Бауэрс Н., Гербер Х., Джанс Д., Несбитт С., Хикман Дж. Актуарная математика

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Приложение

НЕКОТОРЫЕ

МАТЕМАТИЧЕСКИЕ

ФОРМУЛЫ,

ПРИМЕНЯЕМЫЕ

АКТУАРНОЙ

МАТЕМАТИКЕ

{3(t>

F(t)

f(x,

dx,

a:(t)

dF(t)

jf;J(t)

d d

dt =

-8

f(x,

+ f(f3(t),

-d

f3(t)

- f(Q(t),

-d

Q(t).

a:(t)

t t t

Ис'Чuслен.uе

1Сон.е'Чн.'Ых

раз'Н,остеЙ.

(а)

Операторы

сдвига:

Мы

не

ставим

здесь

цели

напомнить

известные

стандартные

формулы

методы,

но

указываем

те

из

них,

которые

могут

быть

менее

привычны

для

студентов-актуариев.

Интеграл,'ьн.ое

ис'Численuе.

Если

то

E(f(x)] =

f(x

1).

(Ь)

Операторы

конечных

разностей:

Ilf(x)

f(x

f(x)

(Е

-1)f(x).

(с)

Кратные

операторы

конечных

разностей:

f(x)

=1l(1l

п-l

f(x») =

(Е

_1)П

f(x)

(

_1)n-k

f(x

+k).

k=O

(d)

Оператор конечных

разностей,

примененный

про:изведению:

Il[j(x)g(x») =

f(x

1)llg(x)

g(x)llf(x).

(е)

Обратный

разностному

оператору:

если

Ilf(x)

у(х),

то

-1

(х)

f(x)

ш(х),

где

ш(х)

ш(х

+1).

Прило:же'Н,u.я..

(а)

Представление

полинома (формула

Ньютона).

Пусть

рп(х)

полином

степени

Тогда

рп

(х)

с:

рп

(а).

(Ь)

Суммирование

рядов:

k=O

если

IlF(x)

f(x),

то

f(1)

Р(2)

Р(1),

f(2)

F(З)

Р(2),

f(n)

Р(n

+1) -

Р(n),

п+l

f(x)

Р(n

Р(1)

-1

f(x)ll

х=1

(с)

Суммирование

по

частям:

n+l

1 I

g(x)llf(x)

f(x)g(x)

1 -

[f(x

1)6.у(х»)

1 .

[Доказательство:

просуммиру:йте

обе

части

равенства

ДЛЯ

Il[f(x)g(x»)

по

от

до

n.]

Распределения

вероятностей

Дискретные

ФУНкция

Ограничения

на

Производящая

функция

Моменты

распределения

параметры

моментов

(э)

СредНее

Дисперсия

Биномиальное

СП

:е

n-:е

< 1, q =

(РеВ

+q)n

пр

npq

:еР

q ,

= , ,

...

Бернулли

частный

случай

n = 1

Отрицательное

cr+:e-l

< 1,

(

)',

qe'

< 1

rq rq

pq,x=,

...

р2

биномиальное

q = 1 -

р,

r >

Геометрическое

частный

случай

n = 1

Пуассона

e->'л

0,1,

...

Л>О

e>.(e·-l)

х!

_е

аn

)

n+1

-1

Равномерное

-,х=1,

...

положительное

) , s

целое

n(1 -

еВ

1 s =

Непрерывные

llлОТНОСТЬ

Ограничения

Производящая

функция

Моменты

распределения

на

параметры

моментов

(s)

Среднее

Дисперсия

ЬЭ

ео.

Ь+а

(Ь

а)2

Равномерное

-Ь--'

(Ь

а)а

' s #

-а

2 12

Нормальное

[(Х_I')2]

0->0

(

•

)

0-2

0-V'2iГ

ехр

2и2

ехр

p.s

-2-

Р.

-00

Гамма~распр~

(30

0-1

-j3;c

О,

(3 >

.)",.<р

а а

Г(а)

(32

деление

Показательное

частный

случай

= 1

Хи-квадрат

u u k

k -

положи-

частныи

случаи

'2)

fЗ=!

Te.zrьнoe

целое

Обратное

-З/2

[(рх_",)2]

О,

exp["'(l-Jl

~)],

а а

..j21ГP

х ехр

2fЗх

- -

гауссовское

(32

Х>О

8 <

Парето

Ха

О,

аХа

ax~

+1 '

Ха

а-1'

(а

2)(а

1)2

)

а>1

а>2

Логнорма.льное

1 [

(logx

т)2]

-00

00,

с72

(е

с72

1)е

2т

с72

ехр

xo-V'2iГ

20-2'

и>О

Х>О

Приложение

ЛИТЕРАТУРА

Actuarial Society of America.

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International

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о/

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о!

Атепса,

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Allen J.

М.

(1907)

Оп

the

Relation between

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and

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Joumal

о/

the Institute

о/

Actuaries, 41:305-337.

Allison G.

О.,

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Н.

Е.

(1975)

The

Interrelationship among Inflation Rates,

Interest

Rates,

and

Pension Costs,

Transactions

о/

the Society

о/

Actuaries,

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Anderson J.

С.

Н.

(1959) Gross

Premium

Calculations

and

Profit Measurement for Nonparticipating Insurance,

Transactions

о!

the Society

о/

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Arrow

К.

(1963)

Uncertainty

and

the

Welfare

Medical

Саге,

Атепсаn

Econoтic

Review, 53:941-

973.

Balducci

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(1921) Correspondence.

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о/

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о/

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Bartlett

К.

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Batten

R. W.

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Е.

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Becker

(1991) Statistical Tests of

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о/

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Beekman J.

А.

(1969)

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48.

Beekman J.

А.

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Beekman J.

А.,

Bowers

(1972) An Approximation

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BeHhouse D. R.,

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Н.

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Applications

Life Contingencies, Jour-

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Risk

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Bellman R.

Е.,

Kalaba

Е.,

Lockett J.

(1966)

Nuтerical

Inversion

о/

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Приложение

615

Bicknell W. S.,

Nesbitt

С.

(1956)

Premiums

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Н.

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М.

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The

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Corporate Liabilities,

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о/

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Есоnоту,

81:637-659.

Boermeester J.

М.

(1956) Frequency Distribution of Mortality Costs, Transactions

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Н.,

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К.

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К.

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Ап

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Introduction

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ТЬе

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616

Приложение

Chalke

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М.

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Chamberlin

С.

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The

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Instrument

New Appraisal of

the

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the

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Work,

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(1976)

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С.

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John

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Sons.

Christiansen

(1992)

Praetical Guide

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the

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(1981) An

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Ordinary Insurance

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1975-80 Basic Tables

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Приложение

617

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Size, Transactions

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[Имеется

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В.

Введение

теорию

вероятностей

ее

приложения.

Т.

М.:

Мир,

1984.]

Feller

(1968)

Аn

Introdu.ction to Probability Theory and Its Applicatians. Vol.

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Sons.

[Имеется

перевод:

Феллер

В.

1Зведение

теорию

вероятностей

ее

приложения.

Т.

М.:

Мир,

1984.]

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С.,

Miller

N., Sternhell

С.

М.

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Theory

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Variable Benefit Life Insurance,

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М.

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Second

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618

Приложение

Gerber

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U., Shiu

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Оп

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оп

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the

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ctuaries, Record of Proceedings, 171-

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Risk

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ХХIII:З91-З99.

Приложение

6 619

Hoskins

Е.

(1929)

New Method of

Computing

Non-Participating Premiums, Transactions

о/

the

Actuarial Society

о/

Aтerica,

ХХХ:140-166.

Hoskins J.

Е.

(1939) Asset Shares

and

Their Relation

Nonforfeiture Va1ues, Transactions

o/the

Actuarial

Society

о/

Aтerica,

XL:379-393.

Huffman

Р.

(1978) Asset Share Mathematics, Transactions

о/

the Society

о/

Actuaries,

ХХХ:277-296.

Institute

Actuaries,

Faculty

of Actuaries,

(1992), Standard Tables

о/

Mortality: The "80"Series.

Institute

of Actuaries, Staple

Inn

НаН,

High Holborn, London

WCIV7QJ,

U.K.;

Faculty

of Actuaries,

St. Andrew Square,

Edinburgh

EH21AQ, U.K.

Jackson R.

Т.

(1959) Some Observations

оп

Ordinary

Dividends, Transactions

о/

the Society

о/

Actuaries,

XI:764-796.

Jenkins

А.

(1943)

Analysis

Self-Selection, among

Annuitants,

Including Comparisons

with

Selec-

tion

among

Insured

Lives, Transactions

о/

the Actuarial Society

о/

Атемса,

XLIV:227-239.

Jetton

М.

(1988)

Interest

Rate

Scenarios, Transactions

o/the

Society

о/

Actuaries, XL,

Part

1:423-437.

Jones

В.

(1994) Actuarial Calculations Using

Markov Model, Transactions

о/

the Society

о/

Actua-

ries, XLVI:227-250.

Jordan

С.

(1952) Li/e Contingencies. 2nd ed., 1967. Schaumburg,

111.:

Society of Actuaries.

Kahn

Р.

М.

(1961) Some

Remarks

оп

Recent

Paper

Ьу

Borch,

ASTIN

Bulletin, 1:265-272.

Kahn

Р.

М.

(1962)

Ап

Introduction

Collective Risk

Theory

and

Its

Application

Stop-Loss Reinsu-

rance,

Transactions

о/

the Society

о/

Actuaries, XIV:400-425.

Keller

В.

(1990) Pricing of Accelerated Benefit Plans, Transactions

о/

the Society

о/

Actuaries,

XLII:259-280.

Kendall

М.,

Stuart

А.

(1977) The Advanced Theory

о/

Statistics. Vol.

4th

ed. New York:

МастШan.

[Имеется

перевод

ИЗД.:

Кендалл

М.,

Стьюарт

А.

Теория

распределений.

М.:

Наука,

1966.]

Keyfitz N.

(1968) Introduction

the

Matheтatics

о/

Population. Reading, Mass.: Addison-Wesley.

Keyfitz

(1977) Applied

Matheтatical

Deтography.

New York:

John

Wiley

and

Sons.

Keyfitz N., Beekman

(1984)

Deтography

through

Probleтs.

New York: Springer.

King G.

(1887) Institute

о/

Actuaries' Textbook.

Part

П.

2nd

ed., 1902. London: Charles

and

Edwin

Layton.

Kischuk R.

К.

(1976) Discussion

"Fundamenta1s of Pension Funding", Transactions

о/

the Society

о/

Actuaries, XXVIII:205-21l.

Klein G.

Е.

(1993)

ТЬе

Sensitivity of Cash-Flow Analysis

the

Choice

Statistical Model for Interest

Rate

Changes, Transactions

о/

the Society

о/

Actuaries, XLV:79-124.

620

Приложение

Kornya

Р.

(1983) Distribution

Aggregate Claims in

the

Individual Risk

Theory

Model, Transactions

о/

the Society

о/

Actuaries,

ХХХУ:823-836.

Lauer J.

А.

(1967) Apportionable Basis for

Net

Premiums

and

Reserves, Transactions

о/

the Society

о/

Actuaries, XIX:13-23.

Linton

М.

А.

(1919) Analysis

ofthe

Endowment

Premium,

Transactions

o/the

Actuarial Society

о/

Атеп

са,

ХХ:430-439.

London D.

(1988)

Suтvival

Models and Their

Estiтation.

Winsted, Conn.:

АСТЕХ

Publications.

Lukacs

Е.

(1948)

Оп

the

Mathematical

Theory

Risk, Journal

о/

the Institute

о/

Actuaries

Students'

Society, 8:20-37.

Lundberg

О.

(1940)

Оп

Random

Processes and Their Application to Sickness and

Accident

Statistics.

Uppsala: Almqvist

and

Wiksells.

Macarchuk

(1969) Some Observations

оп

the

Actuarial Aspects

the

Insured Variable

Annuity,

Trans-

actions

о/

the Society

о/

Actuaries, XXI:529-538.

Maclean

В.,

Marshall

Е.

(1937) Distribution

о/

Surplus. Schaumburg, Ill.: Society

Actuaries.

Makeham W.

М.

(1874)

Оп

the

Application

the

Theory

the

Composition

Decremental Forces, Journal

о/

the Institute

о/

Actuaries, 18:317-322.

МмвЬаll

А.

W., Olkin

(1967),

Multivariate Exponential Distribution, Journal

о/

the

Атепсаn

Statistical Asso-

ciation, 62:30-44.

Marshall

А.

W., Olkin

(1988) Families of Multivariate Distributions,

Joumal

о/

the

Атепсаn

Statistical Associa-

tion, 83:834-841.

McCrory R.

Т.

(1984) Mortality Risk in Life Annuities, Transactions

о/

the Society

Actuaries, XXXVI:

309-338.

Menge

О.

(1932) Forces

ofDecrement

Multiple-Decrement

ТаЫе,

Record

o/the

American

Instit'/J,te

Actuaries, XXI:41-46.

Menge W.

О.

(1946) Commissioners Reserve Valuation Method, Record

olthe

Атепсаn

Institute

Actua-

пеа,

ХХХУ:258-300.

Mereu J.

А.

(1961) Some Observations

оп

Actuarial Approximations, Transactions

о/

the Society

ctuaries,

ХПI:87-102.

Mereu J.

А.

(1962)

Annuity

Values Directly from Makeham Constants, Transactions

о/

the Society

о/

ct'/J,aries,

XIV:269-286.

Mereu

А.

(1972)

AIgorithm for

Computing

Expected Stop-Loss Claims under

Group

Life

Contract,

Transactions

о/

the Society

Actuaries, XXIV:311-320.

Miller

М.

(1951)

Group

Weekly

Indemnity

Continuation

ТаЫе

Study,

Transactions

о/

the Society

о/

Actuaries,

ПI:31-67.