1. Introduction
The ruin probability in infinite time has been extensively studied. Indeed it can be formulated in several cases. The same does not hold in finite time. [1] was the first to consider this problem; [2] considers the general case, which is difficult to compute, and [3] which is slightly less so. An extensive survey is found in [4] .
This paper is focused on exact solutions, or general solutions which do not involve series computations. The ruin problem can be examined under a variety of conditions: with investment income, dividends, or time dependent claims distributions. For example, [5] discusses the situations where the reserve before ruin is different after ruin (but restricted to Erlang and Lindley distributions, say, due to investment income).
An integrated, general and computable approach would be useful, without the use of series expansions. This paper follows the philosophy of [6] .
2. Basic Equations
Consider a risk business involving the following parameters:
· P is the rate of premium received per unit time;
·
is the stochastic variable measuring the amount of claim (given that a claim has occurred) with probability density function
;
· u is the reserve held at any time;
·
is the probability of ruin of the business within time t, where the initial reserve was u at time
;
·
is the corresponding probability of survival, with
.
It will be seen that, since claim amount
is part of the change in reserve u, the symbols
and u will be used interchangeably. The framework is that of [6] .
We define
as the space of Lebesgue integrable functions with finite norm
Throughout this paper all integrals are taken to be defined in the sense of Lebesgue unless otherwise specified. We also consider later (in connection with the inverse Fourier Transform) the space of square integrable functions
with norm.
If
and f is bounded, then it is clear that
.
In general we require that the claim amount density
satisfy the conditions
for
, and
for
. In addition we require that the claim amount density satisfy
These conditions are to ensure that the probability density of claims is sensible, and that it has a finite mean and variance. Additional restrictions on
will be imposed as required.
Without loss of generality we may scale the claim amount
so that the mean claim is 1 and
. In practice this means that we take always the gross premium rate
.
We now consider how the method of [6] for deriving the survival probability
in infinite time may be extended to the finite time case, which satisfies:
This may be expressed in terms of the ruin probability
as follows:
(1)
with
and the FT of
by integration by parts is
where
introduces the role of time.
There are two approaches to the finite time problem. One is the classical of [3] , the other is that of differential equations, relying on the FT wrt u the other on t.
Taking the FT of 1 wrt u:
(2)
where
(3)
This may be written as
(4)
Integrating over
we get
(5)
Now we can express and take the inverse FT:
(ift)
We show that
is Hermitian in u:
(6)
The first term in is easily dealt with:
(7)
where
is the Dirac delta function. Thus the finite time probability rests on computing:
(8)
Since
is bounded we know that
exists for
, but not necessarily for
. The order in which the time derivative
is written is immaterial, as is seen by taking the time derivative as a limit and applying the dominated convergence theorem.
3. An Exponential Example
We have
(9)
with
In this case the FT wrt u becomes
. We need to compute
. This can be accomplished, with
. Further,
(10)
so that
so that
and we have the following general FTs:
(11)
(12)
Hence
(13)
so that since
(14)
(15)
(16)
(17)
Remark 1 Generalized functions are also known as functionals, or distributions in the sense of Schwartz are readily discussed [7] [8] .
4. Conclusion
In this paper we have attempted to demonstrate how complex function theory enables an integrated approach to the solution of ruin probability problems. This has involved a heavy application of the Cauchy theorem for analytic functions. It might be noted that the solutions obtained are complex, but computable. If desired, Appendices A and B provide proof of
as in [1] , but this may be omitted on first reading.
Appendix A
Equation (17) is a PDE of the first order in two dimensions, u and t, so should be solvable under boundary suitable conditions. The natural condition for u is that
.
Letting
in (17) and using the boundary condition
for
, we immediately get
. Hence it may be written as
(18)
The function
must be finite at least for all
. This follows from the fact that
must be bounded by 1, for it to have any physical significance. From Appendix B we know that for all
there exists a unique
such that
.
From the property of FTs we deduce that
(19)
whenever
.
This last result immediately gives us
, as was required in the solution of the infinite time case. For
is implied by 19, whence
This is a bounded function at
, so that
.
Remark 2 Equations (18) and (19), expressed as Laplace Transforms, are attributed to [3] . The difficulty with these equations is that they depend implicitly on the relation
, which is required to be solved in order for the inverse FT to be applied. However it will be seen that Cauchy’s theorem permits us to write these equations in a form which is more amenable to the inverse FT.
In Figure 1 the contour of
passes through the origin since
. As the parameter
we have
, since
is bounded for
. For large z we have approximately:
since
as
by the property of FTs. Thus
for large
. This gives
as an asymptote to
, as depicted above.
As a consequence of the asymptotic property of r above, we have
, since
. This ensures that we can employ inverse FTs, at least for
, in what follows. The solution for
in finite time is now discussed for two distinct cases of interest. The first case corresponds to using the inverse FT to obtain
from (19); the second case to obtain
for
. These cases need to be handled separately because of the discontinuity of
at
.
Taking the inverse FT we then get:
(20)
where we have used the result in (20) and made the transformation
. Both integrals above exist as improper integrals; the second is evaluated along the real axis, whilst the first is evaluated along the contour
, parameterized by
.
We show that the
integral appearing in (20) may be replaced by an integral along the real axis R. For this purpose we consider integration of the function
along the closed contour bounded by the contour
, the vertical lines
, and the horizontal line
, for small
.
Now apply Cauchy’s theorem to the integrand
, which is analytic for
. Let
denote that section of
cut off by
, at the point with real part x. As
the contribution along the vertical lines vanishes, as may be seen from the bounds
,
for all
. Hence we get:
where the integral on
is taken over
.
As
the left hand side of the above equality approaches the improper integral for
. It is also easy to show that the right hand side approaches the real line improper integral
by means of the dominated convergence theorem.
This implies that
may be written solely in terms of real line integrals as:
Then integrating by parts, and noting that
is bounded at
, we get finally
This last expression leads to the formula for ruin, with zero reserve, attributed to [9] by [3] :
Proposition 3 The finite time probability for zero reserve is equal to:
for
(21)
where
is the probability density of total claim amount
in a finite time interval
.
Proof. To prove the equivalence of the equality for
in (20) and the expression above, we show that the appropriate FTs are the same. Since two functions having the same FT must be equal (almost everywhere) this would then prove that the expressions are equal if they are both continuous. Using a well known result for generating functions [1] , the FT of
is given as
for Poisson distributed claim frequency with parameter 1. The FT, with respect to u, of the function
is thus
, which has the value t at
. The FT, with respect to u, of the function
is thus given as
, which is precisely the integrand appearing in the expression for
in §5.l3, after putting
.
Appendix B
Proposition 4 (a) For any
the equation
has a unique solution
.
(b) If
can be analytically continued to a neighborhood of
, then there exists a root of
with
and
. In addition this root has the smallest modulus of all roots in
.
Proof. (a) We first demonstrate the proposition for
. The function
clearly has a root at
. To show that it is unique in
define the function
(22)
We have
and
so that:
for
,
from which the result follows.
If
then the circle
lies completely within
, whereas if
then it touches the real axis at
. In either case,
for
the property of FTs, so that
cannot have a zero outside
.
In the case of
it is clear that a closed curve
may be constructed surrounding
, on which holds the inequality:
Hence by Rouché’s theorem ( [10] , §8,2), the function
has precisely the same number of zeros within
as
. But it is easily shown that the former function has precisely one such zero, from which the result follows for
. (Note that this also gives the proof where
, but only if
.)
In the case of
we use a continuity argument to establish the existence of a root of
. Let
be a sequence such that
. Then from the previous case, there exist
such that
. Now the sequence
is bounded and hence must have a limit point z with
. If necessary we can construct a convergent subsequence so that
say. Since the function
is continuous, we have
, which proves existence of a root. To show uniqueness, let
be another zero, so that we have:
Using the same argument as for the proof of part (a), we consider in place of
the function
and the related function
We have
, which yields the inequality:
This implies that
and thus uniqueness of the zero in the case
.
(b) It is important to note that not all functions p satisfy the condition stated, for example the Pareto distribution
for
does not1
The FT
for
,
corresponds to the moment generating function of p; it may be shown by considering the derivative of
at
[11] that an appropriate root
for
exists. Part of this proof demonstrates that the inequality
holds for
. It is clear the same inequality applies to
since
Thus
has no roots in the region
apart from 0 and
.
NOTES
1In fact a necessary and sufficient condition for
to be analytically continued at the origin
is that the moments
, and that
is uniformly bounded for
. In this sense p is short-tailed, as it must converge to 0 sufficiently fast
.