Insurance: Mathematics and Economics 51 (2012) 303–309
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Insurance: Mathematics and Economics
journal homepage: www.elsevier.com/locate/ime
Optimal investment, consumption and life insurance under mean-reverting
returns: The complete market solutionI
Traian A. Pirvu a,⇤ , Huayue Zhang b
a
Department of Mathematics & Statistics, McMaster University, 1280 Main Street West, Hamilton, ON, L8S 4K1, Canada
b
Department of Finance, Nankai University, 94 Weijin Road, Tianjin, 300071, China
article
info
Article history:
Received January 2012
Received in revised form
May 2012
Accepted 15 May 2012
Keywords:
Portfolio allocation
Life insurance
Mean reverting drift
abstract
This paper considers the problem of optimal investment, consumption and life insurance acquisition for
a wage earner who has CRRA (constant relative risk aversion) preferences. The market model is complete,
continuous, the uncertainty is driven by Brownian motion and the stock price has mean reverting drift. The
problem is solved by dynamic programming approach and the HJB equation is shown to have closed form
solution. Numerical experiments explore the impact market price of risk has on the optimal strategies.
© 2012 Elsevier B.V. All rights reserved.
1. Introduction
The goal of this paper is to provide optimal investment,
consumption and life insurance acquisition strategies for a wage
earner who uses an expected utility criterion with CRRA type
preferences. Existing works on optimal investment, consumption
and life insurance acquisition use a financial model in which risky
assets are modeled as geometric Brownian motions. In order to
make the model more realistic, we allowed the risky assets to have
stochastic drift parameters. Let us review the literature on optimal
investment, consumption and life insurance.
The investment/consumption problem in a stochastic context
was considered by Merton (1969, 1971). His model consists in a
risk-free asset with constant rate of return and one or more stocks,
the prices of which are driven by geometric Brownian motions.
The horizon T is prescribed, the portfolio is self-financing, and the
investor seeks to maximize the expected utility of intertemporal
consumption plus the final wealth. Merton provided a closed form
solution when the utilities are of constant relative risk aversion
(CRRA) or constant absolute risk aversion (CARA) type. It turns out
that for (CRRA) utilities the optimal fraction of wealth invested in
the risky asset is constant through time. Moreover for the case of
(CARA) utilities, the optimal strategy is linear in wealth.
I Work supported by NSERC grant 5-36700, SSHRC grant 5-26758, MITACS grant
5-26761 and the Natural Science Foundation of China (10901086). We thank the
anonymous referee for valuable comments and suggestions.
⇤ Corresponding author.
E-mail addresses: [email protected] (T.A. Pirvu),
[email protected] (H. Zhang).
0167-6687/$ – see front matter © 2012 Elsevier B.V. All rights reserved.
doi:10.1016/j.insmatheco.2012.05.002
Richard (1975) added life insurance to the investor’s portfolio
by assuming an arbitrary but known distribution of death time.
In the same vein Pliska and Ye (2007) studied optimal life
insurance and consumption for an income earner whose lifetime is
random and unbounded. Ye (2008) extended Pliska and Ye (2007)
by considering investments into a financial market. Huang and
Milevsky (2007) solved a portfolio choice problem that includes
mortality–contingent claims and labor income under general
hyperbolic absolute risk aversion (HARA) preferences focusing
on shocks to human capital and financial capital. Horneff and
Maurer (2009) considered the problem in which an investor has
to decide among short and long positions in mortality–contingent
claims. Their analysis revealed when the wage earners demand
for life insurance switches to the demand for annuities. Duarte
et al. (2011) extended Ye (2008) to allow for multiple risky
assets modeled as geometric Brownian motions. More recently
Kwak et al. (2009) looked at the problem of finding optimal
investment, consumption and life insurance acquisition for a
family whose parents receive deterministic labor income until
some deterministic time horizon.
The novelty of our work is that we allow for stochastic drift.
More precisely the stock price is assumed to have mean reverting
drift. This stock price model was used by Kim and Omberg (1969)
who managed to obtain closed form solutions for the optimal
investment strategies. Later, Watcher (2002) added intertemporal
consumption to the model considered by Kim and Omberg.
Furthermore, optimal strategies are obtained in closed form and
an empirical analysis has shown that under some assumptions
the stock drift is mean reverting for realistic parameter values.
We solve within this framework the Hamilton–Jacobi–Bellman
(HJB) equation associated with maximizing the expected utility
T.A. Pirvu, H. Zhang / Insurance: Mathematics and Economics 51 (2012) 303–309
304
of consumption, terminal wealth and legacy and this in turn
provides the optimal investment, consumption and life insurance
acquisition. By looking at the explicit solutions we found out that:
(1) when the wage earner accumulates enough wealth (to leave
to his/her heirs), he/she considers buying pension annuities rather
than life insurance (2) when the value of human capital is small the
wage earner buys pension annuities to supplement his/her income
(3) when the wage earner becomes older (so the hazard rate gets
higher) the wage earner has a higher demand (in absolute terms)
for life insurance/pension annuities.
In our special model, due to the same risk preference for
investment, consumption and legacy, the optimal legacy (wealth
plus insurance benefit) is related to the optimal consumption.
More precisely the ratio between the optimal legacy and optimal
consumption is a deterministic function which depends on the
hazard rate, risk aversion, premium insurance ratio, the weight
on the bequest, and is independent of the financial market
characteristics. In a frictionless market (i.e. hazard rate equals
the premium insurance ratio) it turns out that the optimal legacy
equals the optimal consumption.
Our main motivation in writing this paper is to explore
the impact stochastic market price of risk has on the optimal
investment strategies. This could not be observed by previous
works which assumed a geometric Brownian motion model for the
stock prices. We found out that the optimal investment strategy
(in the stock) is significantly affected by the market price of risk
(MPR). Moreover, the optimal investment in the stock is increasing
in the MPR. As for the optimal insurance we found two patterns
depending on the risk aversion. A risk averse wage earner pays
less for life insurance if MPR is increasing up to a certain threshold;
when the MPR exceeds that threshold, the amount the wage earner
pays for life insurance starts to decrease. A risk seeking wage earner
has a different behaviour; thus he/she pays less for life insurance
when the MPR increases.
Organization of the paper: The remainder of this paper is
organized as follows. In Section 2 we describe the model and
formulate the objective. Section 3 performs the analysis. Numerical
results are discussed in Section 4. Section 5 concludes. The paper
ends with an appendix containing the proofs.
2. The model
In this paper, we assume that a wage earner has to make
decisions regarding consumption, investment and life insurance/pension annuity purchase. Let T > 0 be a finite benchmark
time horizon, and {W (t )}t 2[0,T ] a 1-dimension Brownian motion on
a probability space (⌦ , {Ft }t 2[0,T ] , F , P). The filtration {Ft } is the
completed filtration generated by {W (t )}t 2[0,T ] . Let E denote the
expectation with respect to P. The continuous time economy consists of a financial market and an insurance market.
2.1. The financial market
The financial market contains a risk-free asset earning interest
rate r
0 and one risky asset. By some notational changes we can
address the case of multiple risky assets. The asset prices evolve
according to the following equations:
dB(t ) = rB(t )dt ,
B(0) = 1,
dS (t ) = S (t ) [µ(t ) dt +
dW (t )] .
Moreover, the market price of risk {✓(t )}t 2[0,T ] = {
a mean reverting process, i.e.,
d✓(t ) =
k(✓(t )
¯ dt
✓)
µ(t ) r
}t 2[0,T ] is
✓ dW (t ).
Here , ✓ , k, ✓¯ are positive constants. This model for the stock
price was considered by Kim and Omberg (1969) and Watcher
(2002). In addition, we assume that the wage earner receives
income at the random rate i which follows a geometric Brownian
motion
di(t ) = i(t )(⌫i dt +
i dW
(t )),
with positive constants ⌫i and
i.
2.2. The life insurance
We assume that the wage earner is alive at t = 0 and
his/her lifetime is a non-negative random variable ⌧ defined on
the probability space (⌦ , F , P) and independent of the Brownian
motion {W (t )}t 2[0,T ] . Let us introduce the hazard function (t ) :
[0, T ] ! R+ , that is, the instantaneous death rate, defined by
(t ) = lim
"!0
P(t  ⌧ < t + " | ⌧
t)
"
.
From the above definition it follows that:
P(⌧ < s | ⌧ > t ) = 1
exp
and
P(⌧ > T |⌧ > t ) = exp
⇢ Z
⇢ Z
T
t
s
t
(u) du ,
(u) du .
(2.1)
(2.2)
Denote by f (s; t ) the conditional probability density for the death
at time s conditional upon the wage earner being alive at time
t  s. Thus
f (s; t ) = (s) exp
✓ Z
s
t
◆
(v)dv .
(2.3)
Here F (s; t ) denotes the conditional probability for the wage
earner to be alive at time s conditional upon being alive at time
t  s; consequently
F (s; t ) = exp
✓ Z
s
t
◆
(v)dv .
(2.4)
In our model the wage earner purchases term life insurance/pension annuity with the term being infinitesimally small.
Premium is paid (life insurance) or received (pension annuity) continuously at rate p(t ) given time t. In compensation, if the wage
earner dies at time t when the premium payment rate is p(t ), then
p(t )
either the insurance company pays an insurance amount ⌘(t ) if p(t )
p(t )
is positive, or the amount ⌘(t ) should be paid by wage earner’s
family if p(t ) is negative. Here ⌘ : [0, T ] ! R+ is a continuous,
deterministic, prespecified function called premium-insurance
ratio, and ⌘(1t ) is referred to as loading factor. In a frictionless market ⌘(t ) = (t ), but due to commission fees ⌘(t ) > (t ). In order
to simplify the analysis we assume that ⌘(t ) = (t ).
The insurance market model was pioneered by Yaari (1965)
who considered the problem of optimal financial planning
decisions for an individual with an uncertain lifetime. Later, that
model was extended by Richard (1975). Many works followed
Richard (1975) and Yaari (1965) by considering instantaneous term
life insurance; it means that the investor can only purchase life
insurance for the next instant; if surviving the next instant the
investor has to buy again the instantaneous term life insurance
and so on. The purchase of an annuity could be reversed by buying
a life insurance that guarantees the payment of annuity’s face
amount on the death of the holder. Indeed there are situations of
life annuities when premium is paid only at death (e.g. reverse
mortgage). This brings up the notion of instantaneous pension
annuity as the reverse of instantaneous term life insurance.
T.A. Pirvu, H. Zhang / Insurance: Mathematics and Economics 51 (2012) 303–309
One has to admit that instantaneous term life insurance
products cannot be purchased in the real world and one cannot find
the exact same payoff structure in term life insurance products.
These fictitious instantaneous term life insurance products are
used in portfolio management because they offer a simplified
and tractable model which can be used as a benchmark.
Another drawback of this modelling approach comes from the
irreversibility of pension annuities. In real world, term or lifelong
survival contingent products pay a cash flow for the length of the
term or as long as the buyer is alive. However, the introduction
of life long payments may necessitates complicated numerical
treatments and these models may not be tractable.
2.3. The objective
305
The wage earner’s objective is to find the strategy ↵ ⇤
{⇡ ⇤ , c ⇤ , p⇤ } which maximizes the risk criterion, i.e.,
↵ ⇤ = arg max J (t , x; ↵).
=
(2.9)
3. The analysis
Due to the market completeness we can compute D(t , ✓(t ),
i(t )), the present t value of the income flow at rate i(s) on [t , ⌧ ],
as expectation under the martingale measure Q of the discounted
future income payoff. Imagine that the wage earner buys life
insurance at rate i(s) on [t , ⌧ ], then at death time ⌧ it will pay off
i(⌧ )
. This observation leads to the following result.
⌘(⌧ )
The wage earner starts with wealth x 2 R+ and receives
income at a predictable rate i(t ) during the period [0, min{T , ⌧ }];
it means that the income will be terminated by the investor’s
death or benchmark time T , whichever happens first. At every
time t, the wage earner has wealth X (t ) and makes the following
decisions: he/she chooses ⇡ (t ), the fractions for the wealth process
invested in the risky asset, c (t ) the consumption rate, and p(t ), the
proportion of the wealth to be paid (received) for life insurance
(pension annuity). The wealth process {X ⇡ ,c ,p (t )}t 2[0,T ]
=
{X (t )}t 2[0,T ] satisfies on [0, min{T , ⌧ }] the self-financing equation
Lemma 3.1. The time t value, D(t , ✓(t ), i(t )), of the future [t , ⌧ ]
income is given by
+ i(t )] dt + X (t )⇡ (t ) dW (t ).
(2.5)
The initial wealth X (0) = x is a primitive of the model. The
Et [e
dX (t ) = [X (t )(r + (µ(t )
r )⇡ (t )
c (t )
p(t ))
wage earner total legacy if he/she dies at time t is the wealth plus
insurance amount
✓
I (t ) = X (t ) 1 +
p(t )
⌘(t )
◆
.
(2.6)
In order to evaluate the performance of an investment–
consumption–insurance strategy the wage earner uses an expected utility criterion. For a strategy ↵ = {⇡ , c , p} and its corresponding wealth process, let us define
J (t , x; ↵) = Et ,x
Z
T ^⌧
t
U (s, c (s)X (s)) ds + U (⌧ , I (⌧ ))1{⌧ T }
+ U (T , X (T ))1{⌧ >T } ,
(2.7)
where Et ,x is the conditional expectation operator, given X (t ) =
x, and U is the wage earner utility. It is further assumed that
time preferences are exponential and risk preferences are of CRRA
type, i.e., U (s, x) = e ⇢(s t ) x with
< 1, and ⇢ > 0. Here
1
stands for the coefficient of relative risk aversion. The
positive constant
is the weight on the wage earner legacy’s
utility. A high
reflects the fact that the utility of the legacy
is more important. A more realistic model would perhaps allow
for different consumption and legacy utilities, but in such a case
one may loose tractability. The following lemma shows that the
above expected utility risk criterion with random time horizon is
equivalent to one with a deterministic planning horizon; for the
proof see Ekeland et al. (2012).
Lemma 2.1. The functional J of (2.7) equals
J (t , x; ↵) , Et ,x
Z
T
t
Z
= i(t )
Z
1
t
1
t
and
Q
Here
Rs
i t
✓ (u)du
Rs
i t
=k
✓,
k✓¯
i ((✓ (t )
]=e
r )(s t )
e(⌫i
⇥ EQt [e
Rs
e r (s t ) i(s)e
and ⇣ (x) =
1
✓ (u)du
t
Rs
e
t
(u)du
(u)du
]ds,
(3.1)
¯
)⇣ (s t )+ k✓ (s t ))
x
e
(1
ds
e
2 2
i ✓
2
Rs
t
⇣ 2 (s v)dv
.
).
Appendix A.1 gives the proof. ⇤
Given t, the current time, in light of this lemma we can assume
that the wage earner does not receive income but his/her wealth
is X (t ) + D(t , ✓(t ), i(t )). The amount D is refer to as the value of
human capital. The HJB equation associated with maximizing (2.8)
is the following second-order PDE:
@v
(s, x, ✓, i) (⇢ + (s))v(s, x, ✓, i)
@s
+ sup H (s, x, ✓, i; ↵) = 0
(3.2)
↵
with the boundary condition
v(T , x, ✓, i) =
x
(3.3)
.
Here, the Hamiltonian function H is given by
H (s, x, ✓, i; ↵) = [(r +
c
✓⇡
p)x + i]
@v
@x
@ 2v
@v
1 2 @ 2v
@v
¯
+ ( k(✓ ✓))
+
+ ⌫i i
✓
2
2
@x
@✓
2
@✓ 2
@i
1 2 2 @ 2v
@ 2v
@ 2v
+ ii 2
+ i ⇡ xi
✓⇡x
2
@i
@ x@✓
@ x@ i
⇣
⌘
px
2
x
+
(s)
@ v
(cx)
+
+ (s)
.
(3.4)
i ✓i
@ i@✓
+
1
2
⇡ 2 x2
The first order conditions (FOC) are
[F (s; t )U (s, c (s)X (s)) ds
⇡⇤ =
+ f (s; t )U (s, I (s))] ds
+ F (T ; t )U (T , X (T )) ,
Q
D(t , ✓(t ), i(t )) = Et
(2.8)
where F (s; t ) and f (s; t ) are given by (2.3) and (2.4) respectively.
⇤
c =
@2v
✓ @✓@ x +
2
x @@ x2v
✓ @v
@x
1
@v
@x
x
1
,
@2v
i i @ i@ x
2⇣
6
p = 6
4
⇤
,
1 @v
@x
⌘
x
1
1
3
7
17
5.
(3.5)
T.A. Pirvu, H. Zhang / Insurance: Mathematics and Economics 51 (2012) 303–309
306
We introduce the following PDE which is intimately related
to (3.2):
✓
@h
+
r+
@s
1
✓
¯
+
k(✓
✓)
⇢ + (s)
(s)
✓
1
✓
+ K ( s) = 0
◆
+
✓2
2(1
)
2 2
@h
@ h
+ ✓
@✓
2 @✓ 2
◆
h
(3.6)
with terminal condition
h(T , ✓ ) = 1.
Here K (s) = 1 + (s). We are able to solve this PDE in closed form
by using the Feynman–Kac theorem.
Lemma 3.2. The solution of (3.6) is given by

h(t , ✓ ) = e
RT
t q(u)du
2
e✓ a(t )+b(t ) +
Appendix A.2 gives the proof.
Z
T
t
K (s)e
Rs
t q(u)du
2
e✓ a(s)+b(s) ds
4. Numerical results
⇤
Theorem 3.3. The function
v(s, x, ✓ , i) = h1
(s, ✓)
(x + D(s, ✓, i))
,
(3.7)
solves the HJB equation (3.2)–(3.3). The optimal admissible
investment–consumption–insurance strategy ↵ ⇤ (s) = {⇡ ⇤ (s), c ⇤ (s),
p⇤ (s)}s2[t ,T ] is given by
(X ⇤ (s) + D(s, ✓(s), i(s)))✓(s)
(1
) X ⇤ ( s)
@h
⇤
✓ (X (s) + D(s, ✓(s), i(s))) @✓ (s, ✓(s))
⇤
(1
)X (s)h(s, ✓(s))
@D
@D
✓ @✓ (s, ✓(s), i(s))
i i(s) @ i (s, ✓(s), i(s))
+
,
X ⇤ (s)
X ⇤ (s)
X ⇤ (s) + D(s, ✓(s), i(s))
c ⇤ (s) = [h(s, ✓ (s))] 1
,
X ⇤ (s)

1
p⇤ (s) = (s)
[h(s, ✓(s))] 1
⇡ ⇤ (s) =
⇥
X ⇤ (s) + D(s, ✓(s), i(s))
X ⇤ (s)
Appendix A.3 gives the proof.
1 .
(3.8)
(3.9)
(3.10)
⇤
3.1. Optimal consumption versus legacy
Theorem 3.3 yields the ratio between the optimal legacy I ⇤ (s)
and optimal consumption C ⇤ (s) = c ⇤ (s)X ⇤ (s).
Corollary 3.4. For every
⇤
I (s)
C ⇤ (s)
=
1
somehow expected. Furthermore, a more risk averse wage earner
consumes less in favour of the legacy. If the weight = 1, then the
optimal legacy equals the optimal consumption. Let us now examine the relationship between wealth, consumption and life insurance/pension annuity. If the wage earner buys life insurance then
he/she consumes more; this may be explained by the legacy being supplemented by the insurance payment in case of death (so
he/she may consume more and save less for his/her heirs). On the
other hand if a pension annuity is acquired he/she consumes less
so that there will be enough money at the death time for legacy
and paying off the pension annuity. In real world for the case of
pension annuities, those who survive share in the mortality credit,
money that are added to investment returns from the pension annuity. Let us point out that the finding of this corollary is contingent on the special type of risk preferences considered (CRRA type
with the same coefficient of risk aversion for consumption and
legacy).
< 1, we have
.
This result says that the legacy/consumption ratio is independent of the financial market characteristics. It depends on
the investor’s primitives: the coefficient of risk aversion 1
,
and the bequest weight . If a loading factor is assumed it will
also depend on it as well as on the hazard rate. The higher the
weight
the higher the legacy/consumption ratio. This fact is
In this section, a 35 years old wage earner (the initial time
t = 35), has an initial wealth of 50,000 dollars and a benchmark
time T = 65, invests (in the financial market) consumes and buys
life insurance (pension annuities) as to maximize his/her expected
utility. The financial market parameters are r = 0.01, i(t ) =
4000, ⌫i = 0.02, i = 0.04,
= 0.0436, ✓ = 0.0189, k =
0.0226 and ✓¯ = 0.0788. The weight
= 1, ⇢ = 0.03, and
(t ) = 0.001 + 101.54 exp x 1087.54.24 , (Gompertz hazard function).
We are mainly interested in the impact MPR has on the optimal
insurance strategy.
T.A. Pirvu, H. Zhang / Insurance: Mathematics and Economics 51 (2012) 303–309
307
is a Brownian motion under probability measure Q. Under Q,
D(t , ✓(t ), i(t )) can be computed as follows:

e r (⌧
Q
D(t , ✓(t ), i(t )) = Et
Thus
D(t , ✓(t ), i(t )) =
=
Z
Q
Et
Z
1
Rs
e r (s t ) i(s)e
e r (s t ) e
t
.
(⌧ )
t
1
i(⌧ )
t)
Rs
t
t
(u)du
ds
(u)du Q
Et [i(s)]ds.
(A.1)
The dynamics of stochastic income under Q measure is:
di(t ) = i(t )[(⌫i
i
so
2
i
2
i(s) = i(t )e(⌫i
✓(t ))dt + i dW̃ (t )],
)(s t )
e i (W̃ (s)
W̃ (t ))
Rs
i t
e
The dynamics of MPR ✓(t ) is
d✓(t ) =
✓ )✓(t )dt
(k
=k
Recall that
✓,
so
d(e ✓(t )) = e (k✓¯ dt
t
t
+ k✓¯ dt
✓ (u)du
.
✓ dW̃ (t ).
✓ dW̃ (t )).
Consequently
(u t )
✓u = e
k✓¯
✓t +
(u t )
e
(1
)
Z
✓
t
Integrating, we get:
A risk averse wage earner ( = 1) invests less in the stock
than a risk seeking wage earner ( = 0.1) and this investment is
increasing in the MPR ✓. A risk averse wage earner pays less for life
insurance as long as the MPR ✓ is small. However when MPR ✓ is
large enough the wage earner pays more for life insurance. On the
other hand, if the wage earner is risk seeking then he/she pays less
for life insurance when MPR ✓ gets large.
Z
s
t
✓u du =
✓
k✓¯
+
1
where ⇣ (x) =
(s
D(t , ✓(t ), i(t )) = i(t )
Q
⇣ (s
Z
x
Z
Rs
i t
✓ (u)du
]=e
1
t
(u v)
s
✓ ⇣ (s
t
v)dW̃ (v),
r )(s t )
Rs
i t
2 2
i ✓
2
✓ (u)du
k✓¯
Rs
t
Rs
e
t
]ds,
¯
)⇣ (s t )+ k✓ (s t ))
⇣ 2 (s v)dv
. ⇤
A.2. Proof of Lemma 3.2
For every s 2 [t , T ], we define the following process:
dy(s) =
✓
k
✓
1
◆
y(s)ds + ✓ dŴ (s),
¯
with the initial value y(t ) = ✓ , where Ŵ (s) = k✓ s + W (s). Let
Define the martingale measure Q, such that
W̃t = Wt +
Rs
G(s, y(s)) = e
A.1. Proof of Lemma 3.1
Z
t
0
✓(s)ds
(A.2)
(u)du
✓
Appendix
dW̃ (v),
t)
e(⌫i
i ((✓ (t )
⇥e
e
). Thus
⇥ EQt [e
where
Et [e
◆
t)
e
(1
5. Conclusion
The purpose of life insurance is to provide financial security
for the holders of such contracts and their families. Life insurance
acquisition can be considered in connection with consumption
and investment in the financial market. The problem of optimal
investment, consumption and life insurance acquisition for a wage
earner who has CRRA risk preferences is analysed in this paper. The
wage earner receives income at a stochastic rate, consumes, invests
in a stock and a risk free asset and buys life insurance. The MPR
of the stock is assumed to follow a mean reverting process. The
optimal strategies in this model are characterized by HJB equation
which is solved in closed form. Numerical results explore the effect
of the MPR on the optimal strategies.
k✓¯
✓(t )
u
where
0 r (u,y(u))du
r (s, ⇥ ) = q(s) +
with
q(s) =
1
✓
h(s, y(s)),
⇥ 2,
r+
(s)
⇢ + (s)
◆
,
T.A. Pirvu, H. Zhang / Insurance: Mathematics and Economics 51 (2012) 303–309
308
Case 3: if
and
=
2(1
)
a( t ) =
.
2
Applying Itô formula to G(s, y(s)) leads to
dG(s, y(s)) = e
Rs
⇥
✓
◆
@h
K (s)ds + ✓
(s, y(s))dŴ (s) .
@y
Finally,
Integrating from t to T and taking expectation conditioned on
y(t ) = ✓ , one gets
h(t , ✓ ) = e
RT
t q(u)du
Z
+
T
t
Ê[e
RT
y2 (u)du
t
Rs
K (s)e
t q(u)du
]
Ê[e
Rs
y2 (u)du
t
]ds.
where â = 2(k +
✓ ), b̂
1
y (s), so that
Y (s))ds + ĉ
P (t , y) = Ê[e
RT
Y (u)du
t
2
=
2
Let
2(k+ 1
✓
✓)
, ĉ = 2 ✓ . Let Y (s) =
p
Y (s)dŴ (s).
yP + â(b̂
Therefore
h(t , ✓) =
y)
@P
ĉ 2 y @ 2 P
+
= 0,
@y
2 @ y2
(A.4)
(A.5)
a2 (t ) + â a(t )
+
(cx)
⇤
⇡ ( s) =
2 (e (T
Case 2: if
a(t ) =
b(t ) =
t)
+ (â + )(e
"
1)
(T t )
2 e
= 0, let
Z
T
T
t
t
â+
2
T
t
ĉ 2
,
.
2
e✓ a(t )+b(t )
K (s)e
Rs
t q(u)du
2
e✓ a(s)+b(s) ds . ⇤
2
â
+
âb̂ a(u)du.
â
ĉ 2
(s, ✓)
(x + D(s, ✓, i))
,
s 2 [t , T ].
(s, ✓) x , solves
(A.6)
,
c
✓⇡
p)x
2 2
@V
@ V
+ ✓
@✓
2 @✓ 2
⇣
⌘
px
x + (s)
( s)
.
¯
✓))
+
1
@V
@x
⇡ 2 x2 @ 2 V
2
@ x2
2
@ V
✓⇡x
@ x@✓
@V
+
@x
2
(A.7)
1
x
@V
✓
@x
1)
6
p⇤ (s) = (s) 6
4
,
(T t )
+ (â + )(e
2
2
ĉ 2
Z
2⇣
= â2 2 ĉ 2 .
p
Case 1: if > 0, let =
, and
log
t q(u)du
H (s, x, ✓; ↵) = (r +
Define
ĉ 2
â
@V
(s, x, ✓) (⇢ + (s))V (s, x, ✓)
@s
+ sup H (s, x, ✓; ↵) = 0,
c ⇤ (s) =
a(T ) = b(T ) = 0.
b(t ) =
+
Based on the FOC, we obtain
,
with the boundary conditions
2âb̂
2 a(t )+b(t )
RT
e
+ ( k(✓
âb̂ a(t ),
2
◆◆
By Lemma 3.1 one can think of the wage earner as having an
initial wealth X (t ) + D(t , ✓(t ), i(t )) and no income instead of
having an initial wealth X (t ) and income stream at rate i. Following
Proposition 2 in (Munk and Sorensen, 2010), the solution of the HJB
(3.2) is
Plugging this guess into (A.4) we get the following Riccati system
for the functions a(t ), b(t ):
a(t ) =
] = e✓
+
P (t , y) = exp(a(t )y + b(t )).
b0 (t ) =
â
where
Let us guess that
2
2
✓
↵
P (T , Y (T )) = 1.
ĉ 2

+ tan
1
âb̂ a(u)du.
t
y2 (u)du
, and
t)
The function h is chosen such that V (s, x, ✓) = h1
the HJB
],
with the terminal condition
a0 (t ) =
T
v(s, x, ✓, i) = h1
where Y (t ) = y = ✓ 2 . The function P satisfies the following PDE
@P
+
@t
t
(T
A.3. Proof of Theorem 3.3
y2 (s))ds + ĉy(s)dŴ (s),
dY (s) = â(b̂
RT
Ê[e
(A.3)
Itô formula gives
dy2 (s) = â(b̂
Z
p
=
tan
ĉ 2
b( t ) =
0 r (u,y(u))du
< 0, let
✓
(T t )
1)
#
.
,
@2V
@✓@ x ✓
2
x @@ xV2
1 @V
@x
⌘
1
,
3
1
7
17
5.
x
Substituting (c ⇤ , ⇡ ⇤ , p⇤ ) into the HJB equation (A.6) leads to the
following PDE:
@V
@s
@V
(⇢ + (s))V + (rx + (s)x)
+ ( k(✓
@x
⇣
⌘2
@2V
2 2
✓ @@Vx
✓ @ x@✓
@
V
+ ✓
2
2 @✓ 2
2 @ V2
¯
✓))
@V
@✓
@x
+
1
(1 +
(s))
✓
@V
@x
◆
1
= 0,
(A.8)
T.A. Pirvu, H. Zhang / Insurance: Mathematics and Economics 51 (2012) 303–309
with the terminal condition
V ( T , x, ✓ ) =
x
.
Next, guess the solution of this equation to be of the form
V ( s , x , ✓ ) = h1
(s, ✓)
x
.
Let K (s) = 1 +
(s). Then h solves the following PDE
◆
@h
⇢ + (s)
✓2
+
r+
( s)
+
h
@s
1
2(1
)
✓
◆
@h
¯
+
k(✓
✓)
✓ ✓
1
@✓
✓
+
2
@ 2h
+ K ( s) = 0
2 @✓ 2
(A.9)
✓
with terminal condition
h(T , ✓ ) = 1.
Function h was found in Lemma 3.2.
⇤
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