , L1 L1,n

Ln,n m 1 m n :

1, 11

sup : 1 i ...m

m

i n i i mi

L L L i m

L L F

1 1( , ), . . . , ( , )

n nL L L L F.

( , ) + i i iL L L L ,( + ) i nL L 1( + ) , , ( + ) nL L L L :

1, 11

( ) sup ( ) ... ( ) : 1 ...m

m

i n i i mi

L L L L L L i i m

1 1

sup ... : 1 ...mi i m

L L i i m

1 1sup ... : 1 ...

mi i mL L i i m

, ,1 1

m m

i n i ni i

L L

17

m = n(1-) n, ( ) ( ) ( )ES L L ES L ES L

18

2 :

2.1. (Market risk)

(.. ,

, , ) .

.

2.1.1

1tX

:

1 ( , )t dX N

.

1 [ ] 1( )t t tL X .

:

[ ]( ) ( )t t tx c b x (2.1.1)

tc tb

t.

1tX .

:

1 [ ] 1( ) ( , )t t t t t t tL X N c b b b (2.1.2)

,

1, ,t n tX X .

,

-

19

.

1 1 1( , )t t d t tX F N

1t 1t - .

.

. ,

.

2.1.2

.

[ ] 1( )t tL X

1, ,t n tX X . ,

.

[ ]( : 1,...,s t sL X s t n t (2.1.2)

sL

s.

.

xF

xF .

1, ,t n tL L

[ ]( )t X xF .

.

.

20

2.1.3 Monte-Carlo

1, ,t n tX X .

.

m-

, :

(1) ( )1 1, ,

mt tX X

:

( ) ( )1 [ ] 1( : 1,...,

i it t tL X i m

1tX ,

.

21

2.2. (Operational Risk)

, (, )

.

2.2.1

:

i) () ii) .

2.2.1.1 ()

,

.

,

.

t

:

3

1

1( ) max( ,0)t t iBI

t i

RC OR IZ

3

01

t it Ii

Z I

t iI t i

15% .

2.2.1.2

8 :

, , ,

, , , ,

.

22

.

:

3 8

1 1

1( ) max ,0

3

t t is j j

i i

RC OR I

, t i

j

.

.

2.2.2

,

: ,

, , -

, ,

, -

.

t.

:

, . , ,: 1,..., ; 1,...,8; 1,...,7; 1,...,t i b t i bX i T b N

, .t i bX -

b t i .

, ,t i bN :

T : ( 5T )

23

b

t i :

, ,7, , ,

1 1

t i bNt i b t i b

k

L X

t i :

, ,8,

1 1

t i bNt i t i b

k

L L

tL t

VR

.

( ) ,

:

( )t tAMRC OR L

= 0,99 -0,999.

,

:

8,

1

( )t t bAMb

RC OR L .

24

3 : (Credit Risk)

.

,

, .

-.

Merton (1974),

,

.

,

.

. -

.

-.

25

3.1

.

. ( tX )

tX

0t .

3.1.1 Merton

tV

.

Merton :

T. t

St Bt . (

).

.

Vt=St + Bt, 0 t T .

,

.

:

i) TV B ,

.

St = Vt B.

.

ii) TV B ,

. ,

26

.

, T TB V , ST = 0.

:

max ,0T T TS V B V B (3.1)

min ,T T TB V B B B V (3.2)

Merton (Vt) Brown

:

t t t tdV V dt V dW

, 0 Brown ( tW ).

20

1exp

2T TV V T W

2 20

1ln ln ,

2TV N V T T

:

20

1ln( / )

2ln lnT T

B V T

P V B P V BT (3.4)

.

3.1.2 Merton

Merton

Vt

.

(3.1.1)

i)

ii) 0

27

iii)

.

Th V , (3.1)

(3.2) (3.1.1).

( , )tF t V

t T .

. F(t,Vt) (PDE)

F(t,u) :

2 21, , , ,2

t y uu uF t u u F t u ruF t u rF t u 0,t T (3.5)

f(t,u)=h().

(3.5) Black- Scholes.

. F(t,Vt) Q

( , )tF t V

Q. Q Vt

:

t t t tdV V dt V dW (3.3)

W W Q Brown (

).

:

( )( )( , ) T

Q r T tt V tF t V E e h F (3.6)

28

3.1.3

PDE (3.5)

(3.6) Black-Scholes CBS.

t :

( )

,1 ,2

2

,1 ,2 ,1

( , ; , , , ) ( ) ( )

1ln ln ( )

2

BS r T tt t v t t t

T v

t t t v

v

S C t V r B T V d Be d

V B r T t

d d d T tT t

(3.7)

2 201

ln (ln , )2T v v

V N V r T T

t=0 , :

2

0

0,2 0,2

1ln ln

2( ) 1 ( )

T v

T

v

V V r T

Q V B Q d dT

(d)=1-(-d) 1-(d0,2) .

1-(dt,2)

t.

3.1.4

(ii) (3.11) -

tT

0( , ) exp( ( ))t T r T t

(3.2) :

0( , ) ( , ; , , , )BS

t t vB B t T P t V r B T (3.8)

29

( , ; , , , )BS t vP t V r B T Black-Scholes strike B

( )tV r .

( )

,2 ,1( , ; , , , ) ( ) ( )BS r T t

t v t t tP t V r B T Be d V d (3.9)

,1td ,2td (3.7).

(3.8) (3.9) :

0 ,2 ,1( , ) ( ) ( )t t t tB t T B d V d (3.10)

3.1.5

(3.10)

C(t,T). -

P0(t,T) P1(t,T)

.

11 0

0

( , )1 1( , ) ln ( , ) ln ( , ) ln

( , )

P t TC t T t T T t

T t T t P t T (3.11)

Merton 11

( , ) tP t T BB,

,2 ,1

0

1( , ) ln ( ) ( )

( , )t

t t

VC t T d d

T t B t T (3.12)

20( , ) 1ln ( )2

( ,1)v

t

v

B t TT t

Vd t

T t dt,2

30

T-t, C(t,T) v

0( , )

t

B t Td

V.

3.1.6 KMV

Merton

KMV 1990.

KMV .

KMV

EDF. EDF

KMV.

(3.4) =1 (d)=1-(-d)

2

0

1ln ln

21

v v

Merton

v

V B

EDF (3.13)

KMV EDF Merton.

Merton

.

(3.13).

(DD)

0

0

( )

v

V BDD

V (3.14)

B V0 .

3.1.7

.

31

:

, :

Table 8.2: Probabilities of migrating from one rating quality to another within one year.

Source: Standard & Poors CreditWeek (15 April 1996)

.

, S&P

.

, CCC

.

[0,]. ( ), 0 j np j

.

j

. (0)p

.

t v t v t tdV V dt V dW

32

20

1exp

2T v v v TV V T W

.

0 1 1... n nd d d d

1( ) ( )j T jP d V d P j {0,..., }j n .

.

1d .

VT jd .

:

2

0

1ln ln

2T v v

T

v

V V T

XT

20

1ln ln

2j v v

j

v

d V T

dT

j 1j T jd X d .

3.1.8

.

Merton.

m-

Vt Vt = (Vt,1,, Vt,m)

33

m- Brown v = (v1,, vm)

v = (v1,, vm) .

i VT,I

2

0

1exp

2T v v v TV V T W

v= vi , v= vi WT=WTi

H WT=(WT,1,WT,m) WT~Nm(O,TP)

0 1 1... n nd d d d

.

3.2

(Threshold)

.

i

x,i di

[0,].

x=(x1,,xm).

x.

3.2.1

m-

.

1 i m , Si i

,

Si = ST,i

0,1,...n .

34

t=0, -.

1( ,..., )mY Y Y

1 1( ) ,..., , 0,1m

m mp y P Y y Y y y

.

( 1), 1,...,i iP P Y i m .

:

2 2

( , )( , )

( )( )

i j i ji j

i i j j

E y Y P Pp y Y

P P P P (3.4.1)

2 2 2 2

i i i i i i iY E Y P E Y P P P .

1

m

ii

M Y .

:

1

m

i i ii

L eY

ei i 0i 1

.

S Y.

S

Y, .

S :

1 (1) ( )( , , ) ( , , )d

m mS S S S

35

(1) ( )( ,..., )m (1,..., )m .

1,..., 1m m

-

S .

:

( 1), 1,...,iP Y i m : .

( 1,..., 1)i iP Y Y , 1, , 1,..., ,2i i m m

-.

:

2 1i i iE Y E Y P Y

21, 1i j i jE YY P Y Y

2

2,i jY Y

22

2, ,y i jP P Y Y i j

3.2.2

3.2.2: 1( ,..., )mX X X m-

mxnD dij i ,

i 1

...ni i

d d .

1i

d ( 1)ni

d

( 1)j ji i i iS j d X d , 0,..., , 1,...,j n i m

36

(X,D) 1( ,..., )mS S S .

x

( ) ( )i iF x P x x .

i- D - i.

, ( Si=0)

1i iX d i :

1( )i i ip F d .

( , )i jP Y Y i j

( Xi Xj).

( , )i jP Y Y (1)

1 1

( ) ( , )i j i i j jE YY P X d X d .

, Xi Xj

.

.

, .

.

: d-

[0,1]d .

37

1( ) ( , , )dC u C u u

.

C : [0,1] [0,1]dC

:

1) 1( , , )dC u u iu

2) (1,1,...,1, ,1, ,1)i iC u u 1,..., , [0,1]ii d u

3) 1 1( , , ),( , , ) [0,1]d

d db b i ib :

1

1

2...

1

1 1

... ( 1) ( ,..., ) 0dd

d

i ii di

i i

C u u

1j j

u 2j j

u b 1,...,j d .

(Sklar 1959). F 1, , dF F .

: [0,1] [0,1]dC , 1, , dX X

,

1 1 1( , , ) ( ),..., ( )d d dF X X C F X F X (4.3)

, ,

1 1 1...x x dRanF RanF RanF ( )i iRanF F R

1F . , C 1, , dF F

, F (i)

1, , dF F .

: 1( , , )dC u u Frchet:

1

1

max 1 ,0 ( ) min , ,d

i di

U d C u u u

38

: F 1, , dF F :

1

1

max ( ) 1 ,0 ( ) min ( ), , ( )d

i i di

F x d F x F x F x

F .

S .

3.2.2: ( , )X D ( , )X D

1( ,..., )mS S S 1( ,..., )mS S S .

:

i. S S .

( ) ( )i iP S j P S j , 0,..., , 1,...,j n i m

ii. X X C

.

-

1, , 1,...,i i m 1,...,i ip p . :

1 1 1 1 11 1 ,...,1,..., 1 ,..., ( ,..., )i i i i i i i i i ip Y Y P X d X d C P P ,

1,...,i i

C - C .

. X

Y . :

1,..., ,..., ,2C m

.

39

3.3 - -

()

.

3.3( Bernoulli): p m p-

1( ,..., ) ,

1( ,..., )mY Y Y Bernoulli ,

: 0,1iP , 1 i m , ,

Bernoulli

:

1( ) ( )i iP Y P

y y y1( ,..., )m 0,1m

:

y1

1( ) ( ) (1 ( ))i i

my y

i iiP Y P P

3.4 ( Poisson): p

, 1( ,..., )mY Y Y Poisson

, : 0,i , 1 i m

Y Poisson

i .

1

m

ii

M Y , Poisson i ,

M .

!

1

1

( )

( ) exp ( )

m

imi

ii

P M

40

3.3.1 Bernoulli

..

( ) : 0,1iP

, ,

Bernoulli

1( ) ( )i iP Y P

:

. Bernoulli

iP .

Bernoulli .

1( )Q p

P)S( .

Q q , m

Bernoulli q

q m.

(1 )mm

P M Q q q q

:

(1 ) q)mm

P M q q dS(

.

Bernoulli

.

41

iX ,

( )ip

( ) ( )i ip h x

: (0,1)h

( ) ( )h x x 1

( ) 1 exp( )h x x , =(1,,)

, .

3.3.2 Poisson

Credit Risk Poisson.

i Poisson i

:

i i ik w

0ik , 1,...,

pi i iW W W 1ij

j

W

p ( , )j jS 1,...,

2j j j 0j .

1jE

2

j j ( ) ( )i i i iE k E W k .

:

( 1) ( 0) ( 0 )i i iP Y P Y E P Y

iY Poisson :

( 0 ) (1 exp( )) ( )i i i i i iE P Y E kw k E W k

Poisson, 1

m

ii

M Y

Poisson

42

1

( )m

i ii

M Poi k w

M Poisson

, , S

( , ( 1))bN N .

p, M p

, :

1 1 1 1 1

( )p pm m m

i i i ij j j i iji i j j i

k w k w k w

,...,i pM M

jM

Poisson 1

( )m

i ij ji

k w , j j .

1,..., jM

,

:

1

pd

jj

M M

1

( )m

i ij ji

k w ,

jM .

3.3.3

Bernoulli.

.

.

43

i ie , i iY

i i

(0,1]

.

m :

( )

1

mm

ii

L L i i i iL e Y

:

(1) p- : 0,1pi

, , i iL

:

( )i iE L

(2) : p

( )

1

1 1lim lim ( )

mm

im mi

E Lm m

p .

.

(3) c 2

1

mi

i

ec

i m.

3.5.3: ( )

1

mm

ii

L L

1, 2, 3. ( )P

i iL

.

44

( )1lim mm

Lm

3.3.3: ( )

1

mm

ii

L L

1, 2, 3 S .

S q , . , 0q S .

:

( )1lim ( ) ( )mm

q L qm

.

Bernoulli,

.

3.3.4

Bernoulli. :

i. Bernoulli Monte Carlo.

ii. .

iii. Bernoulli

.

Bernoulli .

3.3.4 : p-

p m p-

45

1( ,..., ) , , 1, , mX X

.

3.3.4: (, ) m-

. p- ,

1i i

i x dY I Bernoulli

, :

1( ) ( )i i ip P X d

46

3.4 Monte Carlo

3.4.1

X , , ,F P ,

f.

( ) ( ) ( )e h x h x f x dx (3.4.1)

h.

( ) x Ah x I A .

(3.4.1)

X , Monte Carlo.

f.

3.1: (Monte Carlo intergration)

(1) 1, , nX X f.

(2) Monte Carlo 1

1( )

MC n

n ii

h Xn

Monte Carlo

( ).

g

r(x) :

( ), ( ) 0

( ) ( )

0, ( ) 0

f x g x

x g x

g x

r

:

47

( ) ( ) ( ) ( ( ) ( ))gh x r x g x dx E h X r X

:

3.2: ( )

(1) 1, , nX X g.

(2) IS : 1

1( ) ( )

IS n

n i ii

h X r Xn

.

IS Monte Carlo.

:

2 2 21var ( ) ( )IS

ng gE h x r xn

2 21var ( )MC

ng gE h xn

2 2( ) ( )gE h x r x

2( )gE h x .

,

IS

n

g.

- h .

( ) ( )( )

( ( ))

f x h xg x

E h x (3.4.2)

( ( ))

( )( )

E h xr x

h x

1 1 1( ) ( ) ( ( ))IS

h x r x E h x .

IS (3.4.2)

( ( ))E h x .

(3.4.2) IS .

48

IS

X .

t :

( ) ( ) ( )txxM t E e f x dx

x, .

( )xM t , IS :

( ) ( ) ( )txt xg x e f x M t

( )

( ) ( )( )

txt x

t

f xr x M t e

g x

t x.

( exp( ))( )

( )t t x

E x txEg x

M t

t , :

( ) : ( )tx

xx cE r x x c E I M t e (3.4.4) .

tx tce e x c 0t , :

( ) ( )tx tcx xx cE I M t e M t e

(3.4.3) t ,

t .

, t ln ( )xM t tc .

(3.4.3) :

( exp( ))ln ( )

( )x tx

d E x txM t tc c c

dt M t.

3.4.2 Bernoulli

)

0,1m

1Y .

49

:

1

1( ) (1 )i i

my y

i iiP Y P P , 0,1

my

iP .

L :

1 11

( ) (exp( )) ( ) ( 1 )i i im m m

teY teL i i i ii i

i

M t E t eY E e e P P

Q(t) : ;( )

tL

t p

t

eQ y E Y y

M t,

1 1

1

expexp( )

(1 )( ) exp( ) 1

i i

m

i i mi y yi i

t i iit i i

t eYteY

Q y P y P PM t te P P

,exp( )

exp( ) 1i i

t i

i i

te Pq y

te P P.

1

, ,1

(1 )i im

y yt t i ti ii

Q y q q , tilting

,t iq .

,t iq 1 t , t .

)

:

( ) ( )P L c .

.

:

3.3: (IS )

(1) ip

50

1

exp( ) ( )

exp( ) ( ) 1 ( )

mi i

ii i i i

te Pe c

te P P

( , )t t c tilting.

(2) n1 1,..., mY Y .

,

i-:

( , )

( , )

exp( ) ( )

exp( ) ( ) 1 ( )

c i i

c i i i

t e P

t e P P

(3) 1

( , ) exp( ) ( ) 1 ( )m

L i i iiM t te P P

L.

n1 1

m

i ii

L eY 1( )(1), , nL L .

IS :

1

,1

( , ) ( , )11

1( ) ( , ) exp

j

IS nj

n L c cL cj

M t I t Ln

) IS

Bernoulli

Gauss ( )ip (0, )pN .

( , )pN

p .

,

.

( )r :

1

1 1

1

1exp ( )

12( ) exp

1 2exp ( ) ( )

2

r

51

IS

3.4: ( IS Gauss)

(1) 1, , ( , )n pN I

(2) i 1

,1

( )IS

n i 1.

(3) IS :

1

,1

1

1( )

IS ISn

n ni ii

rn

: .

1

,1IS

n P L C , IS

P L C

IS g

11( ) ( )exp2

g P L c

.

,

P L C .

g IS .

,

:

11max exp2

P L c

52

3.4.3 Bernoulli

1,...,t n , tm

t tM

.

Qt (0,1), tM

. ,t t q tM Q B m q .

1 2, .

)

1 t n ,1 ,,..., tt t mY Y tm .

1

1

, ,( ,.., ) (1,..., )

,...,t

tt i t i

i i m

MY Y

t .

t tM mE

t tM mE

,

n-.

1 1

( 1)...( 1)1 1( 1)...( 1)

t

n nt t t

t tt t t t

M

M M M

mn n m m m

=1 :

53

1

1n

t

t t

M

n m

:

2

2

2

)

Qt .

, .

tm t , tM -

:

1 1 1

0

,1( ) (1 )

( , ) ( , )m b

b mm mP M q q dq

b b

:

1

,( , , )

( , )

nt t tt

t t

M b m MmL b data

M b

54

4 :

- .

,

, .

4.1

.

Frchet.

Frchet.

1( ,..., )dL L L ,

,

: d .

( )L .

( )L .

.

((L)),

( )L

L .

4.1: iF , 1,..,iL i n

.

, :

min ((L)) max

maxp .

(4.1) (4.1)

.

55

inf : , 1,...,

sup : , 1,...,

i i

i i

p L L F i d

p L L F i d

(4.2)

1,..., dF F (~)

.

4.1 1 VR

= VR.

VR(((L))

iF , 1,..,iL i d .

iL , .

4.1.1 (comonotonicity). 1,..., dX X

comotonic Frchet

1 1( , , ) min , ,d dM u u u u .

4.1.1: 0 1 1,..., dL L comotonic

.

1 1... ( ) ... ( )d dV R L L V R L V R L .

F L.

Fi,

C : F=C(F1,,Fd)

1,..., dL L C.

W C M

W M Frchet .

:

56

(2) oC C oC

1,...,d

dX X X 1 1 1,..., , ,i i i dX X X X X .

1d

dX ( ) sup : ,x d d d ds x X x s

s .

c d , s :

, 1 ( )( , , )c d s cF F L S

1 1

, 1 ( ) 1 1 1 1,...,

( , , ) sup ( ), , ( ), ( )d

d

c d s d d d xx x

F F C F x F x F s

( )dF x dF x .

:

( ) inf : , 1,...,i im s P L s L F i d

, 1 ( )( , , ) :c d s dF F c C

dC - .

4.1.1 ( )

L 1( 2)d d 1,..., dF F copula C.

oC oC C .

: d , s

, 1 ( ) , 1 ( )( , , ) ( , , )c d s c d sF F F F (4.1.1)

, :

57

1

max , ( ) , ,1( )

min ,...,

d

ot

d

t C u u tC u

u u

, 1 ( )( ,..., )oC d st F F (4.4.1)

, 1,max ( ,..., )oC dV R F F

. (4.4.1) :

,maxV R L V R

0 1 , (4.1).

(4.1.2) ( )

F 1 dF F F .

0S 1F F .

( 1)

0,( ) 1 ( )

s d r

rr s dm s d inF F x dx

58

4.2

4.2.1

d

1,..., dL L .

.

:

. (L), 1

d

ii

L L

.

.

. (L)

.

.

,

1,..., dL L (C1,,ACd).

(L) =1

d

ii

AC

.

1,..., dL L , , ,F P ,

d .

\ 0d .

, 1

( )d

i ii

L L .

M

( ) :L .

59

: ( ) ( )Lk k

( )k

.

4.2: k \ 0d ,

1 . :r d

, :

( )

1

( ) ( )d

ri i

i

k

k (4.2)

( )r

i

iL , ( )L .

i iL ( )r

i i

(4.2) ( )

.

4.2.2 Euler

,

.

, :

( ) ( )t tk k , 0,t

( ) : k

.

Euler, k

, :

1

( ) ( )d

iii

rkk (4.2.2)

(4.2) (4.2.2) .

60

4.2.2: ( Euler)

,

, Euler :

:r dk , ( ) ( )

r

i

rk k

(4.2.3)

Euler :

( ) ( )r

rk k .

4.2.2 ( )

( ) ( ( ))SDr v r L

1,..., dL L .

12( )SDr

1

cov( , )cov( , ( ))

( ) ( )( ) var( )

SD

d

i j jr jSD i

iSD

L Lr L L

r L

=1, i-

:

cov( , )(1) , (1)

var( )

SDr ii i

L LAC L L

L

4.2.2 : :

,

. (0, , )dL E ( ). ,

Euler, :

61

1

1

(1)

(1)

d

ri

i ir d

jj

j

AC

AC

k

k , 1 i j d (4.1.9) (4.2.4)

4.2.3 Euler

4.2.3: k , ,

r .

r

, , :

0,( )( ) ( )

( )0,

( )( )

i i

r

i i i

r

E L E L

rE L

r E L E L

r

k

k

k

k

k

i

r

i ,

i i

.

4.2.3 : 4.2.3

Euler.

.

1

( ) ( )d

ii

L Lk k ,

.

,

( )i iAC Lk .

62

4.2.4: ,

, r

,

0,1d

, .

1 1

1

( ) ( ,..., )d

ri i d di

i

rk k (4.2.5)

(4.1.3),

(4.2.5) =1.

,

, 1

( )d

i ii

L L .

:

1 c

d

i i i i i ii i i

L L L

k k k

.

63

5 :

,

.

.

5.1 NSF

5.1: ( )

N(t) 0,t

1 2, ,X X . :

( )

( )

1

N t

N tS X (5.1)

( )( ) ( )NS N tF x P S x

.

, NS NSF .

NS

F

( )x .

5.1 (, )

x

, (0) 0 . ( )x

, (5.1)

. ( ) ( )Np P N ,

0,1,2,... . .

64

5.1 ( ) : NS ,

5.1, 0x :

( )

0

( ) ( ) ( ) ( )NS N

F x P S x x (5.2)

( )( ) ( )Kx P S x - .

(0)( ) 1x

0x (0)( ) 0x 0x .

(5.2) ,

.

.

:

0

( ) ( )sxF s e dF x , 0s

Laplace-Stieltjes, ( )( ) ( )s s .

:

0

( ) ( ) ( ) , 0 NS N N

F s P s M s s

.

5.2 ( )

(5.1) ( ) 2,NE

1( ) ( ) ( )NE S E N E x 2

1 1( ) ( ) ( ) ( )Nv r S v r N E x E N v r x

65

5.2 Poisson

Poisson

.

Poisson

.

0,1

.

, nk 1 np

1,

n n 1,...,n .

.

n

n pn:

( ) (1 ) , 0,...,nn n nn

P N p p n

( )

0

( ) ( ) ( )NS N N

F x P S x p x

.

n lim 0nn

np ,

lim ( ) , 0,1,2,...nn

P N e

66

5.2 ( Poisson)

( , ), 1,...,iN i i

S CPoi i d

, 1

( , )i

d

N Ni

S S CPoi ,

1

d

ii

1

di

ii

5.3

.

NS

F

, .

.

Poisson ( , )nS CPoi .

NS

F

NS

F .

: iX (

) NS iX . :

1

2

1 1

( ) ( ) ( )( )

( )( ) ( ) ( ) ( )

N

N NS

N

S E S x E N E XF x

v r Sv r N E X E N v r X

: NS k Y ,

k ( , )Y . (k, , )

, k Y

NS , :

67

1( )k E X 212

( )E X

31

321

( )2

( )

E X

E X

.

.

Figure 10.4: Simulated CPoi(100,Exp(1)) data together with normal- and translated- gamma

approximations (log-log scale). The 99.9% quantile estimates are also given.

10.4 120x .

90% NS

F .

.

.

5.4 Paujer

.

68

1x

1 0( ) 1P x 1( )g P x , ( )P P N .

0 0g ( )

1( ... )n

ig P x x ,

g .

1( 1) ( )

1

n ni

i

g g g .

:

0 0( 0) ( 0)NS P S P N P (5.3)

( )

1

( ) , 1n N nS P S n p g n

(5.7) ( )ng .

(5.3)

.

(5.4) ( Paujer): (p)

Paujer (, b) ,b , :

1r r

bp P

r 1r .

: ( , )N B n P

(1 )r n rrn

p p pr

0 r n

1

( 1)

1 (1 )r

r

P p n P

P p r p

Paujer (,b) 1

p

p

( 1)

(1 )

n Pb

p

69

5.4 ( Paujer)

Paujer(,b) 0 1( 0) 0g P x ,

0 0S P 1, 11

ri

r i ri

bS g S .

5.5 Poisson

Poisson ( , )CPoi ( )N Poi

F . Poisson

( ) ( )v r N E N ,

( ) ( )v r N E N

.

( )F .

0 0( ) ( ) ( ) ( ) ( )NP P N P N dF e dF

5.5 ( Poisson).

0

( ) ( )NP e dF

Poisson F .

5.5: Poisson

F . ( ) ( )E N E ( ) ( ) ( )v r N E v r .

70

5.6 ( Poisson)

Poisson

( , ) .

( , )1

N NB

5.6 Poisson

[0,1].

, (t)

[0,t] 0t .

(t), 0t

.

5.6 ( ). 0

( )t

N N t

, 0 1 20, , ,...T T T

1

( )kT t

N t I .

5.7 ( Poisson).

0

( )t

N N t Poisson

>0 :

i)

ii) (0)=0 ,

iii)

iv) 0t , ( ) ( )N t Poi t

5.6: iii) iv) 0 u t

()-(u) (t)-() 0 :

71

( )( )

( ) ( ) ( ) uu

P N N u P N u e

()-(u)

(u,] (-u).

5.6 ( Poisson).

.

.

(1) Poisson >0

(2)

( ) 1 0( )P N t t t , 0t

( ) 2 0( )P N t t , 0t

(3) 1 1

( )Exp .

(4) 0t , ( ) ( )N t Poi t ( )N t ,

1 2, ,...,T T T

.

1 1, , / ( ) 1 0 ...

( , , )T T N t t t tf t t It

72

5.7 Poisson

Poisson ,

.

5.7 (- Poisson). -

Poisson , ( ) 0s , :

i) (0) 0N

ii) N

iii) 0t ( ) ( ) 1 ( ) 0( )P N t h N t t h h , 0h

( ) ( ) 2 0( )P N t h N t h , 0h

5.7 ( , ). -

Poisson 0t

1( ) ( )N t N t , N Poisson.

N , .

73

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