Optimal investment strategies for the HARA utility under the constant elasticity of variance model

Insurance: Mathematics and Economics 51 (2012) 667–673

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Optimal investment strategies for the HARA utility under the constant elasticity of variance model Eun Ju Jung, Jai Heui Kim ∗ Department of Mathematics, Pusan National University, Pusan 609-735, Republic of Korea

article

info

Article history: Received February 2012 Received in revised form September 2012 Accepted 22 September 2012 JEL classification: G11

abstract We give an explicit expression for the optimal investment strategy, under the constant elasticity of variance (CEV) model, which maximizes the expected HARA utility of the final value of the surplus at the maturity time. To do this, the corresponding HJB equation will be transformed into a linear partial differential equation by applying a Legendre transform. And we prove that the optimal investment strategy corresponding to the HARA utility function converges a.s. to the one corresponding to the exponential utility function. © 2012 Elsevier B.V. All rights reserved.

Keywords: Stochastic optimal control Constant elasticity of variance model HARA utility function HJB equation Legendre transform

1. Introduction The managers of pension funds buy reinsurance and/or invest their company’s surplus in a financial market to reduce the risk. Two of fundamental aims that an insurance company pursues is to minimize the ruin probability of the company and to maximize the expected utility of the final surplus at the maturity time T . In the case of no reinsurance and no investment during the period [0, T ] before retirement, the surplus process (V (t ))t ∈[0,T ] of the company will be described by the following form: dV (t ) = µ0 dt V (0) = V0 ,



(1.1)

where the constant V0 > 0 is the initial surplus and the constant µ0 > 0 is the continuous rate of contribution. We assume that all of the surplus is invested in a financial market which consists of two securities, named B and S, whose prices are given by the following differential equations: dB(t ) = rB(t )dt

(1.2)

and dS (t ) = µS (t )dt + kS 1+γ (t )dW (t ),

(1.3)

where r , µ, k and γ are some constants with 0 < r < µ and γ ≤ 0, and (W (t ))t ∈[0,T ] is a standard Brownian motion on a complete probability space (Ω , F , P ) with a filtration (Ft )t ∈[0,T ] . Here r is a rate of return of the risk-free asset B, µ is an expected instantaneous rate of return of the risky asset S and γ is the elasticity parameter. In this case we call (B, S ) a financial market with the constant elasticity variance (CEV) model. We denote by β(t ) the proportion invested in the risky security S at time t ∈ [0, T ]. We disallow leverage and short-sales. In this case it holds that 0 ≤ β(t ) ≤ 1 for all t ∈ [0, T ]. Therefore, at any time 0 ≤ t < T , a nominal amount V (t )(1 − β(t )) is allocated to the risk-free asset B. We treat the proportion β(t ) of the surplus at time t as a control parameter. Then the surplus process (V (t ))t ∈[0,T ] is given by the following stochastic differential equations: dV (t ) = [V (t ){β(t )(µ − r ) + r } +µ0 ]dt + V (t )kβ(t )S γ (t )dW (t ) V (0) = V0 .



Given a strategy β(·), the solution (V β (t ))t ∈[0,T ] of (1.4) is called the surplus process corresponding to β(·). In this paper, we are interested in maximizing the expected hyperbolic absolute risk aversion (HARA) utility of the company’s terminal surplus. The HARA utility function with parameters η, p and q is given by U (v) = U (η, p, q; v) =

∗

Corresponding author. Tel.: +82 51 510 2209; fax: +82 51 581 1458. E-mail address: [email protected] (J.H. Kim).

0167-6687/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.insmatheco.2012.09.009

(1.4)

q > 0, p < 1, p ̸= 0.

1−p qp



qv 1−p

+η

p

, (1.5)

668

E.J. Jung, J.H. Kim / Insurance: Mathematics and Economics 51 (2012) 667–673

We use the HARA utility function U (v) = U (η, p, q; v) with parameters η, p and q defined by (1.5). For the surplus process (V β (t ))t ∈[0,T ] given by (1.4), put

We can check that U (0, p, 1 − p; v) = Upower (v) ≡

1 p

vp

(power utility function)

(1.6)

(2.1)

for all (t , s, v) ∈ [0, T ] × R × R , where E [X |A] is the conditional expectation of a random variable X given an event A. In stochastic optimal control theory it is important to find the optimal value function 1

and 1 lim U (1, p, q; v) = Uexp (v) ≡ − e−qv p→−∞ q (exponential utility function).

J β (t , s, v) = E [U (V β (T )) | S (t ) = s, V β (t ) = v]

(1.7)

In the case that γ = 0, Devolder et al. (2003) found an explicit expression for the optimal asset allocation which maximizes the expected power or exponential utility of the final annuity fund at retirement and at the end of the period after retirement. In the case that γ satisfies the more general condition γ ≤ 0, by using the Legendre transform and dual theory, Xiao et al. (2007) solved the same problem as Devolder et al. (2003) with the logarithmic utility function defined as a limit of a family of modified HARA utility functions when p → 0 (cf. Grasselli (2003)). And Gao (2009) extended the work by Devolder et al. (2003) to the case that γ ≤ 0 by the same method as Xiao et al. (2007). In this paper we find an explicit expression for the optimal investment strategy β ∗ (·), under the same condition that γ ≤ 0 as in the works by Xiao et al. (2007) and Gao (2009), which maximizes the expected HARA or power utility of the final value of the surplus process given by the stochastic differential equation (1.4). And we prove that the optimal solution corresponding to the HARA utility function converges a.s. to the solution corresponding to the exponential utility function as p → −∞. Grasselli (2003) investigated these problems in a financial market model which consists with a risk-free asset whose short rate of return follows the CIR model and d risky assets without elasticities. During the period [T , T + N ] after retirement, the surplus process is given by Eq. (1.4) such that µ0 is replaced by the minus of the continuous rate of annuity benefit (−λ0 ). This shows that our problems before and after retirement are mathematically the same. So we consider only the problem before retirement. Xiao et al. (2007) and Gao (2009) derived the optimal strategy by solving the corresponding Hamilton–Jacobi–Bellman (HJB) equation. Usually, in stochastic optimal problem, the corresponding HJB equation is a nonlinear partial differential equation and it is difficult to solve. So they transformed the HJB equation into a linear partial differential equation by applying a Legendre transform. But Gu et al. (2010) gave an optimal reinsurance and investment strategy by directly solving the HJB equation with very complicated calculus. In this paper we use the same methods as the works by Xiao et al. (2007) and Gao (2009). The structure of the paper is as follows. In Section 2 we formulate our problem and give a theory background based on the stochastic optimal control theory (Björk, 1998; Øksendal, 1998) and properties for a Legendre transform (Jonsson and Sircar, 2002; Xiao et al., 2007; Gao, 2009). In Section 3 we give an explicit expression for the optimal investment strategy corresponding to the HARA utility function. In Section 4, we give an explicit expression for the optimal investment strategy corresponding to the power utility function as a special case of the HARA class, and prove that the optimal investment strategy corresponding to the HARA utility function converges a.s. to the one corresponding to the exponential utility function. 2. Formulation of the problem and theory background A control function β(·) in (1.4) is said to be admissible if (β(t ))t ≥0 is a Ft -adapted process satisfying 0 ≤ β(t ) ≤ 1 for all t ∈ [0, T ]. The set of all admissible controls is denoted by A.

1

H (t , s, v) = sup J β (t , s, v)

(2.2)

β∈A

and the optimal strategy β ∗ (·) such that J β (t , s, v) = H (t , s, v). ∗

(2.3)

In this paper we will give an explicit expression of β (t ). The following two theorems are essential to solve our problem. The proofs are standard and can be found in many text books (e.g. Björk (1998) and Øksendal (1998)). ∗

Theorem 2.1 (HJB Equation). (1) Assume that H (t , s, v) defined by (2.2) is twice continuously differentiable on (0, ∞), i.e., ∈ C 1,2 . Then H (t , s, v) satisfies the following HJB equation:



sup Lβ H (t , s, v) = 0

β∈A

(2.4)

H (T , s, v) = U (v)

for all (t , s, v) ∈ [0, T ) × R1 × R1 , where Lβ is the infinitesimal generator corresponding to the diffusion process defined by the stochastic differential equation (1.4), i.e.,

∂ ∂ ∂ + µs + {v[β(t )(µ − r ) + r ] + µ0 } ∂t ∂s ∂v 1 ∂2 1 ∂2 + v 2 β(t )2 k2 s2γ 2 + k2 s2+2γ 2 2 ∂v 2 ∂s 2 ∂ . (2.5) + β(t )k2 s1+2γ v ∂ s∂v (2) Let G(t , s, v) be a solution of the HJB equation (2.4). Then the value function H (t , s, v) to the control problem (2.2) is given by

Lβ =

H (t , s, v) = G(t , s, v).

¯ , Lβ G(t , s, v) = 0 for all (t , s, x) ∈ Moreover if, for some control β(·) ¯ 1 1 [0, T ) × R × R , then it holds G(t , s, v) = J β (t , s, v). In this case ¯β(t ) = β ∗ (t ) and J β¯ (t , s, v) = J β ∗ (t , s, v). ¯

By Theorem 2.1, the HJB equation associated with our optimization problem (2.2) and (2.3) is 0 = Ht + µsHs + (r v + µ0 )Hv +

1 2

ks2γ +2 Hss

 + sup β(µ − r )v Hv + β k2 s2γ +1 v Hsv β

+

1 2

2 2 2γ



β k s v Hvv , 2

(2.6)

where Ht , Hv , Hs , Hvv , Hss , Hsv denote partial derivative of first and second orders with respect to time, stock price and wealth parameters. It is easy to show that the optimal strategy β ∗ is given by

β∗ = −

(µ − r )Hv + k2 s2γ +1 Hsv . v k2 s2γ Hvv

(2.7)

Inserting (2.7) into (2.6), we obtain the following second order partial differential equation for the optimal value function H;

E.J. Jung, J.H. Kim / Insurance: Mathematics and Economics 51 (2012) 667–673

0 = Ht + µsHs + (r v + µ0 )Hv 1

+ k2 s2γ +2 Hss −

and 2 2 γ +1

[(µ − r )Hv + k s

Hsv ]

2k2 s2γ Hvv

2

669

2

.

(2.8)

To get an explicit expression for the optimal strategy β ∗ given by (2.7), we have to solve this nonlinear equation, but it is very difficult. So, by applying a Legendre transform, we transform this equation into a linear partial differential equation of which the solution gives an explicit expression for β ∗ . Let f : Rn → R1 be a convex function. A Legendre transform on R is defined by L(z ) = max{f (x) − zx}.

(2.9)

x

The function L(z ) is called the Legendre dual of the function f (x). If f (x) is strictly convex, the maximum in the above equation will be attained at just one point, which we denote by x0 . It is attained at the unique solution to the first order condition df (x)

− z = 0. dx So we may write

(2.10)

L(z ) = f (x0 ) − zx0 .

(2.11)

Following Jonsson and Sircar (2002), a Legendre transform can be defined by

 H (t , s, z ) = sup{H (t , s, v) − z v|0 < v < ∞}, v>0

0
(2.12)

where z > 0 denotes the dual variable to v . The value of v where this optimum is attained is denoted by g (t , s, z ), so that g (t , s, z ) = inf {v|H (t , s, v) ≥ z v +  H (t , s, z )}, v>0

0 < t < T.

(2.13)

The function  H is related to g by g = − Hz ,

(2.14)

so we can take either one of the two function g and  H as the dual

 H (T , s, z ) = sup{U (v) − z v}, v>0

so that g (T , s, z ) = (U ′ )−1 (z ).

(2.19)

Substituting (2.15)–(2.17) into (2.8) and differentiating  H with respect to z, we get gt − rg − µ0 + rsgs +



(µ − r )2 z

 − rz gz

k2 s2γ

1

+ k2 s2γ +2 gss − (µ − r )szgsz 2   (µ − r )2 z 2 + gzz = 0 2 2γ

(2.20)

2k s

and, from (1.5) and (2.19), we can see that the boundary condition is g ( T , s, z ) =

 1−p  1 z p−1 − η . q

(2.21)

This is a linear boundary problem that we have wanted. Moreover, we have

β∗ =

−(µ − r )zgz + k2 s1+2γ gs k2 s2γ g

.

(2.22)

3. Optimal investment strategies First we solve the partial differential equation (2.20) of which the solution gives an explicit expression for the optimal strategy β ∗ from (2.22). Lemma 3.1. The solution g (t , s, z ) of the PDE (2.20) with the terminal condition (2.21) is given by

of H. As (2.10) describes, we have Hv = z

(2.15)

and hence

 H (t , s, z ) = H (t , s, g ) − zg ,

g (t , s, z ) = v.

Hss =  Hss − Hsv = −

Hs =  Hs , 2  Hsz ,  Hzz

Hv = z ,

Hvv = −

1

 Hzz

 Hz = −g

q

a(t ) = −



1



b(t , s) z p−1 − η + ηc (t , s) + a(t ),

µ0  r

1 − e−r (T −t ) = −µ0 aT −t | ,



−2γ

b(t , s) = A(t )eB(t )s c (t , s) = C (t )s

,

 Hsz .  Hzz

1−p

(3.1)

where (2.16)

By differentiating (2.15) and (2.16) with respect to t, s and z, we obtain Ht =  Ht ,

g ( t , s, z ) =

−2γ

,

(3.2) (3.3)

+ D(t ).

(3.4)

Here (2.17) A(t ) =

At the terminal time, we denote

2γ 2 (λ− −λ+ )(T −t ) λ+  − λ− e 

×e

 U (z ) = sup{U (v) − z v},

 2γ2γ+1

λ+ − λ−



γ (2γ +1)λ− + 1rp −p (T −t )

,

(3.5)

2γ 2 (λ− −λ+ )(T −t )

v>0

λ− − λ− e B(t ) = k−2 = k−2 I (t ) 2 λ 1 − λ− e2γ (λ− −λ+ )(T −t )

G(z ) = inf {v|U (v) ≥ z v +  U (z )}. v>0

(3.6)

+

As a result, we have G(z ) = (U ′ )−1 (z ).

with (2.18)

Since H (T , s, v) = U (v), we can define

λ± =

H (T , s, z )} g (T , s, z ) = inf {v|U (v) ≥ z v + 

and

v>0

(µ − rp) ±

 (1 − p)(µ2 − r 2 p) , 2γ (1 − r )

(3.7)

670

E.J. Jung, J.H. Kim / Insurance: Mathematics and Economics 51 (2012) 667–673

−e(2γ +1)rt



T

Substituting these derivatives into (3.10), we have

−2γ

e−(2γ +1)ru (1 − p)A(u)eB(u)s q t   × B′ (u) − 2γ rB(u) + 2γ 2 k2 B2 (u) du,

C (t ) =



(3.8)

1−p q

T

1 −ru  2γ e (1 − p)eB(u)s  t q  × k2 (2γ 2 + γ )A(u)B(u) − rA(u) + A′ (u)

D(t ) = −ert

+

 − k2 (2γ 2 + γ )qC (u) du.

(3.9)

Proof. We try to find a solution of (2.20) in the form (3.1) with the boundary conditions given by a(T ) = 0, b(T , s) = 1 and c (T , s) = 0. Then gt = gs =

1−p q 1−p q 1

− +

b



−1

,

1−p

gss =

q



1



1 gzz = − b q

1−p

bt +

q



1−p

rsbs +

 −r −

2−p

A



1

z p−1

p−1

−2

.

b

2q



2−p

2 2 γ +2

k s



bss −

(µ − r )

q

2

rb

+

b q



1−p q

rsbs +

2

k2 s2γ

1−p 2q

 −r −

b q

k2 s2γ +2 bss −



2−p



p−1

1−p

B′ +

q

(−2γ )yfy = 0.

(3.14)

(3.15)

2γ (rp − µ)

1−p B(T ) = 0.

B + 2k2 γ 2 B2 +

(3.16)

(µ − r )2 p = 0, 2k2 (1 − p)2

(3.17)

q

(µ − r )2



1

1−p

2

q

rb − rc = 0

µ0 

1 − e−r (T −t ) = −µ0 aT −t | .

r



Finally we solve Eq. (3.11). To do this, put

2k2 s2γ

sbs = 0,

+ k2 s2γ +2 css +

Eqs. (3.16) and (3.17) are just same as (4.13) and (4.12) in Gao (2009), respectively. So we can confirm that the solutions are given by (3.5) and (3.6) from Appendix A.2 in Gao (2009). Next we can easily check that the solution of the Eq. (3.12) is given by a( t ) = −

rb

q 1−p 1−p 1 − p 2 2γ +2 − bt + ct − rsbs + rscs − k s bss q q 2q

c (t , s) = C (t )y + D(t ) and (3.10)

(3.11)

y = s −2 γ

(3.18)

with the boundary condition given by C (T ) = 0 and D(T ) = 0. Then ct = C ′ y + D′ ,

cs = −2γ s−1 yC ,

css = (4γ 2 + 2γ )s−2 yC . Substituting these derivatives and b = AeBy into (3.11), we have

and a′ − ra − µ0 = 0.

(3.12)

First we solve (3.10). To do this, put b(t , s) = f (t , y) and

y = s −2 γ

with the boundary condition given by f (T , y) = 1. Then bt = f t ,

µ−r

q

(µ − r )

µ−r

y+

rp



We can split this equation into the following three equations

−

2k2

p−1

rp + γ (2γ + 1)k2 B + = 0, A 1−p A(T ) = 1.

+ a′ − ra − µ0 = 0.





+ γ (2γ + 1)k2 B + 1−p   2γ (rp − µ) (µ − r )2 p + y B′ + B + 2k2 γ 2 B2 + 2 = 0. 1−p 2k (1 − p)2

 

1 1−p k2 s2γ +2 bss k2 s2γ +2 css + rb − rc 2q 2 q

q

(µ − r )2

 ′ A





1−p

bt +

k2

q



 y−r

We can split this equation into two ordinary differential equations as follows:

2k2 s2γ

p−1

q

1−p

sbs

q

1−p

(µ − r )2

We can try to find a solution of (3.14) in the following form:

A′

bss z p−1 − η + ηcss ,

 1−p 1−p +η − bt + ct − rsbs + rscs −



with A(T ) = 1 and B(T ) = 0. Substituting this function in (3.14) q and multiplying (1−p)A e−By , we obtain

k2 s2γ q

2−p

f



(µ − r )

µ−r

rf −



1

2

q

q





bs z p−1 − η + ηcs , 1

q

f

k2 y−1 s2 4γ 2 y2 s−2 fyy + (4γ 2 + 2γ )ys−2 fy

f (t , y) = A(t )eB(t )y

Substituting these derivatives into (2.20), we have z p−1

q



rs(−2γ )ys−1 fy



bt z p−1 − η + ηct + a′ (t ),

1−p

2q 1−p

q

1

1 1 −1 gsz = − bs z p−1 , q



−

1−p

1−p



gz = − bz p−1 q

1

−

ft +

(3.13)

1−p 2 1 − p ′ By A e + D′ − k (4γ 2 + 2γ )ABeBy q 2q 1−p + (2γ 2 + γ )k2 C + rAeBy − rD q

−

 1 − p ′ By 1−p +y − AB e + C ′ + 2γ rABeBy q

bs = −2γ ys−1 fy

bss = 4γ 2 y2 s−2 fyy + (4γ 2 + 2γ )ys−2 fy .

−

1−p q

q

2k γ AB e 2

2

2 By



− (2γ + 1)rC = 0.

(3.19)

E.J. Jung, J.H. Kim / Insurance: Mathematics and Economics 51 (2012) 667–673

671

We can split this equation into two ordinary differential equations as follows:

Theorem 4.1. Consider the power utility function defined by

 1 − p ′ By 1−p   − AB e + C ′ + 2γ rABeBy   q q 1 − p 2 2 2 By − 2k γ AB e − (2γ + 1)rC = 0,    q  C (T ) = 0.  1−p 2 1 − p ′ By   A e + D′ − k (4γ 2 + 2γ )ABeBy −   q 2q 1−p + (2γ 2 + γ )k2 C + rAeBy − rD = 0,    q  D(T ) = 0.

Upower (v) = (3.20)

(3.21)

It is easy to see that the solutions for (3.20) and (3.21) are given by (3.8) and (3.9). The proof of Lemma 3.1 is complete.  The next result gives us an explicit expression for the optimal strategy β ∗ (t ). Theorem 3.2. The optimal investment strategy β ∗ is given by µ−r q

β ∗ (t ) =

+

ηb +

µ−r (v − 1 −p 2 v k s2γ

η c − a)

−2γ k2 B(v − ηc − a) + k2 s2γ +1 ηcs . v k2 s2γ

1−p q



1

(3.22)



b(t , s) z p−1 − η + ηc (t , s) + a(t ).

(3.23)

From this and (2.16), we get q

1

z p−1 = η +

b( 1 − p )

(v − ηc − a). −2γ

Since b = AeBy = AeBs so bs b

p

vp ,

(3.24)

, we have bs = −2γ s−2γ −1 ABeBs

−2γ

= −2γ s−2γ −1 B.

and

(3.25)

p < 1, p ̸= 0.

(4.1)

Then the optimal investment strategy corresponding to Upower (v) is given by

   µ 0 aT − t | µ−r ∗ , βpo ( t ) = K ( t ) 1 + wer v (1 − p)σt2   2γ (1−p)I (t ) where σt = kS γ (t ) and K (t ) = 1 − . µ−r

(4.2)

Proof. From Theorem 3.2, β ∗ (0, p, 1 − p; t ) is the optimal investment strategy corresponding to the utility function U (0, p, 1 − p; v). And by the definition of the HARA utility function, U (0, p, 1 − p; v) = Upower (v). Thus the optimal investment strategy corresponding to the utility function Upower (v) is given by β ∗ (0, p, 1 − p; t ). But, from (3.22) and Lemma 3.1, we have µ−r (v 1−p

− a) − 2γ k2 B(v − a) β (0, p, 1 − p; t ) = v k2 s2γ     µ 0 aT − t | µ 0 aT − t | µ−r 2γ I (t ) = 2 2γ 1+ − 2 2γ 1 + k s ( 1 − p) v k s v    µ 0 aT − t | µ−r 2γ I (t ) = 1+ − v (1 − p)σt2 σt2     µ 0 aT − t | µ−r 2γ (1 − p)I (t ) 1+ = 1− µ−r v (1 − p)σt2    µ 0 aT − t | µ−r = K (t ) 1 + v (1 − p)σt2 ∗ = βpo w er (t ). ∗

Proof. To prove this theorem, we will find gz and gs and substitute these derivatives in (2.22). In Lemma 3.1, we proved g (t , s, z ) =

1

This completes the proof.



Remark 1. Theorem 4.1 is the same result as Proposition 4.1 in Gao (2009), and which is a special case of Theorem 3.2. Now we will find limp→−∞ β ∗ (1, p, q; t ). To do this, we introduce the following result established by Gao (2009).

From (3.23) to (3.25), we take Proposition 4.2. Consider the exponential utility function defined by

1 1 −1 gz = − z p−1 b q

  1 1 η+ =− q

z

q b(1 − p)

1 Uexp (v) = − e−qv , q



(v − ηc − a) b

(3.26)

and gs = −

(4.3)

Then the optimal investment strategy corresponding to Uexp (v) is given by

µ − r −r (T −t ) ∗ βexp (t ) =  K (t ) e , qvσt2

 1−p  1 bs z p−1 − η bs + ηcs = (v − ηc − a) + ηcs q b

= −2γ s−2γ −1 B(v − ηc − a) + ηcs .

q > 0.

(3.27)

Introducing (3.26) and (3.27) in (2.22), we obtain (3.22) and the proof is complete. 

where γ

σt = kSt ,

 µ−r   K (t ) = 1 + 1 − e−2r γ (T −t ) . 2r

Inserting η = 1 in (3.22) and using (3.2)–(3.4), we have

4. The cases of power and exponential utility functions Since β ∗ (t ) given by (3.22) is the optimal investment strategy corresponding to the HARA utility function U (η, p, q; v) with parameters η, p and q, defined by (1.5), we denote it by β ∗ (η, p, q; t ). In this section we will find β ∗ (0, p, 1 − p; t ) and limp→−∞ β ∗ (1, p, q; t ), and prove that they are the optimal investment strategies corresponding to the power utility function (1.6) and the exponential one (1.7), respectively.

(4.4)

β (1, p, q; t ) ∗

µ−r

=

q

µ−r

b + 1−p (v − c − a) − 2γ k2 B(v − c − a) + k2 s2γ +1 cs v k2 s2γ

= {β1 (p) + β2 (p) + β3 (p) + β4 (p)} where

1

v k2 s2γ

,

(4.5)

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E.J. Jung, J.H. Kim / Insurance: Mathematics and Economics 51 (2012) 667–673

β1 (p) = β2 (p) =

µ−r q

A(t )eB(t )y ,

(4.6)

µ−r {v − C (t )y − D(t ) − a(t )} 1−p

(4.7) (4.8)

β4 (p) = −2γ k2 C (t ).

(4.9)

Remark 2. A(t ) and B(t ) are independent on s = S (t ), but β1 (p) depends on y = s−2γ and so β1 (p) is a random variable. And, since C (t ) and D(t ) contain s = S (t ), βi (p), i = 2, 3, 4, are also random variables for fixed t ∈ [0, T ].

µ−r

e−r (T −t )

q

µ−r

lim β1 (p) = lim

q

p→−∞

a.s.

(4.10)

A(t )eB(t )y =

µ−r q

e−r (T −t ) . 

lim β2 (p) = 0



T

1 q

t



2γ

eru eB(u)s

a.s.

(4.11)

D(t )

is 0, i.e., limp→−∞ 1−p = 0. Therefore it holds limp→−∞ β2 (p) = 0.  Lemma 4.5. It holds (4.15)

Proof. Since, by the same calculation in the proof Lemma 4.4, limp→−∞ {v − C (t )y − D(t ) − a(t )} is a constant in p and limp→−∞ B(t ) = 0, (4.15) is true.  Lemma 4.6. It holds lim β4 (p)

p→−∞

−(µ − r )2 

e(2γ +1)rt −(2γ +1)rT − ert −rT

2rq

(4.12)

lim C (t )

= =



T

−2γ

e(2γ +1)ru (1 − p)A(u)eB(u)s

t   × B′ (u) − 2γ rB(u) + 2γ 2 k2 B2 (u) du  −e(2γ +1)rt T (2γ +1)ru −r (T −u)

q

e

e

2k2

du

Thus we have

p→−∞

a.s.

(4.17)

µ−r q

e−r (T −t ) −

(µ − r )2 2rq

  1 × e(2γ +1)rt −(2γ +1)rT − ert −rT v k2 s2γ    µ − r −r (T −t ) µ−r  = 1+ 1 − e−2r γ (T −t ) e 2r qvσt2 µ − r −r (T −t ) e = K (t ) qvσt2 ∗ = βexp (t ). 



5. Conclusions

(µ − r )2 e(2γ +1)rt −rT −2γ rT {e − e−2γ rt }. 4γ rqk2

lim C (t ) =

(4.16)

e

t

t

=

This completes the proof.

  × lim (1 − p) B′ (u) − 2γ rB(u) + 2γ 2 k2 B2 (u) du p→−∞  −e(2γ +1)rt T (2γ +1)ru −r (T −u) (µ − r )2 q

a.s.

Theorem 4.7. It holds that



p→−∞

e



lim β ∗ (1, p, q; t )

C (t )

=

(4.14)

p→−∞

First we will find limp→−∞ 1−p . Using the facts that limp→−∞ λ+ = r , limp→−∞ λ− = 0, limp→−∞ A(t ) = e−r (T −t ) and γ limp→−∞ B(t ) = 0, we have

p→−∞

du.

Proof. From (4.4) and (4.5) and Lemmas 4.3–4.6, we obtain

µ−r lim β2 (p) = lim {v − C (t )y − D(t ) − a(t )} p→−∞ p→−∞ 1 − p   D(t ) C (t )y − . = (µ − r ) lim − p→−∞ 1−p 1−p

q

1−p

Since limp→−∞ A′ (u) = re−r (T −u) = limp→−∞ rA(u), limp→−∞ C (u) B(u) = 0 and limp→−∞ 1−p = 0, the right hand side of (4.14)

∗ lim β ∗ (1, p, q; t ) = βexp (t )

lim



Proof. Using the definition of β4 (p) and (4.13), the proof is clear. 

Proof. By the definition of β2 (p), it holds

−e(2γ +1)rt

C (u)

2(γ +1)ru

2



p→−∞

=

k2 (2γ 2 + γ )A(u)B(u)



− rA(u) + A (u) + (2γ + γ )e ′

=

Lemma 4.4. It holds p→−∞

= −ert

p→−∞

Proof. By the definition of λ± , we see that limp→−∞ λ+ = γr and limp→−∞ λ− = 0. Introducing these results in (3.5) and (3.6), we have limp→−∞ A(t ) = e−r (T −t ) and limp→−∞ B(t ) = 0, and hence

p→−∞

D(t )

lim β3 (p) = 0 a.s.

Lemma 4.3. It holds lim β1 (p) =

D(t ) 1−p

β3 (p) = −2γ k2 B(t ){v − C (t )y − D(t ) − a(t )}

p→−∞

C (t )

and limp→−∞ 1−p = 0. Now we will find limp→−∞ 1−p . By the expression (3.9) of D(t ),

(µ − r )2 e(2γ +1)rt −rT −2γ rT {e − e−2γ rt } 4γ rqk2

(4.13)

In this paper we were interested in the optimal investment problem, under the CEV market model, to maximize the expected HARA utility of the final surplus at the maturity time. We found an explicit expression of the optimal investment strategy corresponding to the HARA utility function with parameters η, p and q. And we have proved that the optimal investment strategy corresponding to the HARA utility function with parameters 1, p and q converges a.s. to the optimal one corresponding to the exponential utility function with parameter q as p → −∞. The modified HARA utility functions with parameters with η, p and q converge a.s. to the extended logarithmic utility function with η and q as p goes to 0. In future research related to this work, it would be interesting to find an explicit expression of the

E.J. Jung, J.H. Kim / Insurance: Mathematics and Economics 51 (2012) 667–673

optimal solution corresponding to the extended logarithmic utility function (Xiao et al. (2007) proved this problem in the case that η = 0 and q = 1) and to prove the a.s. convergence of the HARA solutions to the extended logarithmic one as p goes to 0. Acknowledgment This work was supported by a 2-Year Research Grant of Pusan National University. References Björk, T., 1998. Arbitrage Theory in Continuous Time. Oxford University Press, New York.

673

Devolder, P., Princep, M.B., Fabian, I.D., 2003. Stochastic optimal control of annuity contracts. Insurance: Mathematics & Economics 33, 227–238. Gao, J., 2009. Optimal investment strategy for annuity contracts under the constant elasticity of variance (CEV) model. Insurance: Mathematics & Economics 45, 9–18. Grasselli, M., 2003. A stability result for the HARA class with stochastic interest rates. Insurance: Mathematics & Economics 33, 611–627. Gu, M., Yang, Y., Li, S., Zhang, J., 2010. Constant elasticity of variance model for proportional reinsurance and investment strategies. Insurance: Mathematics & Economics 46, 580–587. Jonsson, M., Sircar, R., 2002. Optimal investment problems and volatility homogenization approximations. In: Modern Methods in Scientific Computing and Applications NATO Science Series II, Vol. 75. Springer, Germany, pp. 255–281. Øksendal, B., 1998. Stochastic Differential Equations. Springer Verlag, Berlin Heidelberg. Xiao, J., Hong, Z., Qin, C., 2007. The constant elasticity of variance (CEV) model and the Legendre transform-dual solution for annuity contracts. Insurance: Mathematics & Economics 40, 302–310.