202 M. Steffensen Insurance: Mathematics and Economics 27 2000 201–214
unit-linked life insurance, at least if the unit is traded, and this subject of actuarial mathematics has been an aim of financial theory since Brennan and Schwartz 1976 recognized the option structure of a unit-linked life insurance
with a guarantee. For an overview of the existing literature, see Aase and Persson 1994. Aase and Persson 1994 attained a generalized version of TDE for unit-linked insurance contracts. Our model framework covers their set-up,
and we show how the securitization leads to further generalization of TDE by means of purely financial arguments. The fundamental connection between the celebrated TDE and the Black–Scholes differential equation just as
celebrated but in a different forum is indicated by Aase and Persson 1994. Our derivation brings to the surface more directly this connection by treatment of financial risk and insurance risk in equal terms.
The target group of the paper is two-sided. On one hand, we approach an actuarial problem of evaluating an insurance payment process. The tools are imported from financial mathematics, and the reader with a background in
traditional actuarial mathematics will benefit from knowledge of the concept of arbitrage as well as its connection to martingale measures see e.g. Harrison and Pliska, 1981; Delbaen and Schachermayer, 1994. On the other hand,
the paper may also form an introduction to life insurance mathematics for financial mathematicians. A statistical model frequently used in life insurance mathematics is presented, an insurance contract is constructed, and our
main result is specialized to a cornerstone in insurance mathematics, Thiele’s Differential Equation. However, the statistical model and the construction of an insurance contract is hardly motivated, and the reader with a background
only in financial mathematics may consult a textbook on basic life insurance mathematics e.g. Gerber 1990, for such a motivation.
In Section 2, we present the basic stochastic model. In Section 3, we define an index and a market based on this model and in Section 4, we introduce a payment process and an insurance contract based on the index. In Section
5, the price process of an insurance contract is derived, whereas the differential equation imposed by a no arbitrage condition on the market forming this price process is derived in Section 6. Section 7 contains three examples of
which one is the traditional actuarial set-up, whereas the other two treat unit-linked insurance in a simple and a path-dependent set-up, respectively.
2. The basic stochastic environment We take as given a probability space , F , F = {F
t
}
t ≥0
, P . We let X
t t ≥0
be a cadlag i.e. its sample paths are almost surely right continuous with left limits jump process with finite state space J = 1, . . . , J and associate
a marked point process T
n
, 8
n
, where T
n
denotes the time of the nth jump of X
t
, and 8
n
is the state entered at time T
n
, i.e. X
T
n
= 8
n
. We introduce the counting processes N
j t
=
∞
X
n=1
I T
n
≤ t, X
T
n
= j , j ∈ J ,
and the J -dimensional vector N
t
=
N
1 t
.. .
N
J t
.
The process X generates information formalized by the filtration F
X
= {F
X t
}
t ≥0
, where F
X t
= σ X
s
, 0 s ≤ t = σ N
j s
, j ∈ J , 0 s ≤ t. We let W
t t ≥0
be a standard K-dimensional Brownian motion such that W
k t
and W
l t
are independent for k 6= l.
The information generated by W is formalized by the filtration F
W
= {F
W t
}
t ≥0
, where F
W t
= σ W
s
, 0 s ≤ t .
M. Steffensen Insurance: Mathematics and Economics 27 2000 201–214 203
Finally, we let the filtration F = {F
t
}
t ≥0
formalize the flow of information generated by both X and W , i.e. F
t
is the smallest sigma-field containing F
X t
and F
W t
, F
t
= F
X t
∨ F
W t
. For a matrix A, we let A
T
denote the transpose of A and A
i·
and A
·i
the ith row and the ith column of A, respectively. For a vector a, we let diaga denote the diagonal matrix with the components of a in the principal diagonal and 0
elsewhere. We shall write δ
1×J
and δ
J ×1
instead of δ, . . . , δ and δ, . . . , δ
T
, respectively. For derivatives, we shall use the notation ∂
x
= ∂∂x and ∂
xy
= ∂
2
∂x∂y. For a vector a, we let R
a and da mean componentwise integration and componentwise differentiation, respectively.
3. The index and the market
In the subsequent sections, we shall define and study an insurance contract. Instead of letting the payments in the insurance contract be directly driven by the stochastic basis we shall work with an index which is driven by the
stochastic basis and which will form the basis for the payments. We introduce an index S, an I + 1-dimensional vector of processes, the dynamics of which is given by
dS
t
= α
S t
dt + β
S t −
dN
t
+ σ
S t
dW
t
, S
= s ,
where α
S
∈ R
I +1
, β
S
∈ R
I +1×J
, and σ
S
∈ R
I +1×K
are functions of t, S
t
and s ∈ R
I +1
is F -measurable.
We denote by S
i t
, α
Si t
, β
Sij t
, and σ
Sik t
the ith entry of S
t
, the ith entry of α
S t
, the i, j th entry of β
S t
, and i, kth entry of σ
S t
, respectively. The information generated by S is formalized by the filtration F
S
= {F
S t
}
t ≥0
, where F
S t
= σ S
τ
, 0 τ ≤ t ⊆ F
t
. The index will form the basis for the payments to be defined in Section 4. As examples in Section 4, we define
S
1
= X, S
2
is the duration of the sojourn in the current state of X, S
3
the total number of jumps and S
4
the price of a risky asset, and the reader may benefit from having these examples in mind already at this stage.
We assume that S is a Markov process and that there exist deterministic piecewise continuous functions µ
j
t, s,
j ∈ J , s ∈ R
I +1
such that N
j t
admits the F
S t
-intensity process µ
j t
= µ
j
t, S
t
, informally given by µ
j t
dt = EdN
j t
|F
S t −
+ odt = EdN
j t
|S
t −
+ odt, where oh h → 0 as h → 0. We know that the compensated counting processes
M
j t
= N
j t
− Z
t
µ
j s
ds, j ∈ J
are F
S t
-martingales and, more generally, that R
t
H sdN
j s
− µ
j s
ds is an F
S t
-martingale for every j ∈ J , where H is an F
S t
-predictable process such that E R
t
|H s|µ
j s
ds ∞. Introduce the J -dimensional vectors containing the intensity processes and martingales associated with N ,
µ
t
=
µ
1 t
.. .
µ
J t
, M
t
=
M
1 t
.. .
M
J t
. If e.g. X is included in the index S, µt, X
t
candidates to the intensity process corresponding to the classical situation see e.g. Hoem, 1969. However, in general, the intensity process µ may differ from the intensity process
w.r.t. the natural filtration of N .
204 M. Steffensen Insurance: Mathematics and Economics 27 2000 201–214
We introduce a market Z, an n + 1-dimensional vector n ≤ I of price processes assumed to be positive, and denote by Z
i
the ith entry of Z. The market Z consists of exactly those entries of S that are marketed, i.e. traded on a given market. We assume that there exists a short rate of interest such that the market contains a price process
Z with the dynamics given by
dZ
t
= r
t
Z
t
dt, Z
= 1. This price process can be considered as the value process of a unit deposited on a bank account at time 0, and we
shall call this entry for the risk-free asset even though r
t
is allowed to depend on t, S
t
. Furthermore, we assume that the set of martingale measures, Q, i.e. the set of probability measures Q equivalent to P such that Z
i
Z is a
Q-martingale for all i, is non-empty. From the fundamental theory of asset pricing this assumption is known to be essentially equivalent to the assumption that no arbitrage possibilities exist on the market Z.
The entries of an index S will also be called indices, and the indices appearing in Z will then be called marketed indices or assets. With this formulation, the set of marketed indices is a subset of the set of indices and it contains
at least one entry, namely Z .
A Markov trading strategy is a sufficiently integrable predictable process θ ∈ R
n+1
that can be written in the form θ
t
= θ t−, S
t −
, and the value process U corresponding to θ is defined by U
t
= θ
t
· Z
t
=
n
X
i=0
θ
i t
Z
i t
. The Markov trading strategy θ is said to be self-financing if
dU
t
= θ
t
dZ
t
. It is easily shown that if θ is self-financing and Q ∈ Q, then UZ
is a Q-martingale. We say that a trading strategy is admissible if it is a Markov trading strategy such that U
t
0 for all t and complying with whatever institutional requirements there may be. We denote by 2 the set of admissible strategies.
Throughout this paper one can think of θ as the strategy corresponding to a constant relative portfolio, i.e. a strategy θ such that for a constant n + 1-dimensional vector c, θ
i t
Z
i t −
= c
i
U
t −
, i = 0, . . . , n. This strategy reflects an investment profile possibly restricted by the supervising authorities e.g. such that θ
i
is non-negative for all i if short-selling is not allowed. Hereby we point out that θ , in general, is not a strategy aiming at hedging some
contingent claim. As it will be seen, it is rather a part of the contingent claim itself.
4. The payment process and the insurance contract
