ELSEVIER Insurance: Mathematics and Economics 21 (1997) 129-137

U I t l~[ : l [.kt]; [i] ,', ][1~

The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin

Hans U. G e r b e r a'l , El ias S.W. Shiu b'* a Ecole des hautes dtudes commerciales, Universitd de Lausanne, CH-IOI5 Lausanne, Switzerland

b Department of Statistics and Actuarial Science, The University oflowa, Iowa City, Iowa 52242, USA

Received November 1996; received in revised form April 1997

Abstract

We examine the joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. The time of ruin is analyzed in terms of its Laplace transform, which can naturally be interpreted as discounting. We show that, as a function of the initial surplus, the joint density satisfies a certain renewal equation. We generalize Dickson's (I 992) formula, which expresses the joint distribution of the surplus immediately before ruin and the deficit at ruin in terms of the probability of ultimate ruin. © 1997 Elsevier Science B.V.

Keywords: Collective risk theory; Surplus process; Ruin probability; Deficit at ruin; Time of ruin; Lundberg's fundamental equation; Laplace transforms; Beekman's convolution series; Dickson's formula; Renewal equation; Martingales; Optional sampling theorem; Duality

1. Introduction

In classical risk theory, two questions of interest are (a) the deficit at ruin, and (b) the time of ruin, both of which have been treated separately in the literature. Here the two questions are combined. From a mathematical point of view, a crucial role is played by the amount of surplus immediately before ruin occurs. Hence we examine the joint distribution of three random variables: the surplus immediately before ruin, the deficit at ruin, and the time of ruin. The time of ruin is analyzed in terms of its Laplace transform, which can naturally be interpreted as discounting. Using a duality argument, we obtain an explicit formula for the discounted probabili ty that the surplus will ever fall below its initial level u and will be between u - y and u - y - dy when it happens for the first time. With this result we derive a renewal equation for the joint density, as a function of the initial surplus u. Finally, we generalize Dickson's (1992) formula, which expresses the joint distribution of the surplus immediately before ruin and the deficit at ruin in terms of the probabili ty of ultimate ruin.

* Corresponding author. Tel.: 319 335 2580; fax: 319 335 3017; e-mail: eshiu@stat.uiowa.edu. I Tel.: 41 21 692 3371; fax: 41 21 692 3305; e-mail: hgerber@hec.unil.ch.

0167-6687/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PH SO 167-6687(97)00027-9

130 H.U. Gerber, E.S. W. Shiu/lnsurance: Mathematics and Economics 21 (1997) 129-137

2. The joint distribution

We consider the classical model of the collective risk theory. Thus u > 0 is the insurer's initial surplus. The premiums are received continuously at a constant rate c per unit time. For t > 0, the surplus at time t is

N(t)

U(t) = u + ct - ~ Xj, (2.1) j = l

where {N(t)} is a Poisson process with mean per unit time ). and {Xj} are independent random variables with common distribution P(x), P(0) = 0. For simplicity we assume that P(x) is differentiable, with P1(x) = p(x) being the individual claim amount probability density function.

Let T denote the time of ruin,

T = inf{t [U(t) < 0} (2.2)

(T = ec if ruin does not occur). For U(0) = u > 0,

~(u) = Pr[T < cx) I U(0) = u] (2.3)

is the probability of ultimate ruin. We also consider the random variables U ( T - ) , the surplus immediately before ruin, and I U(T) l, the deficit at ruin. For U(0) = u > 0, let f ( x , y, t I u) denote the (defective) joint probability density function of U ( T - ) , I U(T) I and T. Then

~ OO OO

f f f f (x ,y , tlu)dxdydt=Pr[T<~lU(O)=ul=~P(u). (2.4)

0 0 0

We remark that, for x > u + ct,

f ( x , y, t lu) = 0,

and that

f ( u + ct, y, t I u) dx dy dt = e-Xt)~p(u + ct + y) dy dt.

It follows from the conditional probability formula,

e r ( A A B) = Pr (A )Pr (B I A),

that f ( x , y, t I u) is the joint probability density function of U ( T - ) and T at the point (x, t) multiplied by the conditional probability density function of I U(T) I at y, given that U ( T - ) = x and T = t. The latter does not depend on t and is

p(x + y) p(x + y)

f o p(x + y) dy 1 - P ( x ) ' Y > 0.

Hence

FJ 1 f ( x , y , t l u ) = f ( x , z , t l u ) dz p(x + y) . 1 - P ( x )

LO

(2.5)

H. U. Gerber, E.S. W. Shiu /Insurance : Mathematics and Economics 21 (1997) 129-137 131

and

For 3 > 0, define

OO

f ( x , y lu) = f e-~t f ( x , y, t l u )d t

o

(2.6)

O 0

f ( x l u ) = / f ( x , y l u ) dy.

i i

(2.7)

0

Here ~ can be interpreted as a force of interest, or, in the context of Laplace transforms, as a dummy variable. For notational simplicity, the symbols f ( x , y I u) and f ( x [ u) do not exhibit the dependence on 8.

Multiplying (2.5) with e -'~t and then integrating with respect to t yields

f ( x , y lu) = f ( x l u )p(x + y) / [ l - P(x)]. (2.8)

With ~ = 0, (2.8) was pointed out by Dufresne and Gerber (1988, (3)); another proof can be found in Dickson and Egfdio dos Reis (1994).

Let OO OO OO

g(y)=ffe-~' f (x ,y , t lo)atax=ff(x, ylO)dx. (2.9)

0 0 o

With 6 as a force of interest, the differential g(y) dy can be interpreted as the "discounted probability" that the surplus will ever fall below its initial level u and will be between u - y and u - y - dy when it happens for the first time. By distinguishing whether or not ruin occurs at the first time when the surplus falls below the initial value u, we have

U

f(x, y l u ) = f f(x, ylu-z)g(z)dz+f(x-u,y+ulO), O<_u<x, (2.10) o

f ( x , y I u) = j f ( x , y I u - z)g(z) dz, 0 < x < u.

o

(2.11)

and

In the next section we derive an explicit formula for f ( x , y I 0). Integrating f ( x , y I 0) with respect to y and x, we obtain explicit formulas for f ( x 10) and g(y), respectively.

3. The key formula

Let ~ be a number. Because {U(t)}t>_o is a stochastic process with stationary and independent increments, a process of the form

{e-&+~u(t)}t> 0 (3.1)

is a martingale if and only if, for each t > 0, its expectation at time t is equal to its initial value, i.e., if and only if

E[e-~t+~u(t) l u ( o ) : u] = e ~u. (3.2)

132 H. U. Gerber, E.S. W. Shiu /Insurance." Mathematics and Economics 21 (1997) 129-137

In this paper we let f denote the Laplace transform of a function f ( x ) , x > O, i.e.,

O~

f (~) = f e-~X f ( x ) d x . , i

o

Then the left-hand side of (3.2) is

e x p ( - 6 t + ~u + ~ct + ~.t[/3(~) - 1]).

Hence the martingale condition is that

- 6 + c~ + Z[/3(~) - 1] = 0. (3.3)

The function/~(~) is defined for all nonnegative numbers ~. The left-hand side of (3.3) is a convex function of ~; it tends to ~ as ~ tends to e~, and takes the nonpositive value - 6 at ~ = 0; thus it has a unique nonnegative root, say, p. The process {e -St+pU(t) }t>_0 is a martingale.

We remark that, if the individual claim amount density function p is sufficiently regular, Eq. (3.3) has one more root, say ~2, which is negative. With 6 = 0, -~2 is the adjustment coefficient in classical risk theory. Eq. (3.3) is equivalent to Beekman (1974, p. 41, top equation), Panjer and Willmot (1992, (l 1.7.8)), and Seal (1969, (4.24)). Lundberg (1932, p. 144) points out that the equation is "fundamental to the whole of collective risk theory," and Seal (1969, p. 111) calls it "Lundberg's (1928) ' fundamental ' equation." Seal (1969, p. 112) asserts incorrectly that the second root is also positive.

For x > u = U(0), let

Tx = inf{t I U(t) = x} (3.4)

be the first time when the surplus reaches the level x. We can use equality to define the stopping time Tx because the process {U(t)} is jump-free upward. Since

U ( t ) < x f o r 0 < t < T x ,

we can apply the optional sampling theorem to the martingale {e -~t+pU(t) } to obtain

e pu = E[e-~rx+PV(rx) lU(O) = u] = E[e -~Tx I U(0) = u]e px,

or

e -p(x-u) = E[e -STx I U(0) = u]. (3.5)

With 6 interpreted as a force of interest, the quantity e -p(x-u) is the expected discounted value of a payment of 1 due at the time when the surplus reaches the level x for the first time.

We remark that (3.5) remains valid even if u is negative. The required condition is x > u; the condition u > 0 is not needed anywhere in the derivation. Formula (3.5) was probably first given by Kendall (1957, (14)), although he did not provide a complete proof. It can also be found in Prabhu (1961; 1980, p. 79, Theorem 5(i); p. 105, #4), Cox and Miller (1965, p. 245, (184)), Tak~ics (1967, p. 88, Theorem 8), and Gerber (1990, (11)).

For x > u = U(0), let rr(t; u, x), t > 0, denote the probability density function of the random variable Tx. Hence (3.5) is

O@

f e-~tyr(t; u, x) = (3.6) dt e-P(X-U).

o

The differential Jr (t; u, x) dt is the probability that the surplus process upcrosses the level x between t and t + dt and that then this happens for the first time. We remark that the surplus cannot reach the level x before time t = (x - u)/c,

H.U. Gerber, E.S.W. Shiu/Insurance: Mathematics and Economics 21 (1997) 129-137 133

and that it may reach x before the first claim occurs. Hence, for t < (x - u) /c , zr(t; u, x) = 0, and the distribution

of Tx has a point mass at t = (x - u ) / c so that

rc((x - u)/c; u, x) dt = e -~(x-u)/''.

For U(0) = u > 0, x > 0, let Ji'(t; u, x), t > 0, be the function defined by the condition that ~ ( t ; u, x) dt is the probabili ty that ruin does not occur by time t and that there is an upcrossing of the surplus process at level x between t and t + dt. We now show by a duality argument that

zr( t ;0 , x ) = ~ ( t ; 0 , x), x > 0 , t > 0 . (3.7)

Let {u (s)}s_>0 be a sample path of the surplus process with U (0) = 0 contributing to the quantity Jr (t; 0, x). Consider

f f ( s ) = [ x - u ( t - s ) f o r 0 < s < t, \ u(s) for s > t. (3.8)

Then {fi (s)}~ >_0 is a sample path of the same process contributing to the quantity ~ (t; 0, x). Since this is a measure- preserving correspondence, we have (3.7).

Now, the differential f ( x , y, t I u) dt dx dy can be interpreted as the probability of the event that ruin does not take place by time t, that the surplus process upcrosses the level x between time t and time t + dt, but does not attain level x + dx, i.e., that there is a claim within c - l dx time units after Tx, and that the size of this claim is between x + y and x + y + dy.

Thus

f ( x , y, t I u) dt dx dy = [~( t ; u, x) dt][)~c -1 dx][p(x + y) dy], (3.9)

o r

f ( x , y, t l u) = ) w - l p ( x + y)~'( t ; u, x). (3.10)

It follows from (3.10) (with u = 0) and (3.7) that

f ( x , 7 t 10) = A c - l p ( x + y)rr(t ; 0, x). (3.11)

If we multiply (3.11) by e - r t dt, integrate from t = 0 to t = oc, and apply (2.6) and (3.6) with u = 0, we obtain the key formula in this paper:

f ( x , y t O ) = A c - l p ( x + y ) e -px, x > O , y > 0 . (3.12)

Furthermore, by substituting this expression in (2.7) and (2.9), we get the formulas

f ( x l 0 ) = ) ~ c - l [ 1 - P(x ) ]e -px, x > 0 (3.13)

and

g(y) = )~c - ! f p(x + y ) e -px dx,

0

(3.14)

respectively.

134

4. Renewal equations

From (3.12) we obtain

f ( x - u , y +ulO) = Xc - l ep(x-u)p(x + y) = f ( x , y O) e pu.

Hence formulas (2.10) and (2.11) can be written as

f ( x , y [u) = f f ( x , y [u - z)g(z)dz + f ( x , y [0) epUl(u < x),

o

H.U. Gerber, E.S. W. Shiu/lnsurance." Mathematics and Economics 21 (1997) 129-137

(4.1)

where I denotes the indicator function, i.e., I(A) = 1 if A is true and I(A) = 0 if A is false. Integrating (4.1) with respect to y, we obtain by (2.7)

f ( x l u ) = f f ( x l u - z)g(z)dz + f(xlO)epUl(u < x). (4.2) q]

o

As a function of x, f ( x I u) has a discontinuity of amount

f ( u 10)e p" = ~.c-l[1 - P(u)] (4.3)

at x = u. Remarkably, it does not depend on 8. For further discussion, we introduce the function y (u) which is defined as the solution of the equation

U

y(u) = ] y(u - z)g(z)dz + epUI(u < x). (4.4)

0

(The value of x is fixed in the following.) Because the function f ( x [0)Y (u) satisfies (4.2), we gather that

f ( x l u) = f ( x [ 0)y(u) . (4.5)

Also, from this and (2.8) we see that

f ( x , y lu) = f ( x , y lO)y(u). (4.6)

Last, but not least, we introduce the function

~(u) = E[e-~r+pu(r)l(T < ~x~) I U(0) = u], u > 0. (4.7)

For 3 = 0 (and hence p = 0) this is the traditional probability of ruin function. For ~ > 0, it can be interpreted as the expectation of a discounted payment due at the time of ruin. By distinguishing whether or not ruin occurs at the first time when the surplus falls below the initial value u and applying the law of iterated expectations, we obtain

U OO

O(u) = f O(u - z)g(z)dz + f e-°(z-")g(z)dz. (4.8)

0 u

With u = 0, (4.8) becomes

~p(0) = ~(p). (4.9)

The function g (given by (3.14)) is a nonnegative function on [0, ec) and hence may be interpreted as a (not necessarily proper) probability density function. In probability theory, equations such as (4.4) and (4.8) are called

H.U. Gerber, E.S.W. Shiu/lnsurance: Mathematics and Economics 21 (1997) 129-137 135

renewal equations; see Feller (1971) or Resnick (1992). The solution of a renewal equation is unique and can be analyzed by Laplace transforms. As an application and preparation for the following section, we calculate the Laplace transform of the function 4 (u) . Because the Laplace transform of a convolution is the product of the Laplace transforms, taking Laplace transforms in (4.8) yields

7)(~) = ~(~)~(~) + f e -~u e-P(Z-")g(z) dz du. (4.10)

0

By changing the order of integration and finally applying (4.9), we reduce the double integral in (4.10) as follows:

f e-PZg(z) I f e (p-~)u du 0

Hence

~b(~) = ~ (~ ) - 4 ( 0 )

(p - ~)[1 - ~(~)]"

We remark that formula (4.11) can be rewritten as

1 l - 4 ( 0 )

~ - p (~ - - p ) [ 1 - ~ ( ~ ) ] " ~b(~) -

Hence

1

~ - p - - - ~ b ( ~ ) - i - 4 ( 0 ) £

t ~ g 3

k 0 =

dz = [~(~) - ~(p)]/(p - ,e) = [~(e) _ 4(O)] / (p - e).

(4.11)

from which we obtain

e p u - 4 ( u ) = [ 1 - 4 ( 0 ) ] e p"+ e p z Z g * k ( u - z ) d z . (4.12)

0 k= l

In the special case where 6 = 0 (and hence p = 0), this reduces to Beekman's convolution series (Beekman, 1974, p. 68).

5 . D i c k s o n ' s f o r m u l a

The purpose of this section is to show that

/ f (x 10)

f (x l u) -------

\ f (x 10)

e p" - 4 (u ) , X ~ u > O ,

l - 4 ( 0 )

epX4(u - x) - 4(u) O < x < ~ u .

l - 4 ( 0 ) '

(5.1)

This generalizes a result of Dickson (1992), which is for 3 = 0 (and hence p = 0). A proof of (5.1) by duality can be found in Gerber and Shiu (1998).

136 H.U. Gerber, E.S.W. Shiu/Insurance: Mathematics and Economics 21 (1997) 129-137

In view of (4.5), a formula equivalent to (5.1) is

I e p u - ~ ( u ) x > u > 0 , 1 - ~ t ( O ) '

y ( u ) = e p X ~ ( u - x ) - ~ ( u ) 0 < x < u.

1 - ~ p ( O ) ' -

With the definition

e pu, x > u >_ O,

~o(u) = ePXlp(u -- x ) , 0 < x < u,

a third equivalent formula is:

× ( u ) = [,p(u) - 7~(u)] / [1 - ~p(o)],

which we now prove by Laplace transforms. The Laplace transform of (5.3) is

e (p-~)x -- 1 ~b(~ ) - + e ( P - ~ ) ~ ( ~ ) .

p - - ~

Hence

q3(~) -- @(~) = [e (p-~)x -- 1][(p -- ~)-1 + ~.(~)].

Substituting (4.1 I) in the right-hand side of (5.6), we obtain

, ~ ( ~ ) - @ ( ~ ) = e (p-~)x -- 1 1 --~r(O)

p - ~ l - ~ ( ~ )

On the other hand, the Laplace transform of (4.4) is

~ ( ~ ) = ~ ( ~ ) ~ ( ~ ) + e (p-~)x -- 1

p _ ~ '

which yields

9(~) =

e (p-~)x -- 1 1

p - ~ 1 - ~(~)

It follows from (5.9) and (5.7) that

2(~) = [~(~') - ~-(~')]/[1 - ap(O)],

which proves (5.4).

(5 2 )

(5.3)

(5.4)

(5.5)

(5.6)

(5.7)

(5.8)

(5.9)

(5.1o)

Acknowlegement

Elias Shiu gratefully acknowledges the support from the Principal Financial Group Foundation.

H.U. Gerber, E.S. W. Shiu/Insurance: Mathematics and Economics 21 (1997) 129-137 137

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