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Ruin probability by operational calculus
Insurance: Mathematics North-Holland
and Economics
8 (1989) 243-249
243
Ruin probability by operational calculus Elias S.W. SHIU
*
ruin function + is the amount of risk reserve time 0. Let a denote the Lundberg security factor,
University of Mamtoba, Winnipeg, Man., R3T 2N2 Canada
Two series formulas for the probability of eventual ruin in the collective risk model with a Poisson claim-number process are derived by the method of operational calculus. Some recent results of H.U. Gerber are also discussed. Keywords: Probability of eventual ruin, Ultimate ruin probability, Infinite-time ruin probability, Collective risk theory, Operational calculus, Lagrange series. This paper is dedicated to the beautiful subject
to Henk Boom who introduced of Risk Theory.
8=+,x)-‘-1, which should be positive, is called the relative security loading. For k = 1, 2, 3,. . , define +x
For OL2 0, define a
a-
x+-
x,
x20
x
#(a>
(1.6) Formula (1.5) is equivalent to equation (17) on page 61 of Takacs (1967). Formula (1.6) can be found in Prabhu (1965, formula (5.55)) and Gerber (1988a, formula (27)); it can also be obtained by combining formulas (35.22), (16.16) and (19.15) in Takacs (1967). This paper is motivated and stimulated by Gerber’s (1988a) fun. We shall also derive Gerber’s formulas:
,co -yyE[( S, +
x)’
(1.2) that
* Support from the Great-West gratefully acknowledged. 0167-6687/89/$3.50
(1.7)
0.1)
u( t ) = 24+ ct - SN(r) Note
e-“(S;+“)]4,;
and k.
Put S, = 0. The probability of eventual ruin (ultimate ruin probability, infinite-time ruin probability) q(u) is the probability that the risk reserve
is ever negative.
(l-4)
and
Consider the classical collective risk model, in which insurance claims occur according to a Poisson process N(t), t 2 0, and the individual claim amounts Xi, X2, X3, . . . are mutually independent and identically distributed positive random variables, with probability distribution Pr( X 5 x) = P(x) and mean E(X) = p, < co. Assume that the Poisson process N(r) is independent of the random variables { X,}, and E[N(t)] = At. Also assume that premiums are received continuously at a constant rate c. The number
...
(1.3)
me
1. Introduction and notation
s,=x,+x,+
a=Xc?=[(l+B)&‘.
at
the argument
Life Assurance
of the
Company
0 1989, Elsevier Science Publishers
(1.8) 2. A convolution ruin
series for the probability of non-
It is somewhat function
easier to work with the nonruin
is
B.V. (North-Holland)
244
ES. W. Shiu / Ruin probability
Consider a small time interval (0, s). By the Poisson assumption, the probability that a claim will occur in the interval is As + o(s). Hence, for u 2 0, the nonruin function @I(U) satisfies the relation [Takacs (1967, p. 224) Ross (1983, section 6.6.3), Tijms (1986, p. 56)]: +(u)
=AsE[+(u+
cs-A-)]+(I-hs)+(u+cs)
by operational calculus
equation
(2.4) becomes
+)=8(1+8)-‘t-+(u)*+),
We remark that, if the functions f, and the value 0 on the negative axis, then reduced to
f+)*fib)
to(s).
=Jo*fib-dh(v)
(2.8)
f2
take (2.7) is
dy.
(2.9)
(2.1)
Dividing (2.1) by s, rearranging and letting s tend to 0 + , we obtain the integro-differential equation
We now extend entire real line:
O=XE[$(u-X)]+c$+)-Q+A),
&)=8(1+8)-
~'(.)=o(O(u)-E~~(u-X)]~.
u>O,
u>o.
Since + is zero on the negative axis and positive random variable, we have
E[c#+-X)]=jo"~(u-x) u 2 0, integrating
(2.8)
as an equation
‘$++)*k(u).
on the
(2.10)
If we let 6(u) denote the Dirac delta function, may rewrite (2.10) as
(2.2)
For
~20.
X is a
dP(x).
+(u)*[+)-k(u)]
=8(1+8)-r&
we
(2.11)
To solve for G(U) in (2.11), we invert 6(u) - k(u) as the series [cf. Mikusihski (1959) Erdelyi (1962), Yosida (1984)]
(2.3)
(2.2) from 0 to u yields
$+++(O)=U~~@I(U-x)[l-P(x)]
dx.
(2.12)
$ k*‘(u), j=o where
(2.4) [Formula (2.4) is equation (X1.7.2) of Feller (1971) and equation (72) on page 158 of Takacs (1967).] Since the risk reserve U(t) tends to co as t tends to cc with probability 1, we have
k*‘(u)
=8(u)
and, for j = 1, 2, 3,. . . , k*‘(u)
= k*(ip’)(u)*
(2.13)
Hence, G(U) = [6(u)
lim +(24) = 1. U-cc
k(u).
-k(u)]
-’ * [ 8(1+
8)-‘24”,]
Letting u tend to cc in equation (2.4) and applying the Lebesgue dominated convergence theorem, we obtain 1 -+(O)=~j;,~[l
-P(x)]
dx=ap,
= -&$_[k*‘(u)‘~:].
= -!1+8’
or +(O) = 8/(1
+ 8).
(2.5)
Let us define k(u)=a[l-P(u)] = 0
if
~20
if
u < 0,
or +)=a[+P(u)].
(2.6)
With the notation fi(x)*A(x)
= Jca fi(x-Y)./~(Y) --M
dy,
(2.7)
(2.14)
Equation (2.8) is a Volterra integral equation of the second kind [Linz (1985, ch. 3)]. The series (2.12) is sometimes called a Neumann series [Brown and Page (1970, p. 226) Riesz and Sz.Nagy (1955, p. 146)]. The convolution series (2.14) had been given by Dubourdieu (1952, p. 246); I thank Franqois Dufresne for this reference. Formula (2.14) shows that + can be viewed as a compound geometric distribution, for which Dubourdieu (1952, p. 247) had given an interpretation. The formula has been rediscovered several times and in different contexts; see Shiu (1988, p. 42). Indeed, when Kendall
245
E.S. W. Shiu / Ruin probability by operational calculus
(1957, p. 209) presented the formula in the context of storage theory, he knew of ‘no phenomenological interpretation’. J.A. Beekman has consistently promoted the use of the convolution series for evaluation I/J(U). In section 12.6 of Bowers et al. (1986), $I is denoted as FL, where L is the maximal aggregate loss random variable. A major difficulty with (2.14) is the calculation of the convolutions { k *’ }. In the next two sections, we shall present two different expressions for k*j(u)* u”,.
we have .
k*“(y)=o”[l-g(E-‘)].‘(;t-I),
(3.6)
Consequently, k*“(y)*
yo,=a”[I-g(E-l,]‘$
Substituting
(3.7)
(3.7) into (2.14) yields
(P(u) = &;~a$[‘-“(+:.
(3.8)
Put G = g(E-‘);
then
3. First formula
(Gf)(u)
Let g be the generating variable X,
[Feller (1971, section VIII.3) had studied such operators.] Formula (3.8) can be written symbolically as
g(z)
function
of the random
= E(zX).
(3 .I>
(Usually, generating functions are defined for discrete random variables only. However, we do not restrict the random variable X to be discrete.) Let E denote the translation operator (forward-shift operator). We claim that P(Y)
(3.2)
= g(E-‘)YO,
= E[S(u
- X)1.
G,(u) = &e”(‘-“)“+.
(3.9)
Motivated
by the formal
exp[a(I-
G)u+]
identity
=exp(-aGu+)
one may conjecture
exp(au+),
the formula
cc (-a)” c ~G’(ui
+(u)=&
eau+).
(3.10)
J=o
[cf. Hirschman and Widder (1955, p. S), Mikusinski (1959, p. 327) and Erdelyi (1962, p. 57)]. By (3.1), the right-hand side of (3.2) is
To prove (3.10), note that (I-
G)“=
t
(-l)k(
;I)G”
k=O
m E-” dP(x) [j -cc
Y”,= I
J =
_a_(v - x)“+ dP(x)
=
’ dP(x) / --m
It follows from (2.6) and (3.2) that k(Y)=+g(E-‘)]YO,.
(3.3)
It is easy to check that, for each real number
r, (3.4)
E’(f,*~*)=(E~*)*f,=f,*(E’f,).
k*“(y)=a”[I-g(E-‘)]“(yO,*yO,*
Gk ’
(3.11)
Substituting (3.11) into (3.8) and interchanging the order of summation yields (3.10). Since g(z) = E(zx) and the random variables { X;} are independent and identically distributed,
= P(Y).
Hence, for n = 1, 2, 3,. . . , the n-fold of k is
n (-‘Jk “jkFo k!(n -k)!
[g(z)]J=E(zX~+X~+“‘+X,)=E(zs,). Hence,
for a function
(3.12)
f,
G’f(~)=[g(E-‘)]‘f(u)=E[f(u-~J)].
convolution
(3.13) Applying
(3.13) to (3.10) yields
. ..). +(u)
Since
= &
Jfo y
E[ (u - 5”):
eO(yps/)+], (3.14)
(3.5) which is (1.5).
246
E.S. W. Shiu / Ruin probability
Formula (3.14) can also be obtained by applying the method of Laplace transforms to (2.2) or (2.4). See equation (16) on page 61 of Takacs (1967), and also section 5 of Shiu (1988). Let (Y= sup{ x 1P(x) = 0). By assumption, (Y2 0. If (Y> 0, then (3.14) is a finite sum, with the index j ranging from 0 to [u/a]. For a real number Y, we let [r] denote the greatest integer less than or equal to r. Formula (3.14) is valid for continuous or discrete individual-claim-amount random variables X. Willmot (1988) has applied (3.14) to obtain formulas for q(u) when X is Gamma (with arbitrary non-scale parameter) or continuous uniform. Shiu (1988) has assumed that X takes values on positive integers only: If ch*J = Pr
by operational calculus
The coefficient kxe’.‘lm[l
/0
-R(y)]
%Z x zRx,l
dy dx
-P(x)]
dx
BP,
= R
(3.19)
mu eRx [l -P(x)]
/0
dx
[Seal (1969, p. 130) Gerber (1979, p. 124, formula (5.28))]. By an integration by parts, the denominator in the right-hand side of (3.19) can be written as E( X eRX) - (1 + O)p,.
4. Second formula i X,=k i 1=l
=Pr(S,=k), i
then, for u 2 1,
ck*‘[a(k-
1+
k=l
J=l
u)]’
j!
. I
(3.15) The coefficients { czJ } can sively by the formula
be evaluated
recur-
Formula (3.14) is an alternating series. We now derive (1.6) which is a ruin probability formula with only nonnegative terms. As pointed out in the introduction, this formula has been given by Prabhu (1965, formula (5.55)) and Gerber (1988a, formula (27)). Following Gel’fand and Shilov (1964, p. 49) we define, for each nonnegative number (Y,
Ixla, i 0,
xT=
*fm+n) = ch
c
c:mc/**.
(3.16)
r+/=k
28411
x”=
(3.17)
is an effective method for evaluating G(U). In (3.17) R denotes the adjustment coefficient, which is the positive number satisfying the equation 5c eR”k(x) dx = 1. (3.18) J0
E(eRX)=l+(l+O)p,R.
xc0
if
x20’
(4.1)
(-x>:,
unless x = 0 and (Y= 0. Also, for each nonnegative integer n, XT + (- 1)“xl = xn. For clarity, let l(x) denote the constant function that takes on the value 1 for all x. Then, xo,+xY!=l(x). Since (Gf)(x) Cl(x)
J/(u) - C eeRu
way to express
if
Note that
An APL program for evaluating (3.15) can be found in Seah and Shiu (1988). Formula (3.15) is inefficient for large u since the number of terms depends on the size of u. However, for large values of U, Lundberg’s asymptotic formula [Cramer (1955, formula (119)), TakLcs (1967, p. 49, formula (36)), Seal (1969, formula (4.64)), Feller (1971, p. 378), Beekman (1974, p. 52) Gerber (1979, p. 124, formula (5.27)) Tijms (1986, formula (1.78)) Asmussen (1987, p.
Another
C in (3.17) is
(3.18) is the equation
= E[f(x
- X)],
= l(x).
(4.2)
Hence, (I-
G)xO, = (G-1)(-x0,) = (G-l)[l(x) = (G-1)x!?. Similar
XV1 r(o*
to (3.5) XC’
r(p)
-x”t] (4.3)
we have Xa+P-l qtY+p>
(4.4)
E.S. W. Shiu / Ruin probabdity
It follows from (3.3) (4.3)
x’-1
=+‘-I]”
k*“(x)
(3.4) and (4.4) that
(n-l)!’
(4.5)
and consequently, X”, *k*“(x)
= [l(X) =
-X0]
jm
k*“(y)
*k*“(x) dy-a”[G-I]‘&
--r_o
As (1 + 6)nk *“( y) is the n-fold convolution probability density function
of the
G=g(e-“j
=
’
in (4.6) is (1 + e))n.
I
l7=0
Thus . .
Consequently,
the value of the integral
cm (-ij"E(~"jD" n.
I-G=p,D-iE(X*
Y”+- P(Y) Pl
A special case of (5.1) is the well-known formula V”X” = n!, where v = I - E- ’ is the backwarddifference operator. Gerber (1988a) has given two other proofs for (5.1). We now present a fourth ‘proof’. Let D denote the differentiation operator. Then, E-’ = eeD. Assuming that all the moments of X exist, we have
tr! (4.6)
241
bv operational calculus
Since
for nonnegative
(I-G)“x”=n!(p,)” =o
E
(1-t ej_‘=
(1+8)8F,
J=o
(2.14) becomes
(4.7) Hence,
(4.8) from which (1.6) follows. Gerber (1988) obtained (1.6) by considering the last upcrossing of the risk reserve U(t) at the level 0. As the convolution series (2.14) is usually derived by conditioning on the occurrence of the first claim, I did not realize that (1.6) could be derived as a consequence of (2.14) until Professor Gerber pointed it out to me. The formula that corresponds to (3.15) is
if
m=n
if
m>n.
(5.2) m and
n,
We remark that, since pl # 0, the operator I - G is a delta operator [Rota (1975) Roman and Rota (1978) Shiu (1982)]. Multiplying (5.1) with a”/n! and summing over n yields ^13 a”( I - G)nx” n! n=O
1 --=c 1 -up1
for u 2 0,
integers
Symbolically, ea(/&c)_\, or
the e-oG.x
right-hand Hence,
eox.
(5.3)
.
side of (5.3) is one might try pro-
ving that 1++=
cm (-a@’
= Jfo
(x,
eua)
j!
J=o
k$E[
(5.4)
(x - S,)’ e”(-‘ps~)],
which turns out to be Theorem l.(b) (1988a) [also see Gerber (1988b)]. Gerber (1988a) also proved that
of Gerber
~=J~o~E[(x-.S,)Jp’ e”(i-sll],
x # 0.
(5.5)
To prove (5.5) 5. Gerber’s
one may start with the identity
theorems (5.6)
It follows from (3.7) and (4.6) that (I-
G)nx,
= (I-
G)“[x:+(-l)“xY]
= n!( p,)“.
(5.1)
and apply (3.11) and (3.13). After I presented the paper [Shiu (1989)] at the 23rd Actuarial Research Conference, G. Willmot
248
E.S. W. Shiu / Ruin probability
kindly pointed out to me that one should be able to derive (5.4) and (5.5) by the Lagrange series. Let h be an analytic function and let z = b + w/z(z).
(5 -7) By the implicit function theorem, there is a unique root z = z(w) which reduces to b at w = 0. If f is an analytic function, then f(z) =f(z( w)) may be expanded in terms of powers of w by the Lagrange formula f(z)
=f(b)
. (5.8) Melzak (1973, p. 113) has presented an elegant proof of (5.8), which was given by Laplace. The usual proof, which requires the Cauchy integral formula, can be found in Whittaker and Watson (1927). Now, consider b = a, =
We conclude this paper with a proof of (5.5) when X is a degenerate random variable. For each constant cu, the polynomials {y(y-aj)‘+Ij=O,
eYx
1,2,
. ..}
are called Abel’s polynomials. It is known [Comtet (1974, p. 130, Theorem C), Roman and Rota (1978, p. 114), Roman (1984, p. 74)] that each polynomial p can be expanded as p(y)
y=b
f(y)
by operational calculus
= C
‘(’
-Ji,‘-’
,120
p(‘)(
ja>.
(5.13)
Although the exponential function eJ is not a polynomial, we can argue by a limiting process that ey=
E
Y(Y-cfj)‘plda
j=o
j!
(5.14)
Considering y = -ax and X = - a/a, we see that formula (5.14) is a special case of (5.5).
and h(y)
= E(e-“x).
(5.9)
Then [h(y)l’=E[exp(-YS,)] and
=
XE
[
(x -
s,)‘-l
ea@-%)] .
(5.10)
Applying (5.8) we have ~‘-‘=e”‘+x,~~~E[(x-S,)‘-‘eu~~~~~)]. (5.11) If w = -a, then z = 0 and (5.11) becomes (5.5). Formulas similar to (5.11) can be found in Takacs (1962). There is another version of the Lagrange formula [Riordan (1968, p. 146), Polya and Szegij (1972, p. 146) Goulden and Jackson (1983, p. 17)]: f(z)
1 -WV(Z)
=,~~~[~[f(~)[~(~)l~l]~=~ (5.12)
Formula (5.4) follows from (5.12).
References Asmussen, S. (1987). Applied Probability and Queues. Wiley. New York. Beekman, J.A. (1974). Two Stochastic Processes. Almqvist and Wiksell International, Stockholm, and Halsted Press, New York. Bowers, N.L., Jr., H.U. Gerber, J.C. Hickman, D.A. Jones and C.J. Nesbitt (1986). Actuarial Mathematics. Society of Actuaries, Itasca, IL. Brown, A.L. and A. Page (1970). Elements of Functional Analysis. Van Nostrand Reinhold, London. Comtet, L. (1974). Advanced Combinatorics: The Art of Finite and Infinite Expansions. Reidel, Dordrecht. Cram&r, H. (1955). Collective risk theory: A survey of the theory from the point of view of the theory of stochastic processes. Skandia Jubilee Volume, Nordiska Bokhandeln, Stockholm. Dubourdieu, J. (1952). ThPorie mathematique des assurances, I: Theorie mathematique du risque dam les assurances de r&partition. Gauthier-Villars, Paris. ErdClyi, A. (1962). Operational Calculus and Generalized Functions. Holt, Rinehart and Winston, New York. Feller, W. (1971). An Introduction to Probability Theory and Its Applications, II, 2nd ed. Wiley, New York. Gel’fand, I.M. and G.E. Shilov (1964). Generalized Functions, I: Properties and Operations. Academic Press, New York. Gerber, H.U. (1979). An Introduction to Mathematzcal Risk Theory. S.S. Huebner Foundation Monograph Series No. 8. Irwin, Homewood, IL. Gerber, H.U. (1988a). Mathematical fun with ruin theory. Insurance: Mathematics and Economics 7, 15-23.
E.S. W. Shiu / Ruin probability Gerber, H.U. (1988b). Advanced problem 6576. American Mathematical Monthly 95, 665. Goulden, I.P. and D.M. Jackson (1983). Combinatorial Enumeration. Wiley, New York. Hirschman, 1.1. and D.V. Widder (1955). The Convolutton Transform. Princeton University Press, Princeton, NJ. Kendall, D.G. (1957). Some problems in the theory of dams. Journal of the Royal Statistical Society Series B 19, 207-212. Linz, P. (1985). Analytical and Numerical Methods for Volterra Equations. SIAM, Philadelphia, PA. Melzak, Z.A. (1973). Companion to Concrete Mathemattcs: Mathemattcal Techniques and Various Applications. Wiley, New York. Mikusihski, J. (1959). Operational Calculus. Pergamon Press, New York. Phlya, G. and G. SzegG (1972). Problems and Theorems m Analysis, I. Springer-Verlag, New York. A revised and enlarged translation of Aufgabe und Lehrsdtze aus der Analysis, I, 4th ed. Springer-Verlag, Berlin (1970). Prabhu, N.U. (1965). Queues and Inventories: A Study of Their Basic Stochastic Processes. Wiley, New York. Riesz, F. and B. Sz.-Nagy (1955). Functional Analysis. Ungar, New York. An English translation of Leqons d’analyse fonctionelle (deuxibme edition), Akadtmiai Kiadb, Budapest (1953). Riordan, J. (1968). Combinatorial Identities. Wiley, New York. Reprinted with corrections by Krieger, Huntington, NY (1979). Roman, S.M. (1984). The Umbra1 Calculus. Academic Press, New York. Roman, SM. and G.-C. Rota (1978). The umbra1 calculus. Advances in Mathematics 21, 95-188.
by operational calculus
249
Ross, S.M. (1983). Stochastic Processes. Wiley, New York. Rota, G.-C. (1975). Finite Operator Calculus. Academic Press. New York. Seah, E.S. and E.S.W. Shiu (1988). Discussion of D.D. Cody’s Probabilistic concepts in measurement of asset adequacy. Transactions of the Society of Actuaries 40, 164-170. Seal, H.L. (1969). Stochastic Theory of a Risk Business. Wiley. New York. Shiu, E.S.W. (1982). Steffensen’s poweroids. Scandinavian Actuarial Journal, 123-128. Shiu, E.S.W. (1988). Calculation of the probability of eventual ruin by Beekman’s convolution series. Insurance: Mathematics and Economics 7, 41-47. Shiu, E.S.W. (1989). Evaluation of ruin probabilities. Actuarial Research Clearing House, forthcoming. Takacs, L. (1962). Introduction to the Theory of Queues. Oxford University Press, New York. Takacs, L. (1967). Combinatorial Methods m the Theory of Stochastic Processes. Wiley, New York. Reprinted by Krieger, Huntington, NY (1977). Tijms, H.C. (1986). Stochastic Modelhng and Analysis: A Computational Approach. Wiley, New York. Whittaker. E.T. and G.N. Watson (1927). A Course of Modern Analysis, 4th ed. Cambridge University Press, London. Willmot, G.E. (1988). Further use of Shins approach to the evaluation of ultimate ruin probabilities. Insurance: Mathematics and Economics 7, 275-281. Yosida, K. (1984). Operational Calculus: A Theo@ of Hyperfunctions. Springer-Verlag, New York.
and Economics
8 (1989) 243-249
243
Ruin probability by operational calculus Elias S.W. SHIU
*
ruin function + is the amount of risk reserve time 0. Let a denote the Lundberg security factor,
University of Mamtoba, Winnipeg, Man., R3T 2N2 Canada
Two series formulas for the probability of eventual ruin in the collective risk model with a Poisson claim-number process are derived by the method of operational calculus. Some recent results of H.U. Gerber are also discussed. Keywords: Probability of eventual ruin, Ultimate ruin probability, Infinite-time ruin probability, Collective risk theory, Operational calculus, Lagrange series. This paper is dedicated to the beautiful subject
to Henk Boom who introduced of Risk Theory.
8=+,x)-‘-1, which should be positive, is called the relative security loading. For k = 1, 2, 3,. . , define +x
For OL2 0, define a
a-
x+-
x,
x20
x
#(a>
(1.6) Formula (1.5) is equivalent to equation (17) on page 61 of Takacs (1967). Formula (1.6) can be found in Prabhu (1965, formula (5.55)) and Gerber (1988a, formula (27)); it can also be obtained by combining formulas (35.22), (16.16) and (19.15) in Takacs (1967). This paper is motivated and stimulated by Gerber’s (1988a) fun. We shall also derive Gerber’s formulas:
,co -yyE[( S, +
x)’
(1.2) that
* Support from the Great-West gratefully acknowledged. 0167-6687/89/$3.50
(1.7)
0.1)
u( t ) = 24+ ct - SN(r) Note
e-“(S;+“)]4,;
and k.
Put S, = 0. The probability of eventual ruin (ultimate ruin probability, infinite-time ruin probability) q(u) is the probability that the risk reserve
is ever negative.
(l-4)
and
Consider the classical collective risk model, in which insurance claims occur according to a Poisson process N(t), t 2 0, and the individual claim amounts Xi, X2, X3, . . . are mutually independent and identically distributed positive random variables, with probability distribution Pr( X 5 x) = P(x) and mean E(X) = p, < co. Assume that the Poisson process N(r) is independent of the random variables { X,}, and E[N(t)] = At. Also assume that premiums are received continuously at a constant rate c. The number
...
(1.3)
me
1. Introduction and notation
s,=x,+x,+
a=Xc?=[(l+B)&‘.
at
the argument
Life Assurance
of the
Company
0 1989, Elsevier Science Publishers
(1.8) 2. A convolution ruin
series for the probability of non-
It is somewhat function
easier to work with the nonruin
is
B.V. (North-Holland)
244
ES. W. Shiu / Ruin probability
Consider a small time interval (0, s). By the Poisson assumption, the probability that a claim will occur in the interval is As + o(s). Hence, for u 2 0, the nonruin function @I(U) satisfies the relation [Takacs (1967, p. 224) Ross (1983, section 6.6.3), Tijms (1986, p. 56)]: +(u)
=AsE[+(u+
cs-A-)]+(I-hs)+(u+cs)
by operational calculus
equation
(2.4) becomes
+)=8(1+8)-‘t-+(u)*+),
We remark that, if the functions f, and the value 0 on the negative axis, then reduced to
f+)*fib)
to(s).
=Jo*fib-dh(v)
(2.8)
f2
take (2.7) is
dy.
(2.9)
(2.1)
Dividing (2.1) by s, rearranging and letting s tend to 0 + , we obtain the integro-differential equation
We now extend entire real line:
O=XE[$(u-X)]+c$+)-Q+A),
&)=8(1+8)-
~'(.)=o(O(u)-E~~(u-X)]~.
u>O,
u>o.
Since + is zero on the negative axis and positive random variable, we have
E[c#+-X)]=jo"~(u-x) u 2 0, integrating
(2.8)
as an equation
‘$++)*k(u).
on the
(2.10)
If we let 6(u) denote the Dirac delta function, may rewrite (2.10) as
(2.2)
For
~20.
X is a
dP(x).
+(u)*[+)-k(u)]
=8(1+8)-r&
we
(2.11)
To solve for G(U) in (2.11), we invert 6(u) - k(u) as the series [cf. Mikusihski (1959) Erdelyi (1962), Yosida (1984)]
(2.3)
(2.2) from 0 to u yields
$+++(O)=U~~@I(U-x)[l-P(x)]
dx.
(2.12)
$ k*‘(u), j=o where
(2.4) [Formula (2.4) is equation (X1.7.2) of Feller (1971) and equation (72) on page 158 of Takacs (1967).] Since the risk reserve U(t) tends to co as t tends to cc with probability 1, we have
k*‘(u)
=8(u)
and, for j = 1, 2, 3,. . . , k*‘(u)
= k*(ip’)(u)*
(2.13)
Hence, G(U) = [6(u)
lim +(24) = 1. U-cc
k(u).
-k(u)]
-’ * [ 8(1+
8)-‘24”,]
Letting u tend to cc in equation (2.4) and applying the Lebesgue dominated convergence theorem, we obtain 1 -+(O)=~j;,~[l
-P(x)]
dx=ap,
= -&$_[k*‘(u)‘~:].
= -!1+8’
or +(O) = 8/(1
+ 8).
(2.5)
Let us define k(u)=a[l-P(u)] = 0
if
~20
if
u < 0,
or +)=a[+P(u)].
(2.6)
With the notation fi(x)*A(x)
= Jca fi(x-Y)./~(Y) --M
dy,
(2.7)
(2.14)
Equation (2.8) is a Volterra integral equation of the second kind [Linz (1985, ch. 3)]. The series (2.12) is sometimes called a Neumann series [Brown and Page (1970, p. 226) Riesz and Sz.Nagy (1955, p. 146)]. The convolution series (2.14) had been given by Dubourdieu (1952, p. 246); I thank Franqois Dufresne for this reference. Formula (2.14) shows that + can be viewed as a compound geometric distribution, for which Dubourdieu (1952, p. 247) had given an interpretation. The formula has been rediscovered several times and in different contexts; see Shiu (1988, p. 42). Indeed, when Kendall
245
E.S. W. Shiu / Ruin probability by operational calculus
(1957, p. 209) presented the formula in the context of storage theory, he knew of ‘no phenomenological interpretation’. J.A. Beekman has consistently promoted the use of the convolution series for evaluation I/J(U). In section 12.6 of Bowers et al. (1986), $I is denoted as FL, where L is the maximal aggregate loss random variable. A major difficulty with (2.14) is the calculation of the convolutions { k *’ }. In the next two sections, we shall present two different expressions for k*j(u)* u”,.
we have .
k*“(y)=o”[l-g(E-‘)].‘(;t-I),
(3.6)
Consequently, k*“(y)*
yo,=a”[I-g(E-l,]‘$
Substituting
(3.7)
(3.7) into (2.14) yields
(P(u) = &;~a$[‘-“(+:.
(3.8)
Put G = g(E-‘);
then
3. First formula
(Gf)(u)
Let g be the generating variable X,
[Feller (1971, section VIII.3) had studied such operators.] Formula (3.8) can be written symbolically as
g(z)
function
of the random
= E(zX).
(3 .I>
(Usually, generating functions are defined for discrete random variables only. However, we do not restrict the random variable X to be discrete.) Let E denote the translation operator (forward-shift operator). We claim that P(Y)
(3.2)
= g(E-‘)YO,
= E[S(u
- X)1.
G,(u) = &e”(‘-“)“+.
(3.9)
Motivated
by the formal
exp[a(I-
G)u+]
identity
=exp(-aGu+)
one may conjecture
exp(au+),
the formula
cc (-a)” c ~G’(ui
+(u)=&
eau+).
(3.10)
J=o
[cf. Hirschman and Widder (1955, p. S), Mikusinski (1959, p. 327) and Erdelyi (1962, p. 57)]. By (3.1), the right-hand side of (3.2) is
To prove (3.10), note that (I-
G)“=
t
(-l)k(
;I)G”
k=O
m E-” dP(x) [j -cc
Y”,= I
J =
_a_(v - x)“+ dP(x)
=
’ dP(x) / --m
It follows from (2.6) and (3.2) that k(Y)=+g(E-‘)]YO,.
(3.3)
It is easy to check that, for each real number
r, (3.4)
E’(f,*~*)=(E~*)*f,=f,*(E’f,).
k*“(y)=a”[I-g(E-‘)]“(yO,*yO,*
Gk ’
(3.11)
Substituting (3.11) into (3.8) and interchanging the order of summation yields (3.10). Since g(z) = E(zx) and the random variables { X;} are independent and identically distributed,
= P(Y).
Hence, for n = 1, 2, 3,. . . , the n-fold of k is
n (-‘Jk “jkFo k!(n -k)!
[g(z)]J=E(zX~+X~+“‘+X,)=E(zs,). Hence,
for a function
(3.12)
f,
G’f(~)=[g(E-‘)]‘f(u)=E[f(u-~J)].
convolution
(3.13) Applying
(3.13) to (3.10) yields
. ..). +(u)
Since
= &
Jfo y
E[ (u - 5”):
eO(yps/)+], (3.14)
(3.5) which is (1.5).
246
E.S. W. Shiu / Ruin probability
Formula (3.14) can also be obtained by applying the method of Laplace transforms to (2.2) or (2.4). See equation (16) on page 61 of Takacs (1967), and also section 5 of Shiu (1988). Let (Y= sup{ x 1P(x) = 0). By assumption, (Y2 0. If (Y> 0, then (3.14) is a finite sum, with the index j ranging from 0 to [u/a]. For a real number Y, we let [r] denote the greatest integer less than or equal to r. Formula (3.14) is valid for continuous or discrete individual-claim-amount random variables X. Willmot (1988) has applied (3.14) to obtain formulas for q(u) when X is Gamma (with arbitrary non-scale parameter) or continuous uniform. Shiu (1988) has assumed that X takes values on positive integers only: If ch*J = Pr
by operational calculus
The coefficient kxe’.‘lm[l
/0
-R(y)]
%Z x zRx,l
dy dx
-P(x)]
dx
BP,
= R
(3.19)
mu eRx [l -P(x)]
/0
dx
[Seal (1969, p. 130) Gerber (1979, p. 124, formula (5.28))]. By an integration by parts, the denominator in the right-hand side of (3.19) can be written as E( X eRX) - (1 + O)p,.
4. Second formula i X,=k i 1=l
=Pr(S,=k), i
then, for u 2 1,
ck*‘[a(k-
1+
k=l
J=l
u)]’
j!
. I
(3.15) The coefficients { czJ } can sively by the formula
be evaluated
recur-
Formula (3.14) is an alternating series. We now derive (1.6) which is a ruin probability formula with only nonnegative terms. As pointed out in the introduction, this formula has been given by Prabhu (1965, formula (5.55)) and Gerber (1988a, formula (27)). Following Gel’fand and Shilov (1964, p. 49) we define, for each nonnegative number (Y,
Ixla, i 0,
xT=
*fm+n) = ch
c
c:mc/**.
(3.16)
r+/=k
28411
x”=
(3.17)
is an effective method for evaluating G(U). In (3.17) R denotes the adjustment coefficient, which is the positive number satisfying the equation 5c eR”k(x) dx = 1. (3.18) J0
E(eRX)=l+(l+O)p,R.
xc0
if
x20’
(4.1)
(-x>:,
unless x = 0 and (Y= 0. Also, for each nonnegative integer n, XT + (- 1)“xl = xn. For clarity, let l(x) denote the constant function that takes on the value 1 for all x. Then, xo,+xY!=l(x). Since (Gf)(x) Cl(x)
J/(u) - C eeRu
way to express
if
Note that
An APL program for evaluating (3.15) can be found in Seah and Shiu (1988). Formula (3.15) is inefficient for large u since the number of terms depends on the size of u. However, for large values of U, Lundberg’s asymptotic formula [Cramer (1955, formula (119)), TakLcs (1967, p. 49, formula (36)), Seal (1969, formula (4.64)), Feller (1971, p. 378), Beekman (1974, p. 52) Gerber (1979, p. 124, formula (5.27)) Tijms (1986, formula (1.78)) Asmussen (1987, p.
Another
C in (3.17) is
(3.18) is the equation
= E[f(x
- X)],
= l(x).
(4.2)
Hence, (I-
G)xO, = (G-1)(-x0,) = (G-l)[l(x) = (G-1)x!?. Similar
XV1 r(o*
to (3.5) XC’
r(p)
-x”t] (4.3)
we have Xa+P-l qtY+p>
(4.4)
E.S. W. Shiu / Ruin probabdity
It follows from (3.3) (4.3)
x’-1
=+‘-I]”
k*“(x)
(3.4) and (4.4) that
(n-l)!’
(4.5)
and consequently, X”, *k*“(x)
= [l(X) =
-X0]
jm
k*“(y)
*k*“(x) dy-a”[G-I]‘&
--r_o
As (1 + 6)nk *“( y) is the n-fold convolution probability density function
of the
G=g(e-“j
=
’
in (4.6) is (1 + e))n.
I
l7=0
Thus . .
Consequently,
the value of the integral
cm (-ij"E(~"jD" n.
I-G=p,D-iE(X*
Y”+- P(Y) Pl
A special case of (5.1) is the well-known formula V”X” = n!, where v = I - E- ’ is the backwarddifference operator. Gerber (1988a) has given two other proofs for (5.1). We now present a fourth ‘proof’. Let D denote the differentiation operator. Then, E-’ = eeD. Assuming that all the moments of X exist, we have
tr! (4.6)
241
bv operational calculus
Since
for nonnegative
(I-G)“x”=n!(p,)” =o
E
(1-t ej_‘=
(1+8)8F,
J=o
(2.14) becomes
(4.7) Hence,
(4.8) from which (1.6) follows. Gerber (1988) obtained (1.6) by considering the last upcrossing of the risk reserve U(t) at the level 0. As the convolution series (2.14) is usually derived by conditioning on the occurrence of the first claim, I did not realize that (1.6) could be derived as a consequence of (2.14) until Professor Gerber pointed it out to me. The formula that corresponds to (3.15) is
if
m=n
if
m>n.
(5.2) m and
n,
We remark that, since pl # 0, the operator I - G is a delta operator [Rota (1975) Roman and Rota (1978) Shiu (1982)]. Multiplying (5.1) with a”/n! and summing over n yields ^13 a”( I - G)nx” n! n=O
1 --=c 1 -up1
for u 2 0,
integers
Symbolically, ea(/&c)_\, or
the e-oG.x
right-hand Hence,
eox.
(5.3)
.
side of (5.3) is one might try pro-
ving that 1++=
cm (-a@’
= Jfo
(x,
eua)
j!
J=o
k$E[
(5.4)
(x - S,)’ e”(-‘ps~)],
which turns out to be Theorem l.(b) (1988a) [also see Gerber (1988b)]. Gerber (1988a) also proved that
of Gerber
~=J~o~E[(x-.S,)Jp’ e”(i-sll],
x # 0.
(5.5)
To prove (5.5) 5. Gerber’s
one may start with the identity
theorems (5.6)
It follows from (3.7) and (4.6) that (I-
G)nx,
= (I-
G)“[x:+(-l)“xY]
= n!( p,)“.
(5.1)
and apply (3.11) and (3.13). After I presented the paper [Shiu (1989)] at the 23rd Actuarial Research Conference, G. Willmot
248
E.S. W. Shiu / Ruin probability
kindly pointed out to me that one should be able to derive (5.4) and (5.5) by the Lagrange series. Let h be an analytic function and let z = b + w/z(z).
(5 -7) By the implicit function theorem, there is a unique root z = z(w) which reduces to b at w = 0. If f is an analytic function, then f(z) =f(z( w)) may be expanded in terms of powers of w by the Lagrange formula f(z)
=f(b)
. (5.8) Melzak (1973, p. 113) has presented an elegant proof of (5.8), which was given by Laplace. The usual proof, which requires the Cauchy integral formula, can be found in Whittaker and Watson (1927). Now, consider b = a, =
We conclude this paper with a proof of (5.5) when X is a degenerate random variable. For each constant cu, the polynomials {y(y-aj)‘+Ij=O,
eYx
1,2,
. ..}
are called Abel’s polynomials. It is known [Comtet (1974, p. 130, Theorem C), Roman and Rota (1978, p. 114), Roman (1984, p. 74)] that each polynomial p can be expanded as p(y)
y=b
f(y)
by operational calculus
= C
‘(’
-Ji,‘-’
,120
p(‘)(
ja>.
(5.13)
Although the exponential function eJ is not a polynomial, we can argue by a limiting process that ey=
E
Y(Y-cfj)‘plda
j=o
j!
(5.14)
Considering y = -ax and X = - a/a, we see that formula (5.14) is a special case of (5.5).
and h(y)
= E(e-“x).
(5.9)
Then [h(y)l’=E[exp(-YS,)] and
=
XE
[
(x -
s,)‘-l
ea@-%)] .
(5.10)
Applying (5.8) we have ~‘-‘=e”‘+x,~~~E[(x-S,)‘-‘eu~~~~~)]. (5.11) If w = -a, then z = 0 and (5.11) becomes (5.5). Formulas similar to (5.11) can be found in Takacs (1962). There is another version of the Lagrange formula [Riordan (1968, p. 146), Polya and Szegij (1972, p. 146) Goulden and Jackson (1983, p. 17)]: f(z)
1 -WV(Z)
=,~~~[~[f(~)[~(~)l~l]~=~ (5.12)
Formula (5.4) follows from (5.12).
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