Rt = bdt +VtJ −Vt1×J, T D αSt, àt
= bct+rtVt−∂tVt−(∂sVt)T αt−Rtàt−ψt, and abbreviating
αFt = rtFt+T D(αt, àt), βFt = −Rt,
σFt = −(∂sVt)T σt, ρFt = LtZt0
Ut
, we arrive at the simple form
dFt =αFtdt+βFt−dMt+σFt dWt+ρFt−d Ut
Zt0
. (2.6)
The abbreviationsR and T D are motivated by the terms sum atRisk andThiele’s Differential, respectively. In Section 2.6, we shall see that T D, taken in a point different from (αt, àt), equated to 0 constitutes a generalized version of Thiele’s dif- ferential equation. A differential equation for the reserve of a life insurance contract was derived by Thiele in 1875, but we shall refer to Hoem [33] for a classical version presented in probabilistic terms.
Note that (2.6) is not the semimartingale form underP, since ZU0 is not in general a P-martingale. This is, however, a convenient form as the succeeding section will show.
2.6 The set of martingale measures and Thiele’s differential equation
In this section we study the consequences of the no arbitrage condition on the markets Z and (Z, F) by studying the conditions for existence of an equivalent martingale measure on these markets.
For construction of a new measure Q, we shall define a likelihood process Λ by dΛt = Λt−
X
j
gtj−dMtj +X
k
hktdWtk
!
= Λt− gtT−dMt+hTtdWt
, Λ0 = 1,
where we have introduced
gjt =gj(t, St) , hkt =hk(t, St), (2.7)
and
gt =
gt1
...
gtJ
, ht=
h1t
...
hKt
.
Assume that gt and ht are chosen such that the conditions EP[ΛT] = 1,
gj(t, s)>−1, j ∈ J, (2.8) are fulfilled. Then we can change measure fromP toQ on (Ω,FT) by the definition,
ΛT = dQ dP,
and it follows from Girsanov’s Theorems thatWtk underQhas the local drifthkt and that Ntj under Q admits the FtS-intensity process (1 +gtj)àjt. In vector notation, Wt has the local driftht and Nt admits theFtS-intensity process diag 1J×1+gt
àt under Q. Note that by (2.7) we consider only the part of possible measure trans- formations that allow g and h to be stochastic processes in a particular form. This restriction on the measure transformation is imposed by the restriction on the price operator leading to Vt = V (t, St). This will be argued at the end of this section.
Defining the Q-martingales
MtQ = Nt− Z t
0
diag 1J×1+gs àsds, WtQ = Wt−
Z t 0
hsds, we can write the dynamics of (Z, F) as
dZt = αZQt dt+βZt−dMtQ+σZt dWtQ,
dFt = αF Qt dt+βFt−dMtQ+σFtdWtQ+ρFt−d Ut
Zt0
, (2.9)
where
αZQt = αZt +σZtht+βZtdiag 1J×1+gt
àt, αF Qt = rtFt+T D αt+σtht, diag 1J×1+gt
àt . We define the market prices of diffusion and jump risk, respectively, by
ηt = −ht,
ξt = −diag(gt)àt,
and we say that the insurance company is risk-neutral with respect to diffusion risk k or jump riskj if ηkt = 0 or ξjt = 0, respectively.
2.6. THE SET OF MARTINGALE MEASURES 37 Now we determine the set of martingale measuresQin the marketZ by requiring
Z
Z0 to be a martingale under Q. We see that gt and ht should be chosen such that αZt +σZtht+βZt diag 1J×1+gt
àt−rtZt= 0.
We have that also ZU0 is a martingale under Q and (2.9) is seen to be written on semimartingale form under Q. Thus, requiring that also ZF0 is a martingale under Q gives the equation
T D αt+σtht, diag 1J×1+gt
àt
= 0,
which constitutes a generalized version of Thiele’s differential equation (TDE). In Section 2.7, we recognize the classical version from Hoem [33].
Adding to TDE the side conditions V0−= 0 and VT−= ∆BT, we formulate our result as a theorem:
Theorem 1 Assume that the partial derivatives ∂tV, ∂sV, and ∂ssV exist and are continuous. Assume that (g, h) can be chosen such that
αZt +σZtht+βZt diag 1J×1+gt
àt−rtZt= 0. (2.10) Then, if the arbitrage free reserve on an insurance contract B can be written in the form V (t, St), V (t, s) solves for some (g, h) subject to (2.10) the deterministic differential equation (coefficients are (t, s) and (T, s), respectively)
∂tVt = bc+rtVt−(∂sVt)T (αt+σtht)−Rtdiag 1J×1+gt
àt−ψt, VT− = ∆BT.
An arbitrage free insurance contract fulfills the equivalence relation
V0− = 0. (2.11)
Although the semimartingale form of F under P was not needed in our deriva- tion of TDE, it may be interesting for other reasons. After some straightforward calculations one gets
dZt = rtZt+βZtξt+σZtηt
dt+βZt−dMt+σZtdWt, dFt =
rtFt+
βFt +θtLt
Ut
βZt
ξt+
σFt + θtLt
Ut
σZt
ηt
dt (2.12) +
βFt−+θtLt−
Ut−
βZt−
dMt+
σFt + θtLt
Ut
σZt
dWt.
This representation of (Z, F) motivates the term market price of risk and shows how the expected return on the marketed indices, now including F, is increased compared to the return on the asset Z0. From Theorem 1 it is seen that the price process V does not depend on θ, but (2.12) shows that the price process F indeed does. This implies that when it comes to laying down the payment process B,
the only marketed indices of importance are those actually appearing as indices in B. Only if we consider the price process F, the remaining entries of Z, i.e.
those entries of S that play the role of investment possibilities but do not appear as indices in B, are important. In the succeeding section we consider examples of insurance contracts. Since we focus on the process V, we let the market comprise only those assets on which payments depend. The representation in (2.12) may be an appropriate starting point for choice of an admissible strategy θ, taking into account e.g. the preferences of (the owners of) the insurance company.
Here we finish the general study of the process (V, F) by pinning down its stochas- tic representation formula. The traditionally educated life insurance actuary may rejoice at recognizing the reserve as an expected value. We have postponed the representation of the reserve as an expected value in order to emphasize that this is rather a fortunate consequence of the no arbitrage condition than a (measure- adjusted) consequence of traditional actuarial reasoning. In order to prevent arbi- trage possibilities we have constructedQsuch that ZF0,ZU0
underQis a martingale, and then it follows that
Ft
Zt0 = EQ FT
ZT0 FtS
= EQ 1
ZT0 Z t
0
UT
Us
dBs FtS
+EQ
1 ZT0
Z T t
UT
Us
dBs FtS
= EQ UT
ZT0 FtS
Z t 0
1 Us
dBs+ Z T
t
EQ 1
Us
dBsEQ UT
ZT0 FsS
FtS
= Ut
Zt0 Z t
0
1 Us
dBs+ Z T
t
EQ 1
Us
dBsUs
Zs0 FtS
= Lt
Zt0 +EQ Z T
t
1 Zs0dBs
FtS
. Thus,
Vt =EQ
Zt0 Z T
t
1
Zs0d(−Bs) FtS
, (2.13)
and the equivalence relationV0− = 0, can be written EQ
Z T 0−
1 Zs0dBs
= 0.
A special case of this constraint is known in actuarial mathematics as the equiv- alence principle, namely the case of risk-neutrality. In general, market prices of risk are not zero, and here we have taken into account the existence of a marketZ which may contain information of these market prices of risk.
A calculation similar to the one leading to (2.13) shows that the equivalence relationV0− = 0 corresponds to the alternative equivalence relation
EQ LT
ZT0
= 0, (2.14)