286 F. Quittard-Pinon and R. Randrianarivony
price processSis supposed to behaveaccording to the following SDEunder the
chosen equivalent pricing measureQ:
dSt
St−=
(
rt−)
dt+ρσdWt+σ√
1 −ρ^2 dZt+(Y− 1 )dN ̃t. (9)Again,rtis the instantaneous interest rate,represents the fixed proportional insur-
ance risk charge,σis the asset’s volatility,ρis the correlation between the asset and
the interest rate,WandZare two independent standard Brownian motions, and the
last part takes into account the jumps.N ̃is a compensated Poisson process with in-
tensityλ, whileY, a r.v. independent from the former processes, represents the price
change after a jump. The jump size is defined byJ=ln(Y).
Let us emphasise here that the non-drift partM,definedbydMt=ρσdWt+
σ
√
1 −ρ^2 dZt+(Y− 1 )dN ̃t, is a martingale in the considered risk-neutral universe.3.1 Modelling stochastic interest rates and subaccount jumps
Denoting byNtthe Poisson process with intensityλand applying It ̄o’s lemma, the
dynamics ofSwrites as:
St=S 0 e∫t
0 rsds−(+
1
2 σ^2 +λκ)t+ρσWt+σ√
1 −ρ^2 Zt+
∑Nt
i= 1ln(
(Y)i)
, (10)
whereκ=E(Y− 1 ). The zero-coupon bond price obeys the following equation:
P(t,T)=P( 0 ,T)e∫t
0 σP(s,T)dWs−^12∫t
0 σ^2 P(s,T)ds+∫t
0 rsds.The subaccount dynamics can be written as:
St=S 0
P( 0 ,t)e−(+^12 σ^2 +λκ)t+ 21
∫t
0 σP^2 (s,t)ds+∫t
0 [ρσ−σP(s,t)]dWs+σ√
1 −ρ^2 Zt+
∑Nt
i= 1ln(
(Y)i)
.
Let us introduce theT-forward measureQTdefined by
dQT
dQ∣
∣
Ft=δtP(t,T)
P( 0 ,T)=e∫t
0 σP(s,T)dWs−^12∫t
0 σ^2 P(s,T)ds, (11)whereδt is the discount factor defined in (1). Girsanov’s theorem states that the
stochastic processWT,definedbyWtT=Wt−
∫t
0 σP(s,T)ds, is a standard Brownian
motion underQT. Hence, the subaccount price process can be derived under theT-
forward measure:
St=S 0
P( 0 ,t)
eXt, (12)whereXis the process defined by
Xt=−(+^12 σ^2 +λκ)t+∫t0(
σP(s,T)(
ρσ−σP(s,t))
+^12 σP^2 (s,t))
ds+
∫t0(
ρσ−σP(s,t))
dWsT+σ√
1 −ρ^2 Zt+∑Nti= 1ln(
(Y)i