D. Perry, W. Stadje Insurance: Mathematics and Economics 26 2000 25–36 29
ϕα − βE Z
τ
y
e
− αXt −βt
dt = −
e
− αx
+ E
e
− αXτ
y
−βτ
y
= − e
− αx
+ E
e
− βτ
y
1
{ Xτ
y
= 0}
+ P Xτ
y
0Ee
− αXτ
y
−βτ
y
| Xτ
y
0 + e
− αx+y
E e
− βτ
y
1
{ Xτ
y
=x+y}
+ P Xτ
y
x + yE e
− αXτ
y
−βτ
y
|Xτ
y
x + y = −
e
− αx
+ φ
y 1
β + P Xτ
y
ν ν − α
E e
− βτ
y
| Xτ
y
+ e
− αx+y
φ
y 3
β + P Xτ
y
x + y e
− αx+y
ξ ξ + α
E e
− βτ
y
| Xτ
y
x + y = −
e
− αx
+ φ
y 1
β + ν
ν − α φ
y 2
β + e
− αx+y
φ
y 3
β + e
− αx+y
ξ ξ + α
φ
y 4
β. 2.3
Eq. 2.3 holds for every α ∈ −ν, ξ ; but since all terms are analytic functions for α ∈ C\{−ν, ξ }, it follows from the identity theorem for analytic functions that 2.3 holds for all α ∈ C\{−ν, ξ and all β ≥ 0. In particular,
we can set α = α
i
β, i = 1, 2, 3, 4, and obtain the four equations
0 = −e
− α
i
βx
+ φ
y 1
β + ν
ν − α
i
β φ
y 2
β +
e
− α
i
βx+y
φ
y 3
β + e
− α
i
βx+y
ξ ξ + α
i
β φ
y 4
β, i = 1, 2, 3, 4.
2.4 Aβ
is the coefficient matrix of this system of linear equations, and the solution is thus given by 2.2.
3. The revenue functionals
In this section we study the behavior of X until the bankruptcy time T = inf{t ≥ 0|Xt ≤ 0}. We assume from now on that µ − ηξ + λν 0, i.e. the process has a downward drift. This ensures ET ∞. We start from
the martingale relation EMT = EM0, which can be written more explicitly as ϕα − βE
Z
T
e
− αXs−βs
ds = −
e
− αx
+ E
e
− αXT −βT
. 3.1
One can obtain the LT of T by letting y in φ
y
β tend to infinity, but it is easier to use 3.1 and the decomposition
E e
− αXT −βT
= E e
− βT
1
{ XT =
0}
+ ξ
ξ − α E
e
− βT
1
{ XT
0}
= ρ
1
β + ξ
ξ − α ρ
2
β, 3.2
where ρ
i
β = lim
y→∞
φ
y i
β . Setting α = α
i
β, i = 2, 3, in 3.1 yields
0 = −e
− α
i
βx
+ ρ
1
β + ξ
ξ − α
i
β ρ
2
β, i =
2, 3. 3.3
The solution of 3.3 is given by ρ
2
β = ξ − α
2
βξ − α
3
β ξα
2
β − α
3
β e
− α
2
β
− e
− α
3
β
, 3.4
ρ
1
β = e
− α
2
β
− α
2
β ξ − α
2
β ρ
2
β. 3.5
30 D. Perry, W. Stadje Insurance: Mathematics and Economics 26 2000 25–36
Hence, by 3.1 and 3.2, E
Z
T
e
− αXs−βs
ds =
ϕα − β
− 1
ρ
1
β + ξ
ξ − α ρ
2
β − e
− αx
. 3.6
Inserting 2.1 in 3.6, taking the derivative in 3.6 with respect to α and then setting α = 0 gives the desired revenue functional E
R
T
Xs e
− βs
ds . We omit the tedious algebra and state the result.
Theorem 2. The revenue functionals are given by
E e
− βT
= ρ
1
β + ρ
2
β, E
Z
T
Xs e
− βs
ds =
ρ
1
β + ρ
2
β − 1µ − ηξ + λν
β
2
+ x
β +
ρ
2
β βξ
. The quasi-stationary distribution of X can also be derived from 3.6. Its LT is given by
E R
T
e
− αXs
ds ET
= e
− αx
− ρ
1
0 + ξ ξ − αρ
2
ϕαρ
′ 1
0 + ρ
′ 2
. 3.7
The numerator on the left-hand side of 3.7 is computed from 3.6 by setting β = 0.
4. Extension to phase-type and hyperexponential jumps
In order to see how the above martingale approach can be extended to phase-type and hyperexponential jumps, we consider as an example the case that the positive jumps are Erlang ν, 2 while the distribution of the negative
jumps is a mixture of expξ
1
and expξ
2
with weights p and 1 − p, respectively. Thus, the LT of the jump size is ψα =
λ λ + η
ν ν + α
2
+ η
λ + η p
ξ
1
ξ
1
− α
+ 1 − p
ξ
2
ξ
2
− α
, for some p ∈ 0, 1 and ν, ξ
1
, ξ
2
0. The exponent of X is given by ϕα =
α
2
2 −
µα − λ 1 −
ν ν + α
2
− η
1 − pξ
1
ξ
1
− α
− 1 − pξ
2
ξ
2
− α
.
Lemma 2. There is a β
∗
0 such that for the equation ϕα − β = 0 the following holds: 1. If β ∈ 0, β
∗
, there are six distinct real roots. 2. If β ∈ β
∗
, ∞ , there are four distinct real roots and two conjugate non-real roots.
3. If β = β
∗
, there are five distinct real roots, one of them having the multiplicity 2.
Proof. The equation ϕα = β is equivalent to a polynomial equation of degree six so that there are exactly
six complex roots, counted with their multiplicities. Every intersection point of the real curves hα = β + µα − α
2
2 and gα = λ1 − [νν + α]
2
− η 1 − pξ
1
ξ
1
− α −
1 − pξ
2
ξ
2
− α
gives a root. The parabola h satisfies h0 = β ≤ 0, and because of h
′
0 = µ 0, has its maximum at some positive value. Considering the behavior of g at its three vertical asymptotes at α = −ν, ξ
1
, ξ
2
, it is easily seen that there are exactly four intersection points one in each of the intervals −ν, 0, 0, ξ
1
, ξ
1
, ξ
2
, ξ
2
, ∞ , while for
sufficiently large β there are two more in −∞, −ν; only for one value β
∗
of β there is a real double root. In the case of only four real roots i.e. if β β
∗
, there are two more non-real roots, which are conjugate complex and thus distinct.
D. Perry, W. Stadje Insurance: Mathematics and Economics 26 2000 25–36 31
Any downward jump of X can be thought of as the realization of a two-stage experiment: choose ξ
1
or ξ
2
with probability p or 1 − p, respectively, and then take an expξ
1
or on expξ
2
random variable. An upward jump is composed of two independent phases, each being expν-distributed.
Now the process X can leave the interval 0, x + y in six different ways: 1 by an expξ
1
downward jump, 2 by an expξ
2
downward jump, 3 in the first phase of an upward jump, 4 in the second phase of an upward jump, 5 without jump, i.e. by the Brownian component, at 0, 6 without jump at x + y. Define the random variable Z
by setting Z = i if and only if case i occurs, 1 ≤ i ≤ 6. The LT ψ
y
β = E e
− βτ
y
of τ
y
can be written as ψ
y
β = P
6 i=
1
ψ
y i
β , where ψ
y i
β = E e
− βτ
y
1
{ Z=i}
. We now show how to compute ψ
y i
β, 1 ≤ i ≤ 6. The basic observation is that Xτ
y
and τ
y
are condi- tionally independent, given Z; moreover, −Xτ
y
is expξ -distributed given Z = i, i = 1, 2, while Xτ
y
is expν-distributed if Z = 3, and expν ∗ expν-distributed if Z = 4. If Z = 5 or Z = 6, we have Xτ
y
≡ or Xτ
y
≡ x + y , respectively. Therefore, by using again the martingale equation EMτ
y
= EM 0, we
obtain ϕα − βE
Z
τ
y
e
− αXt −βt
dt = −
e
− αx
+ E
e
− αXτ
y
−βτ
y
= − e
− αx
+ ξ
1
ξ
1
− α
ψ
y 1
β + ξ
2
ξ
2
− α
ψ
y 2
β + e
− αx+y
ν ν + α
ψ
y 3
β +
e
− αx+y
ν ν + α
2
ψ
y 4
y + ψ
y 5
β + e
− αx+y
ψ
y 6
β. 4.1
Let β 6= β
∗
. Then ϕα − β = 0 has six distinct roots α
i
β, 1 ≤ i ≤ 6, and inserting them in 4.1 yields a
system of six linear equations for the six unknowns ψ
y j
β, 1 ≤ j ≤ 6. The closed-form solution of these equations
is easily available, e.g. via MATHEMATICA, but not very illuminating. If β = β
∗
, we may of course take limits to obtain ψ
y i
β
∗
= lim
β
∗
6= β→β
∗
ψ
y i
β . But we can also proceed
as follows. Let α
1
β
∗
be the double root of ϕα − β
∗
= 0. Taking the derivative with respect to α in the first
equation of 4.1 and inserting α = α
1
β
∗
we find that the left-hand side is equal to 0, while the right-hand side is given by
x e
− α
1
β
∗
x
− EXτ
y
e
− α
1
β
∗
Xτ
y
−βτ
y
. Again, Xτ
y
and τ
y
are conditionally independent given Z = i, so that x
e
− α
1
β
∗
x
= EXτ
y
e
− α
1
β
∗
Xτ
y
−βτ
y
=
6
X
i= 1
EXτ
y
e
− α
1
β
∗
Xτ
y
| Z = iψ
y i
β
∗
. 4.2
The conditional distribution of Xτ
y
, given Z = i, was given above for i = 1, . . . , 6; it is either a point mass or exponential or Erlang. Thus, 4.2 provides the “missing” sixth linear equation for the ψ
y i
β
∗
. The above technique can be applied in the case of general phase-type or hyperexponential jump size distributions
upwards and downwards. However, there may be more roots of higher multiplicity. Then one has to take higher derivatives to obtain a sufficiently large number of linear equations for the partial LTs.
5. The maximal value of the cash fund before bankruptcy
