.
Expected Exponential Utility Maximization of Insurers
with a Linear Gaussian Stochastic Factor Model
.
(joint work with Hiroaki HATA (Shizuoka Univ.))
Kazuhiro YASUDA
Hosei University
2017/4/26 @ Seoul
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K. Yasuda (Hosei University)
Expected Exponential Utility Maximization
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2017/4/26
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Contents
1.
2.
3.
Introduction and our Problem
Outline
Our Problem
Results
Theoretical Results
Numerical Results
References
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K. Yasuda (Hosei University)
Expected Exponential Utility Maximization
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Introduction and our Problem
Outline
Contents
1.
2.
3.
Introduction and our Problem
Outline
Our Problem
Results
Theoretical Results
Numerical Results
References
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K. Yasuda (Hosei University)
Expected Exponential Utility Maximization
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2017/4/26
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Introduction and our Problem
Outline
Introduction 1 (Risk process : Cramér-Lundberg model)
We consider insurance companies.
▶
▶
▶
They receive a certain amount of money from insurance
policyholders.
They need to pay a certain amount of money when claimed.
Those income and outgo are on standard insurance business.
The risk process Rt :
Rt = x + ct −
Nt
∑
Zi .
i=1
(Cramér-Lundberg model)
▶
▶
▶
▶
x · · · the initial surplus
c · · · the premium rate
Nt · · · the total number of claims until time t. Poisson process with an
intensity λ (> 0)
Zi · · · the claim amounts. i.i.d. positive random variables with
distribution ν.
Rt denotes profit and loss from insurance buisiness. In this talk, we
consider the case that insurers invest to risky assets.
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Expected Exponential Utility Maximization
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Introduction and our Problem
Outline
Introduction 2 (Wealth process and Criterion)
St0 · · · the risk-free asset price at time t.
Sti · · · the i th risky asset price at time t. (i = 1, ..., m)
πit · · · the investment amount to Sti . (i = 1, ..., m)
Xtπ · · · the wealth process of the insurance company. Define



∫ t
m
m
0
i

∑
∑




dS
dS




u
πit i + Xuπ −
πit  0t 
,
Xtπ = Rt +




Su
St 
0
i =1
i =1
X0π = R0 = x .
▶
Wealth process = Risk process + Profit and loss from investments.
Criterion : Let T > 0 be a maturity.
▶
Expected utility maximization :
[
]
sup E U (XTt ,x ,π ) , where U (x ) is a utility function.
π
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Expected Exponential Utility Maximization
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Introduction and our Problem
Outline
Introduction 3 (Previous Works)
There are many papers related to the expected exponential utility
maximization problem:
Ferguson (1965), Browne (1995), Hipp and Plum (2000), Yang and
Zhang (2005), Fernández, Hernández, Meda and Saavedra (2006),
Wang, Xia and Zhang (2007), Wang (2007), Guerra and Centeno
(2008), Zhou (2009), Badaoui and Fernández (2013), Zou and
Cadenillas (2014), ......
Badaoui and Fernández (2013) consider the expected exponential
utility maximization problem in the case that the risky asset follows a
factor model.
dSt0 = rSt0 dt , dSt = µ(Yt )St dt + σ(Yt )St dWt1 , dYt = g (Yt )dt + βdWt2 ,
where Wt1 ⊔ Wt2 and Yt is observable.
▶
Under globally Lipschitz and bounded g, µ(z ) > r (∀z ∈ R) and
(µ(z )−r )2
bounded σ(z )2 with a bounded first derivative, they give the explicit
optimal investment strategy.
.
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Expected Exponential Utility Maximization
2017/4/26
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Introduction and our Problem
Our Problem
Contents
1.
2.
3.
Introduction and our Problem
Outline
Our Problem
Results
Theoretical Results
Numerical Results
References
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K. Yasuda (Hosei University)
Expected Exponential Utility Maximization
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2017/4/26
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Introduction and our Problem
Our Problem
Our Model 1 (Market)
The risk-free asset process : dSt0 = rSt0 dt ,
S00 = s00
The i-th risky asset process (i = 1, ..., m) :
dSti
=
Sti


n∑
+m




 i
ik
k
µ
(
Y
)
dt
+
Σ
dW
,


t
t
P




S0i = s0i
k =1
Factor processes (n-dim.) :
dYt = g (Yt )dt + Σf dWt ,
▶
▶
▶
Y (0) = y ∈ Rn
Wt = (Wtk )k =1,··· ,(n+m) . . . an n + m dimensional standard Brownian
motion process.
r ≥ 0, ΣP ∈ Rm×(n+m) , Σf ∈ Rn×(n+m) , ΣP Σ∗P > 0.
µ(y ) := a + Ay, a ∈ Rm , A ∈ Rm×n , g (y ) := b + By, b ∈ Rn , B ∈ Rn×n
Assume that Yt is observable.
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K. Yasuda (Hosei University)
Expected Exponential Utility Maximization
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Introduction and our Problem
Our Problem
Our Model 2 (Risk Process, Problem Comments)
Rt = x + ct −
The risk process Rt :
Nt
∑
Zi .
i=1
▶
▶
▶
Nt is the Cox process, that is, the intensity is defined as λ(Yt ), where
λ(y ) = y ∗ Λy + λ0 , λ0 > 0 and Λ ∈ Rn×n is non-negative definite.
Zi is independent of Wt and Nt .
Note that our filtration is Ft = σ(Ws , Ns , Zi 1i ≤Ns ; 0 ≤ s ≤ t , i ≥ 1)
The wealth process Xtπ :



∫ t
m
m
0
i

∑
∑




dS
dS




Xtπ = Rt +
πit i u + Xuπ −
πit  0t 
, X0π = R0 = x .





S
S
0
u
t
i =1
i =1
Our problem is the expected exponential utility maximization:
[
π
]
sup E −e −αXT , (α > 0).
π
We give explicit representation of the optimal strategy and the
value function under the above factor model.
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K. Yasuda (Hosei University)
Expected Exponential Utility Maximization
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2017/4/26
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Results
Theoretical Results
Contents
1.
2.
3.
Introduction and our Problem
Outline
Our Problem
Results
Theoretical Results
Numerical Results
References
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K. Yasuda (Hosei University)
Expected Exponential Utility Maximization
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2017/4/26
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Results
Theoretical Results
Formal Derivation of HJB Equation 1
Transform the exponential utility of the wealth as follows:
−e
t ,x ,y ,π
−αXT
−αx er (T −t ) −c α
= − e|
∫T
t
er (T −s ) ds +λ0
∫T∫
{z
t
z >0
(
)
r (T −s )
eαz e
−1 ν(dz )ds
non−random
where set N ([0, t ] × U ) =
∑Nt
i =1
∫T
t
} e|
ℓ(Ys − ,πs )ds
{z
random
0
,T (π),
}E
|t{z
}
R .N .
1(Zi ∈ U ) for U ⊂ R++ ,
∫ (
)
α2 2r (T −t ) ∗
r (T −t )
e
π Σp Σ∗p π − αer (T −t ) π∗ (µ(y ) − r1) + y ∗ Λy
eαz e
− 1 ν(dz ),
2
z >0
)
( ∫ T
∫
α2 T 2r (T −u) ∗
0
r (T −u) ∗
e
πu Σp dWu −
e
πu Σp Σ∗p πu du
Et ,T (π) := exp −α
2 t
t
(∫ T ∫
)
∫ T
∫ (
)
r (T −u )
× exp
αz er (T −u) N (du, dz ) +
λ(Yu− )
1 − eαz e
ν(dz )du .
ℓ(y , π) :=
t
z >0
z >0
t
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K. Yasuda (Hosei University)
Expected Exponential Utility Maximization
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Results
Theoretical Results
Formal Derivation of HJB Equation 2
dP π
0
dP = Et ,T (π).
E [E0t ,T (π)] = 1, then
Set
If
Wsπ := Ws +
P π -Brownian motion.
∫s
t
αer (T −u) Σ∗p πu du is
}
{
Yt follows dYs = g (Ys ) − αer (T −s ) Σf Σ∗p πs ds + Σf dWsπ , Yt = y.
We can rewrite the expectation of the utility:
[
π
]
−αx er (T −t ) −c α
E −e−αXT = −e
|
[
∫T
Set J̃ (t , y ; π) = E (π) e t
∫T
t
er (T −s ) ds +λ0
∫T∫
t
z >0
(
r (T −s )
eαz e
}E
{z
NOT depend on π
ℓ(Ys − ,πs )ds
)
−1 ν(dz )ds
(π)
[ ∫T
]
e t ℓ(Ys− ,πs )ds .
]
which depends on only the
factor Yt , and V (t , y ) := infπ J̃ (t , y ; π). Then we can derive the HJB
equation for V (t , y ):
[
inf
π
}
{
}∗
∂V
1
+ tr(Σf Σ∗f D 2 V ) + g (y ) − αer (T −t ) Σp Σ∗f π DV + ℓ(y , π) = 0, V (T , y ) = 0.
∂t
2
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K. Yasuda (Hosei University)
Expected Exponential Utility Maximization
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Results
Theoretical Results
Solution to HJB eq. 1
Assume that we can write V (t , y ) = e−v (t ,y ) . Then we have
sup Lπt v (t , y ) = 0, v (T , y ) = 0,
(1)
π∈Rm
where Lπt v (t , y ) is defined by
1
∂v
1
+ tr(Σf Σ∗f D 2 v ) − (Dv )∗ Σf Σ∗f Dv + g (y )∗ Dv
∂t
2
2
∫ (
)
1 2 2r (T −t ) ∗
r (T −t )
∗
r (T −t ) ∗
− α e
π Σp Σp π + αe
π {µ(y ) − r1 + Σp Σ∗f Dv } − y ∗ Λy
eαz e
− 1 ν(dz ).
2
z >0
Lπt v (t , y ) :=
Recalling that the maximizer is
π̌(t , y ) :=
1
α
{
}
e−r (T −t ) (Σp Σ∗p )−1 µ(y ) − r1 + Σp Σ∗f Dv (t , y ) ,
we rewrite (1) as
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Expected Exponential Utility Maximization
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Results
Theoretical Results
Solution to HJB eq. 2
{
}∗
∂v
1
1
1
+ tr(Σf Σ∗f D 2 v ) + K1 y + b − Σf Σ∗p (Σp Σ∗p )−1 (a − r1) Dv − (Dv )∗ K2 Dv + y ∗ K0 y
∂t
2
2
2
∫ (
)
1
∗
∗ −1
∗
∗ −1
∗
αz er (T −t )
+ (a − r1) (Σp Σp ) A y + (a − r1) (Σp Σp ) (a − r1) − y Λy
e
− 1 ν(dz ) = 0,
2
z >0
v (T , y ) = 0.
(2)
We obtain an explicit solution of (2).
.
Lemma
.
∫
Assume K0 − 2Λ
z >0
(
)
rT
eαz e − 1 ν(dz ) ≥ 0. Then, (2) has the explicit
solution b
v (t , y ) defined by
.
b
v (t , y ) :=
1 ∗
y P (t )y + q(t )∗ y + k (t ),
2
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Expected Exponential Utility Maximization
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Results
Theoretical Results
Solution to HJB eq. 3
where P (t ) is the solution of the following Riccati equation
Ṗ (t ) + P (t )K1 + K1∗ P (t ) − P (t )K2 P (t ) + K0 − 2Λ
∫
(
r (T −t )
eαz e
z >0
)
− 1 ν(dz ) = 0, P (T ) = 0.
q(t ) and k (t ) are the following linear ordinary differential equations :
q̇(t ) + (K1 − K2 P (t ))∗ q(t ) + P (t )b + (A − Σp Σ∗f P (t ))∗ (Σp Σ∗p )−1 (a − r1) = 0, q(T ) = 0,
and
{
}
1
tr(Σf Σ∗f P (t )) + q(t )∗ b − Σf Σ∗p (Σp Σ∗p )−1 (a − r1)
2
1
1
− q(t )∗ K2 q(t ) + (a − r1)∗ (Σp Σ∗p )−1 (a − r1) = 0, k (T ) = 0.
2
2
k̇ (t ) +
Here constants K2 , K1 and K0 are defined as follows:
{
}
K2 := Σf I − Σ∗p (Σp Σ∗p )−1 Σp Σ∗f ,
K1 := B − Σf Σ∗p (Σp Σ∗p )−1 A , K0 := A ∗ (Σp Σ∗p )−1 A .
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Expected Exponential Utility Maximization
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Results
Theoretical Results
Main Theorem
.
Theorem
.
∫
Assume that
b
π(s , y ) :=
1
α
z >0
e2αz e ν(dz ) < ∞. Define
rT
}
[{
]
e−r (T −s ) (Σp Σ∗p )−1 A − Σp Σ∗f P (s ) y + a − r1 − Σp Σ∗f q(s ) .
Then, b
πs := b
π(s , Ys ) is optimal for our problem.
Moreover, the value function is explicitly given as
(
∫
b
V (t , x , y ) := −exp −b
v (t , y ) − c α
∫
+λ0
.
t
T
∫
(
z >0
T
er (T −s ) ds
t
αz e
e
r (T −s )
)
)
r (T −t )
.
− 1 ν(dz )ds − αx e
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K. Yasuda (Hosei University)
Expected Exponential Utility Maximization
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Results
Numerical Results
Contents
1.
2.
3.
Introduction and our Problem
Outline
Our Problem
Results
Theoretical Results
Numerical Results
References
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K. Yasuda (Hosei University)
Expected Exponential Utility Maximization
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2017/4/26
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Results
Numerical Results
Numerical Experiments 1 (Model: case m = n = 1)
dSt0 = rSt0 dt ,
S00 = s00


2


∑



k
k
dSt = St 
(
a
+
AY
)
dt
+
,
Σ
dW

t
t
P




k =1
dYt = (b + BYt )dt + Σf dWt ,
Rt = x + ct −
Ni
∑
S0 = s0
Y (0) = y ∈ R
Zi .
i =1
Nt is the Cox process with λ(Yt ) = ΛYt2 + λ0 ,
Zi follows the exponential distribution with η, i.e. E [Zi ] = η.
U (x ) = −e −αx
parameter
value
parameter
value
parameter
value
T
a
B
100
0
−1
1
r
0
A
s0
1
100
√
√
(1/ 2, 1/ 2)
Λ
Σp
b
λ0
√
√
(1/ 2, 1/ 2)
0
1
Σf
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Expected Exponential Utility Maximization
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Results
Numerical Results
Numerical Experiments 2 (Value Function)
Under the parameters, especially r = 0 and Zi ∼ exp(η), the value
function is given as
αηT
b(0, x , y ) = −e−αx −c αT + 1−αη − 2 P (0)y
V
1
2
−k (0)
,
where P (0) and k (0) can be calculated explicitly.
value function at time 0 (alpha=0.03)
-1e-007
values of value function at time 0
-2e-007
-1.5e-007
-2e-007
-2.0e-007
-3e-007
-3e-007
-2.5e-007
-4e-007
-3.0e-007
-4e-007
-5e-007
-3.5e-007
-4.0e-007
-4.5e-007
-5.0e-007 0
5
10
initial surplus
15
20-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
initial factor value
−2 ≤ y ≤ 2, c = 2, 0 ≤ x ≤ 20, α = 0.03, η = 1.
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Expected Exponential Utility Maximization
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Results
Numerical Results
Numerical Experiments 3 (Ruin Probability)
Here we consider the ruin probability, which is one of important
criteria for insurance companies.
(
Define the ruin probability with investments: P
)
inf Xtx ≤ 0 .
0≤t ≤T
(
Define the ruin probability without investments: P
)
inf Rtx ≤ 0 .
0≤t ≤T
Unfortunately we cannot explicitly calculate it in our model, so
we simulate them using the Euler-Maruyama approximation and the
Monte-Carlo method.
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Expected Exponential Utility Maximization
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Results
Numerical Results
Numerical Experiments 4 (η and Ruin Probability)
relation between mean value of claim distribution and ruin probability
0.08
with investment
without investment
0.07
ruin probability
0.06
0.05
0.04
0.03
0.02
0.01
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
mean value of claim Z (eta)
y = 0, c = 2, x = 10, α = 0.33, 0.1 ≤ η ≤ 1.
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Expected Exponential Utility Maximization
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Results
Numerical Results
Numerical Experiments 5 (x and Ruin Probability)
relation between initial surplus and ruin probability
0.6
with investment
without investment
ruin probability
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
12
14
16
initial surplus
y = 0, c = 2, 1 ≤ x ≤ 15, α = 0.33, η = 1. (ruin with investment ≈ e −0.46x )
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Expected Exponential Utility Maximization
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Results
Numerical Results
Numerical Experiments 6 (α and Ruin Probability)
relation between alpha and ruin probability
0.5
with investment
without investment
0.45
ruin probability
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
alpha
y = 0, c = 2, x = 10, 0.03 ≤ α ≤ 0.33, η = 1.
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Expected Exponential Utility Maximization
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Results
Numerical Results
Numerical Experiments 7 (α and Ruin Probability)
Table: Expected utility.
α
0.03
0.06
0.09
Expect. utility with invest.
Expect. utility without invest.
10−7
−3.365 ×
−3.768 × 10−8
−5.534 × 10−9
−0.193
−0.0532
−0.0219
As is obvious, “Expected utility with investments" > “Expected
utility without investment" holds.
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Expected Exponential Utility Maximization
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Results
Numerical Results
Numerical Experiments 8 (c and Ruin Probability)
relation between premium rate and ruin probability
1
with investment
without investment
0.9
ruin probability
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
premium rate
y = 0, 0.1 ≤ c ≤ 3, x = 10, α = 0.33, η = 1.
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Expected Exponential Utility Maximization
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References
References 1
Browne S (1995) “Optimal investment policies for a firm with a random
risk process: Exponential utility and minimizing the probability of ruin”
, Math. Oper. Res. 20(4), 937–1958.
Badaoui, M and Fernández, B (2013) “An optimal investment strategy
with maximal risk aversion and its ruin probability in the presence of
stochastic volatility on investments.” , Insurance Math. Econom. 53(1),
1–13.
Ferguson, TS (1965) “Betting systems which minimize the probability
of ruin” , J Soc Indust Appl Math. 13(3), 795–818.
Fernández, B, Hernández, D, Meda, A and Saavedra, P. (2008) “An
optimal investment strategy with maximal risk aversion and its ruin
probability” , Math. Methods Oper. Res. 68(1), 159–179.
Guerra, M and Centeno M de L (2008) “Optimal reinsurance policy: the
adjustment coefficient and the expected utility criteria” , Insurance
Math Econom. 42(2), Expected
529–539.
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Exponential Utility Maximization
2017/4/26
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References
References 2
Hipp, C and Plum, M (2000) “Optimal investment for insurers” ,
Insurance Math Econom. 27(2), 215–228.
Wang, N (2007) “Optimal investment for an insurer with exponential
utility preference” , Insurance Math Econom. 40(1), 77–84.
Wang, Z, Xia, J and Zhang, L (2007) “Optimal investment for an insurer:
the martingale approach” , Insurance Math Econom. 40(2), 322–334.
Yang, H and Zhang, L (2005) “Optimal investment for insurer with
jump-diffusion risk process", Insurance Math Econom. 37(3),
615–634.
Zhou, Q (2009) “Optimal investment for an insurer in the Lévy market:
The martingale approach” , Statist. Probab. Lett. 79(14), 1602–1607.
Zou, B and Cadenillas, A (2014) “Optimal investment and risk control
policies for an insurer: expected utility maximization” , Insurance Math
Econom. 58, 57–67.
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K. Yasuda (Hosei University)
Expected Exponential Utility Maximization
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2017/4/26
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References
Thank you
Thank you for your attention.
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K. Yasuda (Hosei University)
Expected Exponential Utility Maximization
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2017/4/26
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