http://dx.doi.org/10.4236/jmf.2013.33040 Published Online August 2013 (http://www.scirp.org/journal/jmf)
A Liability Tracking Approach to Long Term
Management of Pension Funds
Masashi Ieda1, Takashi Yamashita2, Yumiharu Nakano1 1
Graduate School of Innovation Management, Tokyo Institute of Technology, Tokyo, Japan
2
The Government Pension Investment Fund, Tokyo, Japan Email: ieda@craft.titech.ac.jp
Received May 28, 2013; revised July 2, 2013; accepted July 11, 2013
Copyright © 2013 Masashi Ieda et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
We propose a long term portfolio management method which takes into account a liability. Our approach is based on the LQG (Linear, Quadratic cost, Gaussian) control problem framework and then the optimal portfolio strategy hedges the liability by directly tracking a benchmark process which represents the liability. Two numerical results using em- pirical data published by Japanese organizations are served: simulations tracking an artificial liability and an estimated liability of Japanese organization. The latter one demonstrates that our optimal portfolio strategy can hedge his or her liability.
Keywords: Pension Fund Management; Long Term Portfolio Optimization; Quadratic Hedging; Stochastic Optimal Control; Hamilton-Jacobi-Bellman Equations; LQG Control
1. Introduction
In the management of pension funds, a long term portfo- lio strategy taking into account a liability is one of the most significant issues. The main reason is the demo- graphic changes in the developed countries: if the work- ing-age population is enough to provide for old age, the liability is a minor issue in the portfolio management of pension funds. Since the life expectancy has increased in recent decades, it becomes insufficient to provide for old age. Furthermore, the low birth rate continues and drives up this problem for decades. Thus pension funds face a challenging phase to construct long term portfolio strate- gies which hedge their liabilities.
A lot of pension funds except a few ones [1] determine their portfolio strategies by the traditional single time period mean variance approach which excludes an eva- luation of a liability. Its intuitive criterion attracts man- agers of pension funds. However the single time period approach is unsuitable for a long term portfolio manage- ment in the sense that it is unable to change the strategy excepting the initial time. The multi time period ap- proach which arrows the change of the strategy has a problem that the computational complexity grows expo- nentially. Hence if we employ this approach, we are usu- ally unable to obtain the optimal portfolio strategy in rea- listic time.
Therefore the aim of this paper is to propose a long term portfolio strategy which 1) involves an evaluation of a liability, 2) admits changes of the strategy at any time, and 3) is obtained in realistic time. To tackle this problem, we employ the LQG (Linear, Quadratic cost, Gaussian) control problem (see, e.g., Fleming and Rishel [2]). The LQG control problem is a class of stochastic control problem and is able to provide the control mini- mizing the mean square error of a benchmark process and a controlled process. Roughly speaking our tactic is that we compute the optimal portfolio strategy with the benchmark process which represents the liability. Then we can track the liability by using our optimal portfolio strategy. Although it is difficult to obtain the solution of stochastic control problem in general, the LQG control problem has the analytical solution which assures that we are able to obtain the solution in realistic time and thus it meets our purpose.
sion scheme including the de-cumulation phase. Our stu- dy is on the cutting edge in the sense that deals with tracking liabilities directly and constructs a suitable long term portfolio at the same time.
The organization of the present paper is as follows. We introduce continuous time models of assets and a benchmark in Section 2. To fit in the LQG control prob- lem, they are defined by the linear stochastic differential equations (SDEs). We mention that our portfolio strategy is represented by the amounts of assets. In Section 3, we define a criterion of the investment performance and provide the optimal portfolio strategy explicitly. Several numerical results are served in Section 4 Throughout the section the parameters related to the assets are deter- mined by an empirical data provided by the Government Pension Investment Fund in Japan. The simulation using an artificial data is discussed in Section 4.1 and this re-sult gives conditions that our optimal portfolio strategy works well. Section 4.2 provides a case study using an empirical estimation published by the Japanese Ministry of Health, Labour and Welfare. It demonstrates that our strategy is able to hedge the liability well.
2. Continuous Time Models of Assets and a
Benchmark
In this section, we present mathematical models of assets and a benchmark. The market which we are considering consists of only one risk-free asset and -risky assets and we have -benchmark component processes.
n m
Let t0 be a filtered probability space be a -dimensional Brownian motion where
and be a space of stochastic
processes which satisfy
, , t ,0 d
m 2
T
Zt t0
Wt td n
2
0 d < . T t Z t
We denote price process of the risk-free asset, those of the risky assets and the benchmark component processes
by , and respec-
tively, where the asterisk means transposition. To fit in the LQG control problem, we assume that , and
are governed by the following SDEs:
0 t
S St
S1t,,Stn
*
* 1
, , m
t t t
Y Y Y
0 t
S St
t Y
0 0 0 0 0 d d , , t t Sr t t S S s (1)
1 0 0 dd d , 1, 2, , ,
,
i d
i ij j
t S t i j t i i S
b t t t W i n
S S s
(2)
0 0 d d ,t t Y
m
Y t Y h t t t
Y y
: 0,r T , b: 0,
T n, S: 0,
T n d ,dWt,
(3)
where
: 0 ,T m m , h: 0,
T m and
: 0,T m d
Y
nd T <
are deterministic continuous func-
tions a r
r, bi and S
epresents the maturity. Coefficients
pected return rate o
stand for the risk-free rate and the ex-
ss of ed
f the i-th asset and the volatility. Let a cla portfolio strategy be the collection of n -valu t -adapted process
ut 0 t T whichsatisfies
2
0 d < , T t u t
n t
nvestor at t
be the amount of the risky asset held by an
i ime t, and Xt
ount
be the va e of our portfolio . Then e am of the ris ree asset held by th
lu k-f at time t
e invest th
or is represented by Xt i1t ni
. Hence,
Xt 0 t T is gov rned bye
0 0 1 1 0 * d d d , , i n ni t i t
0
0 0 0 0
,
t t t t T
X x
where . To emphasize the initial w
and th may write
t t
i
i t i t
S S X X S S s s
1 (4)
*1, ,1 n
1
e control variable, we
ealth 0,
x t t
X X .
The solution Xt of the SDE (4) is w:
s
given as follo
d d 0 t t
r u u
0 d *
0 0
e e d
t
sr u u
t
*
e d .
t s
t r u u
s S s
X x b s r s s
s W
1Moreover since r, b, and S are continuous func-
tions on
0,T and , Xt is in
2 T :
0 2 d 2 * d 0 0 2 d * 0 02 2 * * * *
0 0 1 0 2 0
e d
e d d
e d d
d d < , t t s t s
T T r u u
T t r u u
s
T t r u u
s S s
T T
s s s s
b s r s s t
s W t
TK x T K s TK s
2 0
0Xtdt 0 x t
1 11 11
where K0, K1 and K2 are constants.
3. Optimal Investment Strategy
ine he n erformance
We def t criterio of investment p J
by
T
* x,
2 2 , * 0 2 d , , x T T 0 10 t t
J a t Y X t
A Y X
are constants, Am
where 1 0 and 2 0 is a
constant vector, and : 0
Hence, m
a T
our invest is
contin ction. ment
find the l
a determin p
istic roblem is to uous fun
contro ˆ s.t. J ˆ J
, by. Sinc quadratic f
LQG co e the
performance cri unc-
ur inve
terion is re t probl e
presented
a t ,
tions, o stmen em becomes the ntrol
problem. We determin AT
t
and the parameters of Yt to be able to regard
*
a t Y and A YT* T as a li-
ability.
The optimal portfolio strategy is represented in the following form:
Theorem 1 We define the p tfolio strategy or ˆ as follows:
1 * 00
1 ˆ
2
t S t S t
F t
b t r t
00 0* 0
* 0
2 2
2
t t
S Y
F t X F t Y G t
t t F t
where
1
(6)
00, 0: 0,
F G T and F0: 0,
Tnary differential equations
m
are
solutions
(ODEs):
of following ordi
*
*
1
00 00
d
0,
S S
b t t t b t F
(7)
00 00
1
2
d
2
,
F r t F
t
F T
*
0 0
1
1
* * 0
0
2
d d
0, 2 ,
S S
0
F t a t r t F t t
t
b t t t b t F t
F T A
(8)
F t
*
0 0 0
1
* * 0
1
* * * 0
0
d
2 d
0,
0.
S S
S S S Y
G t r t G t h t F t t
b t t t b t G t
b t t t t t F t
G T
(9)
Here we have written b t
b t
r t 1.Then ˆ satisfies ˆ and J ˆ J
, . The proof of Theorem 1 is given in the appendix. We note that ˆt has feedback terms of Xt andmal strategy has to catch up the the benchmark process . Hence the pref-
er tu ying o tegy is the case th
4.
R
pirical section
cial liab ted by the estima-
nistry of Health, Labour and er one suggests the situation
t
Y.
This implies that our opti delays
* ta t Y
able si ation appl ur stra at
*
a t Yt does not fluctuate violently.
Numerical esults
We apply our method to an em data provided by the Japanese organizations. This is divided to two subsections according to the type of liabilities: an artifi-
ility and the liability construc tions published by the Mi
Welfare of Japan. The form
that our optimal strategy works well and the latter one demonstrates that our portfolio strategy is able to hedge the liability.
Before we move on the each subsection, we determine the common parameters in following subsections. The first task is to determine the parameters relating to the benchmark component processes. They consist of the income of a pension fund Ct and his or her expense
t
B and thus n2 and
,B
*. We set the pa- rat t t
Y C
meters constructing the benchmark process as follows:
*
*1 2 1, 1,1 , a t A 1,1 .
Hence, the benchmark process is BtCt which
represents a shortfall of the come and then we regard
t is shortfall as liabilit s the performance
of the strategy, we introduce a hedging error function of the i-th sample path i
and its average in
h the y. To discus
t
E Et as fol-
lows:
1
, ,
N
i i
t t t t t
i
E
E B C X E
N
where ti
i t
X is the i-th sample path of Xt and N
is the number of the sample paths. We set N1000
except as otherwise noted The
.
next task is to determine the risk-free rate and the expected return rates and volatilities of risky assets. We invest the following four assets: indices of the domestic
he ond he for
ording to the Gov- bond, t domestic stock, the foreign b and t - eign stock; we number them sequentially. Acc
the estimations of return rate and volatilities by
ernment Pension Investment Fund in Japan [7], we con- struct b t
and S
t as follows:
1
3%
b t ,
2
4.8%
b t , b3
t 3.5% and b4
t 5.0%;
,
1, ,
,0 otherwise,
ij ij
S
i j n
t
(10)
where the Cholesky decomposition of , a vari-
ance-co ance m f the assets:
29.7 4.39
vari atrix o
4
10 . 18.2 5.41 18.2 495 77.8 119
4.39 77.8 181 147 5.41 119 147 394
We choose a money market account as the risk-free asset and we set
4.1. Simulation with an Artificial Liability
In this subsection, we consider the following an artificial deterministic liability model:
(11)
i.e., we set
0.0%r t .
0
0.01 d , 80 trillion yen ,
t t
C
dC C t
0 100 trillion yen ,
t t
B
(12)
dB 0.01 d ,B t
t 0.01 ijij
wealth coincides wi
0 0 0
and . We assume
that our th the
tial time:
0h t
benchmark at the ini-
X B C . We over three deca
functions 00
const ptimal port-
folio strategy des, . Then we
determ
ruct the o
i.e., T: 30
ine the F , F0
d simulate and G0
by solving
the ODEs (7)-(9) numerically. an N paths of
S Yt, t
on
0,T accordinFigure 1 de
g to E (1)-(3) using
a standard Euler-Maruyama scheme with time-step
0.25
t
. scribes an investment result of a
sample path. The black and red lines ure 1 re- present BtCt and t
quations
in Fig
X respectively.
The most significant issue it indicates is that the per-
form rategy is quite poor near the maturity.
Figure 2 describing ance of the st
t
E implies that this poor perform- a not depend on the sample path. Figure 3 sug-gests a key factor of this phenomenon: values of func- tions 00
nce does
F , F0 and G0 change drastically between
25 and t30; this time period coincides with the term the hedging error becomes large rapidly. Figure 3
also implies that the existence of the stationary solutions of the ODEs (7)-(9). As described in Figure 2, the strat- egy relatively works well on the time period when the
functi 00
t
ons F , F0
an G0
reach the stationary state.
H the st will be improved by using the sta-
tionary solutions of the ODEs (7)-(9) on entire region. d
ence rategy
Figure 1. A sample path of BtCt and Xt
Figure 2. Averaged hedging error E . t
Figure 3. . The
lack, red
The time evolution of F , F and , green and blue lines represent
00 0 0
G 01
b 00
F , F , 02 F and respectively.
To obtain the stationary solutions of the ODEs (7)-(9), we replace to a value large enough. We denote it by
and set
0 G
T T
T 50. Figure 4 shows values of F00, F0
pa-and
ram
0
G
eter
obtained by solving the ODEs can find that the functions
(9) with
00
T. We F , 0
F
and G0 take the stationary solutions on
0,T .proved strategy are Results of simulations using the im
described as follows.
Figures 5 and 6 indicate that the performance near the maturity is improved and it does not depend on the sam
the onc th
the ol
-ple paths. This result leads us to c lusion at we should construct the strategy with stationary s utions of the functions F00, F0 and G0 if ey xi .
At e end of this s
th e sts
th b n, ntion about our
portfolio com tio
st ed by ec
th rke
bench-m
d the foreign stock, high risk
u sectio we me
position. Figure 7 displays the asset alloca- n on the sample path described in Figure 5. The mon-ey market account, the domestic bond and the for- eign
ock indicat light blue, black and blue lines resp -tively dominate our portfolio. The optimal strategy is that we keep e most part of the wealth as the money ma t account and compensate for the increment of the
. The black and red lines represent BtCt and Xt
ark by the investment for the domestic bond, low risk and low return asset, an
Figure 4. The time evolution of 00 F , 0
F and 0
G (improved case). The black, red, green and blue lines represent F00
, F ,01 02
[image:5.595.100.247.277.419.2]F and G respectively. 0
Figure 5. A sample path of BtCt and Xt (improved
version). The ackbl and red lin represent es BtCt and t
X respectively.
gure 6. Averaged
Fi hedging error E (improved version). t
nd high return asset. If
a Xt is deficient in BtCt, the
strategy increases the p tion of the do and the foreign stock.
4.2. Simulation with an Empirical Liability
According to the Japanese actuarial valuation published
in 2009 [8], the estimated income and expense of the welfare pension are showed in the Figure 8.
We regard these estimations as and and
si-mulate the three decades investme n optimal
strategy from 2040 when the shortfall of the p fund starts to expand drastically. The following r s sup- port that this situation is a valid dy: 1 a phase expanding
ropor mestic bond
t
C
nts usi
case stu
t
B
g our ension eason
)
t t
B C , the shortfall of pe fund, is
the most typical one expressing the dem ;
2) the beha
the nsion ographic changes viour of BtCt
optima on. Th
in this meets the con-
dition to apply l strategy: increas-
ing in the entire roughout t we
2 term
h 0 our
regi
t t
B C is is subsection set the start point as the year 40, i.e., t0 and
15
t represent the year 2040 and the year 2055 respec- tively.
To construct the optimal strategy, we first calibrate
t , Y
t and h t
to fit the estimations. Setting
t Y
t 0 and h t
as a numerical differentia- plish cades tion of the estimations is a simple method to accom the purpose. Since we are discussing the three de portfolio, we determine T 30. As suggested in SectionFigure 7. An amount of each asset on the sample path de- d in Figure 5. The black, red, green, blue and light blue lin
scribe
es represent the amount of a domestic bond, a do- m
mark c n tive
estic stock, a foreign bond, a foreign stock and money et a cou t respec ly.
[image:5.595.313.537.352.479.2] [image:5.595.95.250.475.622.2] [image:5.595.354.493.551.685.2]4.1, we set T50 to obtain the stationary F00 and
0
F . We are unable to expect the stationary G0 because
h t explicitly depends on t . We assume that our wealth coincide with the benchmark at the initial time:
0 0 0
X B C . Then we simu te la N paths of
S Yt, t
on
0,T
Euler-according to Equations
a sc e-ste
(1)-(3) using a p
stan-
0.25
dard Maruyam heme with tim t
which means that we can rearrange our portfolio every quarter. Results of the simulations are as follows.
We are able to argue that our strategy hedges the shortfall well since Figure 9 suggests that Et, the aver-
aged hedging error, is approximately 3% of BtCt, the
shortfall, in every quarter.
Figure 10 displays the asset allocation on the sample path described in Figure 11. In the same manner as in the case of the artificial liabilities discussed in Section 4.1, our optimal portfolio is dominated by the money
ncr
ing p
[image:6.595.350.495.86.222.2]market account, the domestic bond and the foreign stock. However the proportion of the domestic bond and the foreign stock is much higher. We can understand this phenomenon intuitively: since the shortfall i eases more rapid than that discussed in Section 4.1, the hedg- ortfolio is rearranged to become more profitable. The practical suggestion from this fact is that we have to take a risk to track the increasing liability and this is
Figure 9. Averaged hedging error E . t
Figure 10. An amount of each asset on the sample path de- scribed in Figure 11. The black, red, green, blue and light blue lines represent the amount of a domestic bond, a do- mestic stock, a foreign bond, a foreign stock and money market account respectively.
Figure 11. A sample path of BtCt and Xt. The black
and red lines represent BtCt and Xt res ectively. quite natural.
5. Summary
We have proposed a long term portfolio management method which takes into account a liability. The LQG control approach allows us to construct a more suitable long term portfolio strategy than myopic one obtained by the single time period mean variance approach in th
d hence it is intuitive. Two numerical simulations are served: the former one sug- gests the situation that our portfolio strategy works well; the latter one provides the result with the empirical data published by Japanese organizations. The result demon- strates that our portfolio strategy is able to hedge the li- ability, the shortfall of the income of the Japanese wel- fare pension, over three decades.
This study leaves ample scope for further research. Since our criterion is the mean square error, our portfolio strategy inhibits that our wealth exceeds a liability. This is the similar problem with the traditional mean variance
to avoi ha
liability. Then he
putability as mentioned in the introduc-
gements
p
e sense that we are able to change the strategy at any time. Our optimal portfolio strategy hedges the liability by directly tracking the benchmark process which represents the liability. The strategy is evaluated by the mean square error from the benchmark an
approach. One of approaches d it is t t we extend criterion which is able to hedge only the case that our
wealth goes under the we again face t
problem of com tory section.
6. Acknowled
This work is partially supported by a collaboration re- search project with the Government Pension Investment Fund in Japan in 2011-2012.
REFERENCES
[image:6.595.63.284.396.675.2]http://www.cppib.ca/Investments/Total_Portfolio_View/ [2] W. H. Fleming and R. W. Rishel, “Deterministic and Sto-
chastic Optimal Control,” Springer-Verlag, New York, 1975. doi:10.1007/978-1-4612-6380-7
[3] G. Deelstra, M. Grasselli and P.-F. Koehl, “Optimal In- vestment Strategies in the Presence of a Minimum Guar- antee,” Insurance: Mathematics and Economics, Vol. 33, No. 1, 2003, pp. 189-207.
doi:10.1016/S0167-6687(03)00153-7
[4] M. Di Giacinto, S. Federico and F. Gozzi, “Pension Funds with a Minimum Guarantee: A Stochastic Control Ap- proach,” Finance and Stochastics, Vol. 15, No. 2, 2010, pp. 297-342. doi:10.1007/s00780-010-0127-7
[5] F. Menoncin and O. Scaillet, “Optimal Asset Manage-
ment for Pension Funds,” Manageri
No. 4, 2006, pp. 347.
al Finance, Vol. 32,
doi:10.1108/03074350610652260
[6] R. Gerrard, B. Højgaard and E. Vigna, “Choosing the Optimal Annuitization Time Postretirement,” Quantita- tive Finance, Vol. 12, No. 7, 2012, pp. 1143-1159.
doi:10.1080/14697680903358248
[7] The Government Pension Investment Fund in Japan, 2013. http://www.gpif.go.jp/operation/committee/pdf/h191001_ appendix_05.pdf
[8] The Japanese Ministry of Health, Labour and Welfare, 2013.
Appendix
roof of Theorem 1
he value function corresponding to our problem (5) is efined by P T d
0 0 0 2 * , 1 2 , , * 2, inf d
, .
T x
t t s s
x x
T T t t
V x y a s Y X s
A Y X X x Y y
Hence the corresponding Hamilton-Jacobi-Bellman JB) equation is given by
0,
(13) (H
* 2
1
, ,
inf tV x yt V x yt a t y x
L
with terminal condition
m
2* 2
,
T
V x y A yx ,
x y, , where t is partial differential opera-tor with respect to t and L is the infinitesimal gen- erator of the process
X Yt, t
:
* 2 * , 2 t S t S t t x x y
(i)
* * * * 2 , , , 1 2 ,Tr , ,
t x
y
t S Y y x
Y Y y
x y r t x b t x y
t y h t x y
t t x y
t t x y
L
for . Here
*
2 m
C
j
x
and yj are the
l operators with respect
j-th
order partial differentia to x
and y. As S
t S t
*
is positive ni fim
defi te, the in- um of (13) is attained at
*
, , S S x tx t S Y y x t
t t
b t V x y t t V x y
and hence (13) can be written as
1 * 2 1 ˆ , t
V x y
,
* 1 2 * * 2 * 1 2 2 * * * * * 2 * , , 1 , 2 , , , 2 , , , 2 0t t x t
S S x t
x t x t
S S x t
x t
y x t Y S S Y y x t
V x y r t x V x y t y h t
b t t t b t V x y
V x y V x y
t
b t t t b t V x y
V x y
V x y t t t t V x
c x xa t y y a t a t y
. (ii)
Let us try a value function of the form
where 2 1 * * * * , , 1 x t
S S S Y y x t
V x y
b t t t t V x y
,
yV x yt
y
*
00 2 0 *
* 0
, 2
,
t
V x y F t x xF t y y F t y
G t x G t y g t
00, 0, : 0,
F G g T , F G0, : 0,
T mand
F is a time-dependant symmetric matrix. It is straight- forward to see that
00 00
1
* * 00
00 d 2 d 0, , S S
F t c r t F t
t
b t t t b t F t
F T C
(14)
* 0 0 1* * 0
0 d d 0, 2 , S S 0
F t ca t r t F t t F t
t
b t t t b t F t
F T CA
(15)
*0 0 0
1
* * 0
1 * * 0 d 2 d 0, 0, S S
G t r t G t h t F t
t
b t t t b t r t G t
t G T
* 0
S S S Y
b t t t t t F
1 (16)
* * * 0 0 d 2 d 1 0, F
* , 00t ca t a t t F t
t
F t F t
F t
F T CAA
(17)
* * 0 1* * 0
00 0 1 * * 00 * 0 d 2 d 0, 0, S S S S S Y
G t r t G t t G t h t F t
t
G t
b t t t b t G t
F t
F t
b t t t
F t
t t F t
* 2 0 1 * * 00 0 1 * * 00 * 0 d Tr d 4 2 0, 0, Y Y S S S S S Yg t h t G t F t
t
G t
b t t t b t
F t
G
b t t t
F t
t t F t
g T (19)
and then the associated candidate strategy is represented by
1 * 0000 0 0
* 0
1 ˆ
2
( ) 2 2
2 .
t S S
S Y
t t
F t
b t F t F t G t
t t F t
(iii)
Since (14) and (15) are linear ODEs, they have solutions on
unique
0,T . The existence of unique F0
tnce (16) suggests that (1 d (17) are linear ODEs. He
and (17) also nique solutions on
6) an
have u
0,Tutions si
. In the of (18) nce t
same ma existence of unique sol
and (19) are nteed. Therefore
nner, the
guara ˆt X
ow that
and in
the verification, t Y We
. now start 2 T i.e., we sh
0 0, 0
V x y J , and V0
x y0, 0
J ˆnce of stopping tim . To
que es
this end, we
l l s.t.introduce a se
u 2 u 2
0 0
inf 0 : d , d .
l u Xt t t l Yt t t l
Applying the Ito formula to
l
taking expectation, we have
d
,
T l T l T
V X Y and
0 0 0 0 * 0 * 0 , , , , , d
, d .
T l
t t t t t t t
T l
x t t t S t T l
x t t t Y t
V x y
V X Y L V X Y t
V X Y
V X Y t W
V X Y t W
T Tl Tl (iv)
As Xt and Y are in
2 T
and Y ,
0
F ,
F and G are continuous functions on
0,T , the last term vanishes: * 0 , T l
0 0 * 0 * 0 2 d d 0.
d 2*
d
y t t t Y t
T l
t Y t
t Y t
T l
Y t
V X Y t W
X F t W
Y F t t W
G t t W
T l t
(v)
By the definition of l, the continuity of the functions 00
F , F0
and G0
, and the fact that
so vanishes:
, the al
maining stochastic integral term
* 0 00 * 0 * 0 * 0 0 * 0 , d 2 d 2 d d 0. T lx t t t t S t
T l
t t S t
T l
t t S t
T l
t S t
V X Y t W
F t X t W
F t Y t W
G t t W
(vi) eSinc l when l goes to infinity, we get
0, 0
T
V x y
0
0 , ,
, .
t t t t t t t
T T T
V X Y L V X Y t
V X Y
d (vii)
By the HJB Equation (13) and its term we obtain
inal condition,
*
2
*
20 0, 0 0 1 d 2 T
V x y a t yx t A yx ,
which means that V0
x y0, 0
J
, . In theme manner we find that V0
x y0, 0
J ˆsa and then
