published in SIAM Journal of Control and Optimization34, no. 1, 1996, 329--364
ON AN INVESTMENT-CONSUMPTION MODEL WITH
TRANSACTION COSTS
MARIANNE AKIANy x , JOSE LUIS MENALDIz , AND AGNE S SULEMy{
Abstract. This paper considers the optimal consumption and investment policy for an investor
who has available one bank account paying a xed interest rate and n risky assets whose prices are
log-normal diusions. We suppose that transactions between the assets incur a cost proportional to
the size of the transaction. The problem is to maximize the total utility of consumption. Dynamic
Programming leads to a Variational Inequality for the value function. Existence and uniqueness of
a viscosity solution is proved. The Variational Inequality is solved by using a numerical algorithm
based on policies iterations and multigrid methods. Numerical results are displayed for n = 1 and
n = 2.
Key words. portfolio selection, transaction costs, viscosity solution, variational inequality,
multigrid methods.
AMS subject classications. 90A09, 93E20, 49L20, 49L25, 65N55, 35R45
1. Introduction. This paper concerns the theoretical and numerical study of a
portfolio selection problem. Consider an investor who has available one riskless bank
account paying a xed rate of interest r and n risky assets modeled by log-normal
diusions with expected rates of return i > r and rates of return variation i2 .
The investor consumes at rate c(t) from the bank account. Any movement of money
between the assets incurs a transaction cost proportional to the size of the transaction,
paid from the bank account. The investor is allowed to have a short position in one
of the holdings, but his position vector must remain in the closed solvency region S
dened as the set of positions for which the net wealth is nonnegative. The investor's
objective is to maximize over an innite horizon the expected discounted utility of
consumption with a HARA (Hyperbolic Absolute Risk Aversion) type utility function.
This problem was formulated by Magill and Constantinides [21] for n = 1 who
conjectured that the no-transaction region is a cone in the two-dimensional space of
position vectors. This fact was proved in a discrete-time setting by Constantinides
[8] who proposed an approximate solution based on making some assumptions on
the consumption process. Davis and Norman proved, in continuous time, without
this restriction, that the optimal strategy connes indeed the investor's portfolio to
a wedge-shaped region in the portfolio plane [10]. An analysis of the optimal strategy together with regularity results for the value function can be found in Fleming
and Soner [13, chapter 8.7] and Shreve and Soner [28]. Taksar, Klass and Assaf [31]
consider a model without consumption and study the problem of maximizing the
long-run average growth of wealth. A deterministic model is solved by Shreve, Soner
and Xu [29] with a general utility function which is not necessarily a HARA type
function. A stochastic model driven by a nite state Markov chain rather than a
Brownian motion and with a general but bounded utility function has been investigated in Zariphopoulou [32]. She supposes that the amount of money allocated in
the assets must remain non-negative and shows that the value function is the unique
y INRIA, Domaine de Voluceau Rocquencourt, BP105, 78153 Le Chesnay Cedex, France
z Wayne State University, Dept. of Mathematics, Detroit, Michigan, 48202, USA
([email protected]). Supported in part under NSF grant DMS-9101360.
x [email protected]
{ [email protected]
1
2
M. AKIAN, J.L. MENALDI AND A. SULEM
constrained viscosity solution of a system of variational inequalities with gradient
constraints. Fitzpatrick and Fleming [12] study numerical methods for the optimal
investment-consumption model with possible borrowing. They examine a Markov
chain discretization of the original continuous problem similar to Kushner's numerical schemes [18]. The convergence arguments rely on viscosity solution techniques.
We consider here Davis and Norman's model [10] in the case where more than
one risky asset is allowed. We restrict to power utility functions of the form c with
0 < < 1.
The purpose of the paper is to prove an existence and uniqueness result for the
dynamic programming equation associated to this problem and then solve this equation by using an ecient numerical method, the convergence of which is ensured by
the uniqueness result.
The mathematical formulation of the problem is given in x2. In x3, we prove
that the value function is the unique viscosity solution of a variational inequality.
Since the utility and the drift functions are not bounded, uniqueness does not derive from classical results. For the numerical study, an adequate change of variables
is performed in x4 which reduces the dimension of the problem. Then, in x5, the
variational inequality is discretized by nite dierence schemes and solved by using
an algorithm based on the \Howard algorithm" (policy iteration) and the multigrid
method. Numerical results are presented in x6 in the case of one bank account and
one or two risky asset(s). They provide the optimal strategy and indicate the shape
of the transaction and no-transaction regions. Finally, in x7, a theoretical study of
the optimal strategy is done by using properties of the variational inequality; this
analysis corroborates the numerical results.
2. Formulation of the problem. Let (
; F ; P) be a xed complete probability
space and (Ft)t0 a given ltration. We denote by s0 (t) (resp. si (t) for i = 1 : : :n)
the amount of money in the bank account (resp. in the ith risky asset) at time t and
refer as s(t) = (si (t))i=0;:::;n the investor position at time t. We suppose that the
evolution equations of the investor holdings are
(1)
8
n
>
< ds (t) = (rs (t) ? c(t))dt + X(?(1 + )dL (t) + (1 ? )dM (t));
i i
i
i
0
0
i=1
>
: ds (t) = s (t)dt + s (t)dW
i
i i
i i
i (t) + dLi(t) ? dMi(t); i = 1; : : :; n;
with initial values
(2)
si (0? ) = xi ; i = 0; : : :; n
where Wi (t), i = 1; : : :; n, are independent Wiener processes, Li (t) and Mi (t) represent cumulative purchase and sale of stock i on [0; t] respectively and s(t? ) denotes
the left hand limit of the process s at time t. The coecients i and i represent the
proportional transaction costs.
A policy for investment and consumption is a set (c(t); (Li(t); Mi (t))i=1;:::;n ) of
adapted processes such
Z tthat
1. c(t; !) 0;
c(s; !)ds < 1, for a.e. (t; !),
0
2. Li(t), Mi(t) are right-continuous, non-decreasing and Li (0? ) = Mi (0? ) = 0.
The process s(t) is thus right continuous with left-hand limit and equations (1) and
INVESTMENT-CONSUMPTION MODEL WITH TRANSACTION FEES
3
(2) are equivalent to :
8
>
>
>
<
>
>
>
:
s0 (t) = x0 +
si (t) = xi +
Zt
Z 0t
0
(rs0 () ? c())d +
i si ()d +
Zt
0
n
X
i=1
(?(1 + i )Li(t) + (1 ? i )Mi (t)) ;
isi ()dWi () + Li(t) ? Mi (t); i = 1; : : :; n
for t 0.
We dene the solvency region as :
S = fx = (x0 ; x1; : : :; xn) 2 Rn+1; W (x) 0g
where
(3)
W (x) = x0 +
n
X
i=1
min((1 ? i )xi; (1 + i )xi)
represents the net wealth, that is the amount of money obtained in the bank account
after performing the transactions which bring the holdings in the risky assets to zero.
Suppose that the investor is given an initial endowment x in S . A policy is
admissible if the bankruptcy time dened as
(4)
= inf ft 0; s(t) 62 Sg
is innite. We denote by U (x) the set of admissible policies. The investor's objective
is to maximize over all policies P in U (x) the discounted utility of consumption
(5)
Jx (P ) = Ex
Z1
0
e?t u(c(t))dt
where Ex denotes expectation given the initial endowment x, is a positive discount
factor and u(c) is a utility function dened by
(6)
u(c) = c ; 0 < < 1:
We dene the value function V as
(7)
V (x) = sup Jx (P ):
P2U (x)
We are facing here a singular control problem. We refer to Menaldi and Robin [23],
Chow, Menaldi and Robin [7] for various treatments of singular stochastic control
problems.
Remark 2.1. When the process s(t) reaches the boundary @ S at time t, i.e.
s(t? ) 2 @ S , the only admissible policy is to jump immediately to the origin and
remain there with a null consumption (see Shreve, Soner and Xu [29]). Consequently,
if the initial endowment x is on the boundary, then V (x) = 0.
Remark 2.2. Let denote the exit time of the interior of S , dened as
(8)
n
o
= inf t 0; s(t) 62S :
4
M. AKIAN, J.L. MENALDI AND A. SULEM
For all admissible policies P , we have
(9)
Jx (P ) = Ex
Z
0
e?t u(c(t))dt:
On the other hand, for any policy P , we can construct an admissible policy which
coincides with P until time (it is such that the process s(t) jumps to the origin at
time ). The value function can then be rewritten as
(10)
V (x) = sup Ex
where U is the set of all policies.
We make the assumptions
(A:1)
P2U
Z
0
e?t u(c(t))dt
n ?r !
X
> r + 2(1 ? ) ( i )2 ;
i
i=1
1
0 i < 1; i 0; i + i > 0; 8i = 1; : : :; n:
Remark 2.3. When the transaction costs are equal to zero (Merton's problem),
the value function V is nite i assumption (A.1) is satised (see Davis and Norman
[10] for n = 1, Karatzas, Lehoczky, Sethi and Shreve [17], and x7 below).
3. The Variational Inequality. We state the main theorem.
Theorem 3.1. Under assumptions (A.1) and (A.2),
(i) the value function V dened in (7) or (10) is -Holder continuous and concave in S and non decreasing with respect to xi for i = 0; : : :n.
(ii) V is the unique viscosity solution of the variational inequality (VI) :
@V ); max L V; max M V = 0 in max AV + u ( @x
(11)
S
i 1in i
0 1in
(A:2)
(12)
where
V =0
on @ S
(13)
n
n
X
@V + rx @V ? V;
@2V + X
ixi @x
AV = 12 i2 x2i @x
0 @x
2
i
i
0
i=1
i=1
(14)
@V + @V ;
Li V = ?(1 + i ) @x
0 @xi
@V ? @V ;
Mi V = (1 ? i ) @x
0 @xi
and u is the convex Legendre transform of u dened by
u () = max
(?c + u(c))
c0
(16)
= ( 1 ? 1) ?1 :
(15)
INVESTMENT-CONSUMPTION MODEL WITH TRANSACTION FEES
5
The solvency region S is divided as follows :
(17)
Bi = fx 2 S ; Li V (x) = 0g;
(18)
Si = fx 2 S ; Mi V (x) = 0g;
(19)
NTi = S n (Bi [ Si );
(20)
NT =
n
\
i=1
NTi :
NT is the no transaction region. Outside NT, an instantaneous transaction brings
the position to the boundary of NT : buy stock i in Bi , sell stock i in Si . After the
initial transaction, the agent position remains in
@V
NT = fx 2 S ; AV + u @x = 0g;
0
and further transactions occur only at the boundary (see [10]).
We shall rst recall the denition of viscosity solutions, then prove points (i) and
(ii) of Theorem 3.1 in x3.2 and x3.3 respectively.
3.1. Viscosity solutions of non linear elliptic equations. For simplicity,
we restrict ourselves to equations with Dirichlet boundary conditions. Consider fully
nonlinear elliptic equations of the form
(21)
F(D2 v; Dv; v; x) = 0 in O
(22)
v = 0 on @ O
where F is a given continuous function in S N RN R O, S N is the space of
symmetric N N matrices, O is an open domain of RN , and the ellipticity of (21) is
expressed by
(23) F(A; p; v; x) F(B; p; v; x) if A B; A; B 2 S N ; p 2 RN ; v 2 R; x 2 O:
A special case of (21) is given by
(24) F(X; p; v; x) = max
f
2U
N
X
i;j =1
aij (x; )Xij +
N
X
i=1
bi (x; )pi ? (x; )v + u(x; )g
where (23) is satised when the matrix (aij (x; ))i;j is symmetric non negative in
O U.
Bellman equations are clearly equations of this type, whereas variational inequalities like (11-12) can also be formulated in this form by using an additive discrete
control which selects the equation which satises the maximum.
Definition 3.2. Let v 2 C (O ). Then v is a viscosity solution of (21-22) if the
following relations hold, together with (22) :
(25)
F(X; p; v(x); x) 0
8(p; X) 2 J 2;+ v(x); 8x 2 O
(26)
F(X; p; v(x); x) 0
8(p; X) 2 J 2;?v(x); 8x 2 O
where J 2;+ and J 2;? are the second order \superjets" dened by :
J 2;+ v(x) = f(p; X) 2 RN S N ;
limy!sup
[v(y) ? v(x) ? (p; y ? x) ? 21 (X(y ? x); y ? x)]jy ? xj?2 0g
x
y 2O
6
M. AKIAN, J.L. MENALDI AND A. SULEM
and
J 2;?v(x) = f(p; X) 2 RN S N ;
lim
inf[v(y) ? v(x) ? (p; y ? x) ? 21 (X(y ? x); y ? x)]jy ? xj?2 0g:
y !x
y 2O
A viscosity subsolution (resp. supersolution) of (21) is similarly dened as an
upper semicontinuous function satisfying (25) (resp. a lower semicontinuous function
satisfying (26)) (see Crandall, Ishii and Lions [9]).
3.2. Properties of the value function.
Proposition 3.3. The value function V is concave in S .
Proof. The dynamic (1) is linear and the solvency region S is convex. Hence, for
any in [0; 1], x and x0 in S , P in U (x) and P 0 in U (x0 ), we have P +(1 ? )P 0 2 U (y)
for y = x + (1 ? )x0 and
V (y) Jy (P + (1 ? )P 0 ) = E
Z +1
0
e?t u(c(t) + (1 ? )c0 (t))dt:
Since u is concave we infer
V (y) Jx (P ) + (1 ? )Jx0 (P 0 ):
Taking now the supremum over all P and P 0 , we obtain that
V is concave.
As a consequence, V is locally Lipschitz continuous in S . The continuity of V at
the boundary is a consequence of the Proposition 3.5 below. First, let us state the
following lemma.
Lemma 3.4. Suppose (A.1) holds. Then, there exists a positive constant, a, such
that the functions
(27) ' (x) = a x0 +
n
X
i=1
(1 ? i)xi
!
with = (1 ; : : :; n); i = ?i or i
are classical supersolutions of equation (11). Consequently, the function
(28)
'(x) = a x0 +
n
X
i=1
min((1 ? i )xi ; (1 + i )xi)
!
is a viscosity supersolution of equation (11) such that ' = 0 on @ S .
Proof. Denote
(29)
Then
(30)
and we have
(31)
W (x) = x0 +
n
X
i=1
(1 ? i)xi :
' (x) = aW (x)
Li ' (x) = ?(i + i )a W (x) ?1 0;
INVESTMENT-CONSUMPTION MODEL WITH TRANSACTION FEES
7
Mi ' (x) = (i ? i)a W (x) ?1 0;
(32)
A' (x) = a W (x) ?2 G(x)
with
n
X
G(x) = 21 i2 x2i (1 ? i)2 ( ? 1)
i=1
+ W (x)
n
X
i=1
!
(i ? r)xi (1 ? i)
2
+ W (x) r ? :
From Assumption (A.1), there exists > 0 such that
n ? r 2
?r? 1 X
i
:
2(1 ? ) i=1 i
This implies
G(x) ?W (x)2
and
(33)
A' (x) ?aW (x) = ?' (x):
Moreover
(x)) = u (a W (x) ?1) = ( 1 ? 1)(a) ? 1 W (x) = (1 ? )(a) ?1 1 ' (x):
u( @'
@x0
The constant a can then be chosen such that
) 0 in :
(34)
A' + u ( @'
S
@x0
Since (31), (32), (34) hold and ' 0 on @ S , ' is a classical
supersolution of (11)
(continuous in S and twice continuously dierentiable in S ).
Now, ' can be rewritten as
' = min
':
Consequently, ' is a viscosity supersolution of (11) as the minimum of continuous
supersolutions and clearly vanishes on @ S .
Proposition 3.5. Suppose (A.1) holds. The value function V satises
(35)
0 V (x) '(x) 8x 2 S ;
where ' is the supersolution dened by (28). Consequently, V is continuous in S .
Proof. Consider x 2 S and P 2 U and denote by the rst exit time of S of
the process s(t) dened by (1) with s(0? ) = x. The function ' dened in (27)
8
M. AKIAN, J.L. MENALDI AND A. SULEM
has C 2-regularity and is a classical supersolution of (11). Denote by Ac the operator
A ? c @x@ 0 and by Lci(t) and Mci (t) the continuous parts of Li (t) and Mi(t). We apply
Ito's formula for cadlag processes (see Meyer [25]) to e?t ' (s(t)). For any stopping
time , the process
e?(^ ) ' (s( ^ ))
)
(
?
?
Z ^
0
X
e?t Ac(t)' (s(t))dt +
0t^
n
X
i=1
[Li ' (s(t))dLci (t) + Mi ' (s(t))dMci (t)]
e?t [' (s(t)) ? ' (s(t? ))]
is a martingale.
Since s(t) has a jump only when Li (t) or Mi(t) is discontinuous, we have
W (s(t)) = W (s(t? ))
?
n
X
[(i + i )(Li(t) ? Li (t? )) + (i ? i)(Mi (t) ? Mi (t? ))]
i=1
W (s(t? )):
Hence,
' (s(t)) ' (s(t? )):
In addition, Lci and Mci are non decreasing. Consequently
(36)
Mt = e?(t^ ) ' (s(t ^ )) +
Z t^
0
e? u(c())d
is a supermartingale, as well as the process min
Mt . Therefore
E
Z
0
e?t u(c(t))dt '(x):
Taking the supremum over all policies P 2 U , we get V (x) '(x). As V (x) 0, we
conclude that V (x) = 0on @S and that V is continuous on @S. Since V is locally
Lipschitz continuous in S , V is continuous in S .
The regularity of V can be precised as follows:
Proposition 3.6. Suppose (A.1) holds. Then, V is uniformly -Holder continuous in S , that is
(37)
9C > 0; jV (x) ? V (x0)j C kx ? x0 k
8x; x0 2 S :
Proof. Consider two initial positions x and x0, and denote by (resp. 0) the rst
exit time of S of the process s(t) (resp. s0 (t)) dened by (1) and s(0? ) = x (resp.
s0 (0? ) = x0). We have
V (x) ? V (x0 ) = sup E
P2U
Z
0
e?t u(c(t))dt ? sup E
P2U
Z 0
0
e?t u(c(t))dt
INVESTMENT-CONSUMPTION MODEL WITH TRANSACTION FEES
!
Z
Z 0
?
t
?
t
sup E
e u(c(t))dt ?
e u(c(t))dt
P2U
0
0
Z
sup E 0 e?t u(c(t))dt:
P2U ^
Using the supermartingale property of min
Mt dened in (36), we get
Z
E 0 e?t u(c(t))dt E(e?( ^ 0 ) '(s( ^ 0)) ? e? '(s()))
^
9
and since ' vanishes on @ S , we have
0
V (x) ? V (x0) sup E(e? ('(s( 0 )) ? '(s0 ( 0 )))1 0 < )
P2U
where 1A denotes the characteristic function of the set A.
Let us x for instance kxk = supi=0;:::;n jxij. The function ' is -Holder continuous, that is
j'(x) ? '(x0 )j C kx ? x0k
for some positive constant C. We thus get
(38)
V (x) ? V (x0 ) C sup E(e? 0 k(s( 0 ) ? s0 ( 0 )k 1 0 < ):
P2U
The process (t) = s(t) ? s0 (t) is a diusion process with generator A + I and
initial value (0) = x ? x0.
If the function (x) = kxk would satisfy A 0, then (( ^ t))e?( ^t)
would be a supermartingale whichPwould
readily lead to (37). As is not smooth,
we consider the function (x) = ni=0(x2i + )=2 with > 0. We have :
n
X
A = (x20 + )( 2 ?1)((r ? )x20 ? ) + (x2i + )( 2 ?2)fi
i=1
with
2
fi = x4i ( 21 i2 ( ? 1) + i ? ) + x2i ( 21 i2 + i ? 2
)? :
Assumption (A.1) implies
r ? < 0 and 21 i2 ( ? 1) + i ? < 0:
Consequently, there exists a positive constant C such that
A C =2 :
Applying now Ito's formula to , we obtain
(39)
E(e?( 0 ^ ) (( 0 ^ ))) (x ? x0) + C =2 :
10
M. AKIAN, J.L. MENALDI AND A. SULEM
Taking the limit of (39) when goes to zero and using
(x) 0(x) (n + 1) (x)
we get
0
E(e?( ^ ) (( 0 ^ ))) (n + 1) (x ? x0)
which leads together with (38) to the desired estimate (37).
Proposition 3.7. V is non decreasing with respect to xi, for i = 0; : : :; n.
Proof. Let us denote explicitly by s(t; x) the process s(t) dened in (1) with initial
value x and by x the exit time of S of s(t; x). As
V (x) = sup E
P2U
Z x
0
e?t u(c(t))dt
and u is positive, it is enough to prove the non decreasing property of the stopping
time x for any control process P .
Dene y(t; x) by
8
< y0 (t; x) = e?rt s0 (t; x)
: y (t; x) = e?(i ? 12 i2 )t?i Wi (t)
si (t; x) i = 1; : : :; n:
i
The process y(t; x) evolves according to
!
8
n
>
< dy (t; x) = e?rt ?c(t)dt + X(?(1 + i )dLi(t) + (1 ? i )dMi(t))
(40) > 0
i=1
:
dyi (t; x) = e?(i ? 21 i2 )t?i Wi (t)(dLi (t) ? dMi(t))
and satises y(0; x) = x. Hence, we can write
y(t; x) = x + Y (t; P )
where Y (t; P ) depends only on P . Consequently,
(41)
s(t; x) = (ert x0; (e(i ? 21 i2 )t+i Wi (t)xi )i=1;:::;n ) + S(t; P )
where S(t; P ) is a process which is independent of x.
Consider now x~ x (i.e. x~i xi, 8i = 0; : : :; n) and x P in U . We have from
(41),
s(t; x) s(t; x~)
and
W (s(t; x)) W (s(t; x~))
where W is dened in (3).
Since
x~ = inf ft 0; W (s(t; x~)) 0g;
for any t > x~ , there exists t0 such that x~ < t0 < t and W (s(t0 ; x~)) 0. This implies
W (s(t0 ; x)) 0 and t > t0 x . Consequently, x~ x and V (~x) V (x).
INVESTMENT-CONSUMPTION MODEL WITH TRANSACTION FEES
11
3.3. Existence and Uniqueness results. First, we show that the value function V is a viscosity solution of the variational inequality (11). The problem is reduced
to prove a weak dynamic programming principle (see Fleming and Soner [13]).
Lemma 3.8. There exists C > 0 such that
(42)
jJx (P ) ? Jx0 (P )j C kx ? x0 k 8x; x0 2 S ; 8P 2 U ;
where Jx (P ) is given in (9).
Proof. Estimate (42) is readily obtained from the proof of Proposition 3.6.
Proposition 3.9. The weak dynamic programming principle is satised for the
value function V , that is
(43) V (x) = sup E
P2U
Z ^
0
!
?
t
?
(
^
)
?
e u(c(t))dt + e
V (s(( ^ ) ))
8x 2 S ;
for any stopping time .
Proof. By means of the Markov property, we have for all P in U
E F ^
Z
0
e?t u(c(t))dt =
Z ^
0
e?t u(c(t))dt + e?(^ ) Js((^ )? ) (P 0 );
with P 0 equal to P \shifted" by ^ . Note that P 0 may not be admissible. The correct
method would be to proceed with admissible systems composed with a ltration
(
; Ft; P), a Wiener process W = (Wi )i=1;:::;n in Rn, and an admissible control
process P and consider V as the supremum of Jx (P ) over all admissible systems
instead of the supremum over all admissible policies. We give here a formal proof.
Rigorous proofs are given in Fleming and Soner [13], Nisio [26], El Karoui [11], Lions
[19]. Thus,
Jx (P ) = E
E
Z ^
0
Z ^
0
!
?
t
?
(
^
)
0
e u(c(t))dt + e
Js((^ )? ) (P )
!
?
t
?
(
^
)
?
e u(c(t))dt + e
V (s(( ^ ) )) :
By taking the supremum over all policies P , we deduce one inequality side of (43).
For the reverse inequality, we need to construct nearly optimal controls for each initial
state x in a measurable way. To that purpose, consider " > 0 and fS k g1
k=1 a sequence
of disjoint subsets of S such that
1
[
S k = S ; diameter(S k ) < ":
k=1
For any k, take xk in S k and P k = (ck ; (Lki ; Mki )i=1;:::;n ) in U such that
(44)
V (xk ) ? " Jxk (P k ):
Now, for a given stopping time and an arbitrary policy P in U , we dene
P = (c ; (Li ; Mi )i=1;:::;n) with
c (t) = c(t)1t< + ck (t ? )1t ;
Li (t) = Li (t)1t< + (Li (? ) + Lki (t ? ))1t ;
Mi (t) = Mi (t)1t< + (Mi (? ) + Mki (t ? ))1t ;
12
M. AKIAN, J.L. MENALDI AND A. SULEM
for s(? ) 2 S k . Using (42) and (44) we have
Js(? ) (P k ) = (Js(? ) (P k ) ? Jxk (P k )) + Jxk (P k )
?C" ? " + V (xk )
?2C" ? " + V (s(? )):
Denote by I the right-hand side of (43). There exists a policy P such that
I?"E
Z ^
0
e?t u(c(t))dt + e?(^ ) V (s(( ^ )? ))
!
and using the Markov property, we get
I ? " Jx (P ^ ) + (2C" + ")
and
I ? 2C" ? 2" Jx (P ^ ) V (x)
which leads to (43).
Corollary 3.10. The value function V (x) dened by (10) is a viscosity solution
of the variational inequality (11-12).
In the case of pure diusion processes, this is a standard result of the theory of
viscosity solutions (see Lions [20]). For singular stochastic control problems, we refer
to Fleming and Soner [13, chapter 8, Theorem 5.1].
Proposition 3.11. Under assumptions (A.1) and (A.2), the value function V
is the unique viscosity solution of the variational inequality (11-12) in the class of
continuous functions in S which satisfy
(45)
jV (x)j C(1 + kxk )
8x 2 S :
Proof. By Corollary 3.10 and equation (35), the value function V is a viscosity
solution of (11-12) and satises (45). Uniqueness is a consequence of the following
maximum principle :
Lemma 3.12. if v is a viscosity subsolution and v0 is a viscosity supersolution of
(11) which satisfy (45) and v v0 on @ S , then v v0 in S .
Indeed, a viscosity solution of (11-12) is both a subsolution and a supersolution
with the boundary condition v = 0 on @ S . We prove Lemma 3.12 by using the Ishii
technique, in particular we adapt the proofs of Theorems 3.3 and 5.1 of Crandall, Ishii
and Lions [9]. They are themselves based on the following corollary of Theorem 3.2
of [9] :
Lemma 3.13. Let V be an upper semicontinuous function and V 0 a lower semicontinuous function in an open domain O of RN . Consider W(x; y) = V (x) ? V 0 (y) ?
k jx ? yj2 with k > 0 and suppose that (^x; y^) is a local maximum of W . Then, there
2
exist two matrices X and Y in S N such that
(k(^x ? y^); X) 2 J 2;+ V (^x); (k(^x ? y^); Y ) 2 J 2;?V 0 (^y )
and
(46)
X 0
0 ?Y
3k ?II ?II :
INVESTMENT-CONSUMPTION MODEL WITH TRANSACTION FEES
13
In this statement, j:j denotes the euclidian norm, I the identity N N matrix and
J 2;+ is dened as follows :
J 2;+ v(x) = f(p; X) 2 RN S N ; 9(xn; pn; Xn) 2 O RN S N ;
(pn ; Xn) 2 J 2;+ v(xn ) and (xn ; v(xn); pn; Xn ) n!1
! (x; v(x); p; X)g:
J 2;? is similarly dened. If F is a continuous function in S N RN RO satisfying
the elliptic condition (23), and v is a viscosity subsolution of (21), we have
F(X; p; v(x); x) 0 8(p; X) 2 J 2;+ v(x); 8x 2 O:
Consider now v and v0 as in Lemma 3.12 and argue by contradiction in order to
prove v v0 in S . Suppose that there exists z in S such that v(z) ? v0 (z) > 0. For
k > 0, dene the function wk in S S as :
wk (x; y) = v(x) ? v0 (y) ? k2 jx ? yj2 ? "(W (x) 0 + W (y) 0 )
where
(47)
W (x) = x0 +
n
X
i=1
(1 ? i)xi ;
and ; "; 0 are parameters which will be chosen further. In addition, denote
mk = sup wk (x; y):
(x;y)2SS
In the following, C; C1; C2 denote generic constants.
Lemma 3.14. For = (i )i=1;:::n with ?i < i < i, there exist C1 and C2 > 0
such that
C1jxj W (x) C2jxj 8x 2 S :
(48)
Proof. The second inequality of (48) is straightforward. To obtain the rst inequality, we use the non negativity of W (dened in (3)) in S :
W (x) = W (x) ?
Moreover,
jx0j = jW (x) ?
Consequently,
n
X
min((i ? i)xi ; (i + i )xi) C
i=1
n
X
n
X
i=1
i=1
(1 ? i)xi j W (x) + C
jxj C W (x):
n
X
i=1
jxij 0:
jxij C W (x):
Fix 0 > such that the assumption (A.1) is still valid with 0 instead of , and
as in Lemma 3.14. This guarantees mk < +1 (see Lemma 3.15). On the other
hand, we have :
mk m = supfv(x) ? v0 (x) ? 2"W (x) 0 g v(z) ? v0 (z) ? 2"W (z) 0 :
x2S
14
M. AKIAN, J.L. MENALDI AND A. SULEM
As v(z) > v0 (z), there exists " > 0 such that mk m > 0 for any k; in the following,
we consider such ".
Lemma 3.15. Consider 0 > and as in Lemma 3.14. There exist xk ; yk in S
such that
mk = wk (xk ; yk ) < +1;
kjxk ? yk j2 k!1
! 0;
(49)
and
mk k!1
! m supfv(x) ? v0 (x) ? 2"W (x) 0 g:
(50)
x2S
Proof. Since v and v0 satisfy (45), we have
mk C + sup(C1jxj ? C2jxj 0 ) < +1:
x2S
Let (xn ; yn ) be a maximizing sequence :
which implies that
wk (xn ; yn ) mk ? n1 m ? n1
C2jxnj 0 ? C1jxnj C:
Hence, xn is bounded and similarly yn is bounded. Consequently, there exists a
converging subsequence of (xn ; yn) and the limit (xk ; yk ) 2 SS realizes the maximum
of wk . As
v(xk ) ? v0 (yk ) ? "(W (xk ) 0 + W (yk ) 0 ) = mk + k2 jxk ? yk j2 0
for any k,we conclude that xk , yk and kjxk ? yk j2 are bounded. Moreover, for any
subsequence of (xk ; yk ) converging to (^x; y^) when k goes to innity, we have x^ = y^
and using mk m, we get
lim sup k2 jxk ? yk j2 v(^x) ? v0 (^x) ? 2"W (^x) 0 ? m 0:
k!1
Consequently, (49) and (50) are satised.
Now, since m > 0 and v v0 on @ S , the limit x^ of xk and yk is in S ; then for any
converging subsequence of (xk ; yk ), we have (xk ; yk ) 2 S S for large k. Applying
0
0
Lemma 3.13 with V = v ? "W and V 0 = v0 + "W at the point (xk ; yk ) in S S ,
we obtain that there exist X; Y in S n+1 satisfying (46) such that
(pk ; Xk ) k(xk ? yk ) + " 0 W (xk ) 0 ?1p^; X + " 0 ( 0 ? 1)W (xk ) 0 ?2 A
2 J 2;+ v(xk )
(51)
INVESTMENT-CONSUMPTION MODEL WITH TRANSACTION FEES
15
(p0k ; Yk ) k(xk ? yk ) ? " 0 W (yk ) 0 ?1 p^; Y ? " 0 ( 0 ? 1)W (yk ) 0 ?2A
(52)
2 J 2;?v0 (yk )
with p^ = (1; 1 ? 1; : : :; 1 ? n ) and A = p^tp^:
Denote
F(X; p; v; x) = max F0(X; p; v; x) + u (p0); 1max
G (p); max H (p) ;
in i 1in i
n
n
X
X
F0 (X; p; v; x) = 21 i2x2i Xii + i xipi + rx0p0 ? v;
i=1
i=1
Gi(p) = ?(1 + i )p0 + pi ;
Hi(p) = (1 ? i )p0 ? pi ;
where X = (Xij )i;j =0;:::;n ; p = (pi)i=0;:::;n:
Note that although F is continuous, F takes its values in R[f+1g, since F = +1
when p0 0. This leads to a diculty to obtain a uniform continuity property similar
to [9, eq. (3.14)] and consequently straightforward application of the results of [9] can
not be used. Moreover, as the discount factor appears only in the F0 component of
F and not in Gi and Fi , property [9, eq. (3.13)], that is
F(X; p; v; x) ? F(X; p; v0; x) ?(v ? v0 ) for v v0 ; with > 0;
is not satised.
Using that v is a viscosity subsolution
and v0 is a viscosity supersolution of (11)
(that is of F(D2 v; Dv; v; x) = 0 in S ), and using (51) and (52), we get :
F(Xk ; pk; v(xk ); xk ) 0;
F(Yk ; p0k ; v0(yk ); yk ) 0:
This last inequality implies Gi(p0k ) 0 and Hi (p0k ) 0 and by linearity of Gi and Hi,
we obtain
Gi(pk ) = Gi(p0k ) ? " 0 (W (xk ) 0 ?1 + W (yk ) 0 ?1)(i + i ) < 0
and
Hi(pk ) = Hi(p0k ) + " 0 (W (xk ) 0 ?1 + W (yk ) 0 ?1 )(i ? i ) < 0:
This leads to
F0(Xk ; pk ; v(xk ); xk ) + u((pk )0 ) 0 F0 (Yk ; p0k ; v0 (yk ); yk ) + u ((p0k )0 ):
Using now that u is non increasing and (p0k )0 < (pk )0 , we obtain
F0(Xk ; pk ; v(xk ); xk ) ? F0(Yk ; p0k ; v0(yk ); yk ) 0:
16
M. AKIAN, J.L. MENALDI AND A. SULEM
Hence,
(53)
n
X
0 12 i2 ((xk )2i Xii ? (yk )2i Yii )
i=1
n
X
+k( i((xk )i ? (yk )i )2 + r((xk )0 ? (yk )0 )2 )
i=1
?(v(xk ) ? v0 (yk ) ? "(W (xk ) 0 + W (yk ) 0 ))
+"(f(xk ) + f(yk ))
where
n
X
f(x) = 21 i2 x2i 0 ( 0 ? 1)W (x) 0 ?2Aii
i=1
n
X
0
0
+ i xi 0 W (x) ?1p^i + rx0 0 W (x) ?1p^0
i=1
? W (x) 0
n
n
0
X
X
= 0 W (x) 0 [r ? 0 + (i ? r)yi + ( 2? 1) i2 yi2 ]
with
We have
i=1
i=1
i)xi :
yi = (1W? (x)
n
X
f(x) 0 W (x) 0 [r ? 0 + 2(1 ?1 0 ) ( i? r )2]
i=1
i
and since 0 is such that (A.1) is satised, f(x) 0 8x 2 Rn+1: Using (46), we see
that the rst term of the r.h.s. of (53) is bounded by Ckjxk ? yk j2 . Hence,
0 Ckjxk ? yk j2 ? mk k!1
! ?m < 0:
We thus get a contradiction and Lemma 3.12 and Proposition 3.11 are proven.
4. Change of variables.
4.1. Reduction of the state dimension. The value function V dened by (7)
has the homothetic property (see [10])
(54)
8 > 0; V (x) = V (x):
Consequently, the (n + 1)-dimensional VI (11-12) satised by V can be reduced to a
n-dimensional VI by using the following homogeneous model, that is by considering
the new state variables :
8
n
X
>
>
(net wealth)
< = x0 + (1 ? i )xi
i=1
(55)>
>
: yi = (1 ? i )xi ; i = 1; : : :; n (fraction of net wealth invested in stock i)
INVESTMENT-CONSUMPTION MODEL WITH TRANSACTION FEES
and the new control variable
(56)
C = c (fraction of net wealth dedicated to consumption).
The function V (x) can be written as
n
X
1 ; : : :; yn )
yi ); (1 y
V
(x)
=
V
((1
?
(57)
?
1 )
(1 ? n )
i=1
= W(y)
where the function
n
X
W(y) = V (1 ? yi ; (1 ?y1 ) ; : : :; (1 ?yn ) )
(58)
1
n
i=1
is dened in
n
X
S~ = fy = (y1 ; : : :; yn ) 2 Rn; 1 ? 1i ?+i fyi g? 0g
i
i=1
?
with fyg = max(0; ?y).
Using inequality (35) we deduce that the function W is bounded in S~ :
n
X
0 W(y) '(1 ? yi ; (1 ?y1 ) ; : : :; (1 ?yn ) ) a:
(59)
1
n
i=1
The function W is the unique viscosity solution of
~ + u (BW); max L~ i W; max M~ i W) = 0 in S~;
(60)
max(AW
1in
1in
(61)
where
(62)
(63)
(64)
(65)
and
(66)
(67)
(68)
W =0
n
n
2
~ = X ajk @ W + X bj @W ? W;
AW
j;k=1 @yj @yk j =1 @yj
n
X
BW = W ? yj @W
;
j =1 @yj
~Li W = @W ? i + i BW;
@yi
1 ? i
M~ i W = ? @W
@y
i
n
X
ajk = yj2yk i2 (ki ? yi )(ji ? yi );
i=1
n
X
bj = yj [( ? 1)i2 yi + i ? r](ij ? yi );
i=1
= ? (r +
n
X
i=1
[(i ? r)yi + ?2 1 i2yi2 ]):
on @ S~;
17
18
M. AKIAN, J.L. MENALDI AND A. SULEM
The symbol ij denotes the Kronecker index which is equal to 0 when i 6= j and equal
to 1 when i = j.
Using the properties of V and (60), we deduce that W is concave, non negative
and non decreasing with respect to each coordinate yi .
Remark 4.1. Equation (60-61) only depends on = (i)i=1:::n with i = (i +
i )=(1 ? i ), and so does the function W. Denote by V; the value function (7) in
order to express explicitly the dependency of V on the transaction costs and by W;
the solution of (60-61). We have :
W; (y) = W;0(y)
n
X
(69)
= V;0(1 ? yi ; y1; : : :; yn ):
i=1
Using (54), we get
(70)
V; (x) = V;0(x0 ; (1 ? 1)x1 ; : : :; (1 ? n)xn ):
Consequently, it is sucient to compute the value function V when the transaction
costs on sale are equal to zero.
This remark could have been observed directly from the model. Indeed, the
quantity si (t) represents the amount of money in the ith risky asset at time t, that
is the quantity of the ith asset multiplied by the reference price Pi (t). This reference
price is useless in practice unless the transaction costs are time-dependent. What
matters for the investor is the buying price (1 + i )Pi and the selling price (or net
price) (1 ? i )Pi . The relevant quantity to consider is the net value of the ith asset,
that is (1 ? i )si . Purchase of dLi units of the ith asset increases the net value of this
asset by dL0i = (1 ? i )dLi and requires a payment of (1 + i )dL0i , whereas sale of
dMi units reduces the net value by dM 0i = (1 ? i )dMi and realizes eectively dM 0i
in cash. Consequently, by using a formulation of the problem based on the net values
(1 ? i )si of the assets, the value function only depends on the coecients i where
i represents the proportional transaction cost on purchase with respect to the net
price of the ith asset.
4.2. Additional treatment for numerical purpose. Our purpose is now to
solve equation (60-61).
In order to simplify the numerical computation, we restrict the admissible region
S to
S + = fx 2 Rn+1; x1; : : :; xn 0; x0 +
n
X
i=1
(1 ? i )xi 0g
that is, we suppose that the amounts of money allocated in the risky assets are non
negative, while the amount of money in the bank account can be negative as long as
the net wealth remains non negative. This is not restrictive since when i > r the
no-transaction cone is inside S + and a trajectory which starts in S + remains in S +
(see [10] for n = 1).
This leads to the study of VI (60) in the domain (R+)n :
~ + u (BW); max L~ i W; max M~ i W) = 0 in (R+)n
max(AW
(71)
1in
1in; y >0
i
This VI degenerates at the boundary and is valid up to the boundary, but the
controls which make the trajectory go out of the domain are not admissible. Note
that the function W has bounded derivatives in (R+)n .
INVESTMENT-CONSUMPTION MODEL WITH TRANSACTION FEES
19
We proceed with a technical change of variables which brings (R+)n to [0; 1]n,
namely :
8
>
> (z) = (z)W(y);
>
>
>
>
>
n
>
Y
<
(z) = (1 ? zi );
>
i=1
>
>
>
>
>
y
>
>
: zi = i ; i = 1; : : :; n:
1 + yi
(72)
The function is bounded and concave with respect to zi , i = 1; : : :; n, has bounded
derivatives, and satises :
8
< max(A + sup (?CB + (z) C ); max Li ; max M i ) = 0 in [0; 1)n
i; zi >0
1in
(73):
C 0
n
= 0 on [0; 1] \ fzi = 1g 8i = 1; : : :; n
where
A
B
Li
Mi
with
n
n @
2
X
X
@
0
=
ajk @z @z + b0j @z ? 0 ;
j k j =1
j
j;k=1
n
n
X
X
@ ;
= ( ? zj ) ? zj (1 ? zj ) @z
j
j =1
j =1
@ )? B ;
= (1 ? zi )( + (1 ? zi ) @z
i
i
@ );
= ?(1 ? zi )( + (1 ? zi ) @z
i
a0jk = 21 zj (1 ? zj )zk (1 ? zk )ajk ;
n
X
b0j = zj (1 ? zj )(bj + zk ajk);
k=1
k6=j
n
n
X
X
0
ajk zj2zk ;
= ? zj bj ?
j;k=1
j =1
j 6=k
n
X
zi
z
i
2
i (ki ? 1 ? z )(ji ? 1 ? z );
i
i
(( ? 1)i2 1 ?zi z + i ? r)(ij ? 1 ?zi z );
bj =
i
i
i=1
ajk =
i=1
n
X
and dened in (68).
The numerical study is organized as follows : equation (73) is solved by using
the numerical methods explained in x5 below. Then a reverse change of variable is
performed in order to display the numerical results for equation (71) (see x6).
20
M. AKIAN, J.L. MENALDI AND A. SULEM
5. Numerical methods. We consider equations of the form :
8
< max (AP W + u(P)) = 0 in = [0; 1]m n ?
P 2 Pad
(74)
: W =0
on ?
where AP is a second order degenerate elliptic operator
AP W(x) =
m
X
i;j =1
m
2
X
bi(x; P) @W
(x)
+
aij (x; P) @x@ W
@x
@x (x) ? (x; P)W(x)
i
j
i
i=1
with
m
X
i;j =1
aij (x; P)ij 0;
(x; P) 0
8x 2 ; 2 Rm; P 2 Pad :
Pad is a closed subset of Rk (which may depend on x) and ? is a part of the
boundary @
, which consists of faces of the m-cube [0; 1]m. On @
n ?, the operator
AP is degenerate.
In x3, we have proven that the value function (7), within a change of variables, is
the unique viscosity solution of an equation of type (74). This solution can be approximate by the following numerical method : (i) discretize (74) by using a consistent
nite dierence approximation which satises the Discrete MaximumPrinciple (DMP)
(recalled below), (ii) solve the discrete equation by means of the value iteration (successive approximation) algorithm or the Howard algorithm (policy iteration). This
method does not require any stronger regularity condition on the viscosity solution
(see Barles and Souganidis [3], Fleming and Soner [13]). The algorithms mentioned in
(ii) may be replaced by the (Full) Multigrid-Howard algorithm (FMGH), introduced
in Akian [1, 2], based on the \Howard algorithm" and the multigrid method. This
algorithm is more ecient but proof of convergence has been obtained only when
the DMP is satised, the feedbacks are regular and the Bellman equation is strongly
elliptic.
For the numerical solution of (74), we use a classical nite-dierence discretization in a regular grid and the FMGH algorithm. Convergence arguments used in
[1, 2] cannot be applied here since the DMP is not satised (presence of mixed derivatives), the equation is degenerate and the control is singular. Nevertheless, numerical
experiments show that this numerical method converges.
This procedure and the computer implementation are treated by using the expert
system Pandore (see Chancelier, Gomez, Quadrat and Sulem [6], Akian [2]) that has
been developed to automate studies in stochastic control.
5.1. Discretization. Let h = 1=N (N 2 N) denote the nite dierence step
in each coordinate direction, ei the unit vector in the ith coordinate direction, and
x = (x1 ; : : :; xm ) a point of the uniform grid h = \ (hZ)m. Equation (74) is
discretized by replacing the rst and second order derivatives of W by the following
approximation :
(75)
@W W(x + hei ) ? W(x ? hei )
@xi
2h
INVESTMENT-CONSUMPTION MODEL WITH TRANSACTION FEES
or
(76)
(77)
(78)
21
8
W(x + hei ) ? W(x) when b (x; P) 0
>
>
>
i
h
@W <
>
@xi >
>
: W(x) ? W(x ? hei ) when bi (x; P) < 0:
h
@ 2 W (x) W(x + hei ) ? 2W(x) + W(x ? hei ) ;
@x2i
h2
@ 2 W (x) W(x + hei + hej ) ? W(x + hei ? hej )
@xi @xj
4h2
+ W(x ? hei ? hej )4h?2W(x ? hei + hej )
for i 6= j:
Approximation (75) may be used when A is uniformly elliptic, whereas (76) has to be
used when A is degenerate (see Kushner [18]). These dierences are computed in the
entire grid h , by extending W to the \boundary" of h in (hZ)m :
W(x) = 0
8x 2 ? \ (hZ)m
W(x ? hei ) = W(x) 8x 2 fxi = 0g \ h
W(x + hei ) = W(x) 8x 2 fxi = 1g \ h :
We obtain a system of Nh non linear equations of Nh unknowns fWh (x); x 2 h g :
(79)
max (AP W + u(P))(x) = 0 8x 2 h
P 2 Pad h h
where Nh = ]
h 1=hm . Let Ph denote the set of control functions P : h ! Pad
and Vh the set of functions from h into R. Equation (79) can be rewritten :
max (AP Wh + u(P)) = 0; Wh 2 Vh :
P 2 Ph h
Then, the operator APh , depending on P in Ph , maps Vh into itself (or is a Nh Nh
matrix).
Because of the degeneracy of the operator AP at some points of the closed mcube and the presence of mixed derivatives, APh does not satisfy the usual Discrete
Maximum Principle (i.e. (APh Wh (x) 0 8x 2 h ) ) (Wh (x) 0 8x 2 h )).
Consequently, equation (79) may not be stable, even for small step h. However APh
can be written as the sum of a symmetric negative denite operator and an operator
which satises the Discrete Maximum Principle; we thus infer the stability of APh
which is conrmed by numerical experiments.
We describe below the available algorithms to solve equation (79).
5.2. The Value Iteration method. Suppose that the Nh Nh matrix APh
satises
(80)
(APh )ij 0 8i 6= j;
Nh
X
(APh )ij = ? < 0 8i
j =1
22
M. AKIAN, J.L. MENALDI AND A. SULEM
which implies that APh satises the Discrete Maximum Principle. Equation (79) can
be rewritten as
Wh = 1 +1 k Pmax
(81)
(M P Wh + ku(P));
2Ph
where k > 0 and M P = I + k(APh + I) is a Markov matrix (I is the Nh Nh
identity matrix). Equation (79) can then be interpreted as the Dynamic Programming equation of a control problem of Markov chain with discount factor 1=(1 + k),
instantaneous cost ku(P) and transition matrix M P
max
(P )
n
X
k
n+1 u(Xn ; Pn):
n=0 (1 + k)
The value iteration method (see Bellman [5]) consists in the contraction iteration
W n+1 = 1 +1 k Pmax
(82)
(M P W n + ku(P)):
2Ph
The contracting factor is 1=(1 + k) = 1 ? O(h2 ) and the complexity1 of the method
is
h N ) = O(?h?(2+m) log h) = O(N 1+2=m log N ):
Ch = O( ? log
h
h
h
2
h
When the operator APh does not satisfy the Discrete Maximum Principle, equation
(79) cannot be interpreted as a discrete Bellman equation. Nevertheless, the iterative
method (82) can still be used if we nd and k such that the L2 norm of M P (which
is no more a Markov matrix) is lower than 1 for all P. This condition may be obtained
for instance when the discount factor (x; P) is large enough.
An example of the use of the value iteration algorithm is given in Sulem [30] for
solving the one dimensional investment-consumption problem.
5.3. The Multigrid-Howard algorithm. Another classical algorithm is the
Howard algorithm (see Howard [16], Bellman [4, 5]) also named policy iteration. It
consists in an iteration algorithm on the control and value functions (starting from
P 0 or W 0) :
(83)
for n 1;
P n 2 Argmax(APh W n?1 + u(P))
P 2Ph
(84)
for n 0;
is solution of APh n W + u(P n) = 0:
When APh satises the Discrete Maximum Principle, the sequence W n decreases and
converges to the solution of (79) and the convergence is in general superlinear [4, 5,
1, 2].
The exact computation of step (84) is expensive in dimension m 2 (the complexity of a direct method is O(Nh3?2=m )). We thus use the multigrid-Howard algorithm
introduced in [1, 2] : in (84), W n is computed by a multigrid method with initial
value W n?1. The advantage is that each multigrid iteration takes a computing time
of O(Nh ) and contracts the error by a factor independent of the discretization step
Wn
1 The number of elementary operations for computing an approximation of the solution of (79)
with an error in the order of the discretization error.
INVESTMENT-CONSUMPTION MODEL WITH TRANSACTION FEES
23
h. For a detailed description of the multigrid algorithm, see for example McCormick
[22], Hackbusch [14], Hackbusch and Trottenberg [15].
Let MP denote the operator of an iteration of the multigrid method associated to
the equation APh W +u(P) = 0. Starting from W 0, we proceed the following iteration :
(85)
8
(83)
>
>
8 n;0
<
< W = W n?1
for n 1 : for i = 1 to mn , W n;i = MP n (W n;i?1)
>
>
:
W n = W n;mn
This algorithm converges to the solution Wh of (79) if W 0 is suciently close to Wh
and mn is large enough (independently of the step h) [1, 2].
We introduce now the FMGH algorithm which solves equation (79) from any
initial value W 0 .
5.4. The Full-Multigrid-Howard algorithm. This algorithm [1, 2] fully uses
the idea of the Full-Multigrid method (FMG) (see for example Hackbusch and Trottenberg [15]).
Consider the sequence of grids (
k )k1 of steps hk = 2?k and denote by Ikk+1
the operator of the m-linear interpolation from Vhk into Vhk+1 .
If Wk 2 Vhk , Wk+1 = Ikk+1Wk is dened by :
8
Wk+1(x) = Wk (x)
8x 2 k k+1
>
>
>
<
x + y ) = Wk+1(x) + Wk+1 (y) 8x; y 2 such that x+y 2 W
(
>
k+1
k
+1
k+1
>
2
>
2
2
:
and x, y are in the same cell of k ;
where a cell of h is a m-cube of width h included in and with vertices in (hZ)m.
The Full-Multigrid-Howard (FMGH) algorithm is dened as :
For 1 k k,
Wkn is the n-th iteration of the sequence dened by (85) in
the grid k of initial value Wk0 .
For 1 k < k,
Wk0+1 = Ikk+1Wkn .
Under appropriate assumptions (strong ellipticity, Discrete Maximum Principle,
regularity of the feedback, see [1, 2]), the error between Wkn and the solution Wk of
(79) with h = hk is in the order of the discretization error, for any k. This property
is realized for any initial value W10, if the numbers mn and n are large enough (but
independent of the level k). Consequently, this algorithm solves equation (79) (with
an error in the order of the discretization error) with a computing time of O(Nh ).
6. Numerical results. Equation (71) is solved in (R+)n by using the FMGH
algorithm for n = 1 and n = 2 and various numerical values of the parameters.
Remark 6.1. The regions Bi and Si dened in (17) and (18) are characterized by
Bi = fx 2 S ;
Si = fx 2 S ;
L~ i W(y) = 0; y given by (55)g;
M~ i W(y) = 0; y given by (55)g
24
M. AKIAN, J.L. MENALDI AND A. SULEM
where the operators L~ i and M~ i are dened in (64) and (65). By extension we use the
notation :
Bi = fy 2 (R+)n ; L~ i W(y) = 0g;
Si = fy 2 (R+)n ; M~ i W(y) = 0g;
(86)
NTi = (R+)n n(Bi [ Si );
NT =
n
\
i=1
NTi :
6.1. One risky asset. Numerical tests are performed with = 0:3, = 10%,
r = 7%, 1 = 11%, 1 = 30%, = 1 = (1 +1 )=(1 ? 1 ) = 0:1; 0:3; 0:5; 1; 2;3 or 4%.
These values of are obtained for example when 1 = 1 ' =2 = 0:05; 0:15; 0:25;
0:5; 1; 1:5; 2%.
When > 0, the regions B1 and S1 are of the form (see x7) : B1 = [0; ?] and
S1 = [+ ; +1) with 0 < ? < + . When = 0 (no transaction costs), the optimal
policy is to keep a constant proportion of risky asset equal to 1 (given by (90) below),
that is + = ? = 1 . In our example, 1 = 0:635. The values of + and ? are
given in the table below and displayed in Figure 1 as functions of .
(%) 0.1 0.3 0.5 1
2
3
4
?
0.56 0.54 0.52 0.47 0.42 0.39 0.36
+
0.68 0.68 0.68 0.68 0.68 0.68 0.68
The graphs of + and ? are similar to those obtained by Davis and Norman [10]
who already observed that the \sell-barrier" is very insensitive to the transaction cost
while the \buy-barrier" decreases rapidly as increases. Indeed, even if the selling
cost is high, the risky asset must be sold before it can be realized for consumption. On
the other hand, it may not be worthwhile to invest in the risky asset if the transaction
costs are too high.
The value function W, solution of (71), and the optimal consumption C are
displayed in Figures 2 and 3.
From equations (71) and (86), we obtain W(y) = c(1 + y) in B1 , where c
is a constant depending on . In S1 , W is constant and seems insensitive to the
transaction costs. This means that when the initial proportion in the risky asset is
in S1 , the probability of a future purchase is small. On the other hand, if the initial
proportion invested in stock is in B1 , loss of prot (when increases) is due mainly
to the rst transaction.
The values of C are not relevant in B1 and S1 since the investor makes transactions
and thus does not consume. As expected, C decreases in [? ; + ] as well as the
fraction of wealth in cash.
6.2. Two risky assets. We set = 0:3, = 10%, r = 7% and x the parameters
of the rst risky asset to 1 = 11%, 1 = 30% and 1 = (1 + 1 )=(1 ? 1) = 1%.
Four tests are performed with
test 1 : 2 = 15% 2 = 35% 2 = 2%,
test 2 : 2 = 15% 2 = 35% 2 = 0:5%,
test 3 : 2 = 15% 2 = 35% 2 = 1%,
test 4 : 2 = 20% 2 = 50% 2 = 1%.
INVESTMENT-CONSUMPTION MODEL WITH TRANSACTION FEES
25
For test 1, the value function W, the optimal consumption C and their contour
lines are displayed in Figures 4, 5, 6 and 7.
The partition of the domain is displayed for each test in Figures 8, 9, 10 and 11.
As expected, 9 regions appear : buy (resp. sell) asset i when yi is below (resp. above)
a critical level i? (resp. i+ ) depending on yj (j 6= i), and no transaction between
i? and i+ .
After the rst transaction, the position of the investor evolves as a diusion
process with reection on the boundary of NT. The direction of the reection is
given by the equation Li W = 0 on the frontier with Bi and Mi W = 0 on the frontier
with Si .
Note that the no transaction interval for the rst asset NT1 \ fy2 = constantg '
[0:39; 0:78] is much larger than the no transaction interval [0:47; 0:68] obtained in
dimension 1, when only one asset (with same parameters) is available. This is not
surprising since the second asset has larger expected rate of return; it is thus more
interesting to make transactions on the second asset.
We observe that the boundaries of the regions Bi and Si seem at rst sight to
be straight lines (yi = constant). This would mean that the investment policies are
decoupled although the dynamics are correlated. In fact, when the cost for purchase
2 grows, the region NT2 grows as expected but the boundaries of S1 and B1 are
also perturbed. Moreover, a variation of 2 and 2 aect both NT2 and NT1 . A
theoretical study of the boundaries is done below in order to conrm these remarks.
0.700
0.665
0.630
0.595
0.560
0.525
0.490
0.455
0.420
0.385
ν in %
0.350
0.0
0.4
π−
π+
0.8
1.2
1.6
2.0
2.4
2.8
3.2
3.6
4.0
Fig. 1. Graph of + and ? for n = 1, = 0:3, = 10%, r = 7%, 1 = 11%, 1 = 30%.
26
M. AKIAN, J.L. MENALDI AND A. SULEM
W
15.889
15.880
15.870
15.860
15.851
15.841
15.832
15.822
15.813
15.803
y1
15.794
0.0
0.1
0.2
ν=0.1%
ν=0.3%
ν=0.5%
0.3
0.4
0.5
ν=1%
ν=2%
ν=3%
0.6
0.7
0.8
ν=4%
0.9
1.0
Fig. 2. Value function W for n = 1, = 0:3, = 10%, r = 7%, 1 = 11%, 1 = 30%.
C
0.109921
0.109672
0.109423
0.109173
0.108924
0.108674
0.108425
0.108176
0.107926
0.107677
y1
0.107427
0.0
0.1
0.2
ν=0.1%
ν=0.3%
ν=0.5%
0.3
0.4
0.5
ν=1%
ν=2%
ν=3%
0.6
0.7
0.8
ν=4%
0.9
1.0
Fig. 3. Optimal consumption C for n = 1, = 0:3, = 10%, r = 7%, 1 = 11%, 1 = 30%.
INVESTMENT-CONSUMPTION MODEL WITH TRANSACTION FEES
27
17.71
17.65
1.13
17.59
1.13
0.57
0.57
y1
y2
0.00
0.00
Fig. 4. Value function W for = 0:3, = 10%, r = 7%, = (11%; 15%), = (30%; 35%),
= (1%; 2%).
1.13
1.02
0.91
0.79
17.696
17.685
0.68
17.675
0.57
17.664
0.45
17.653
0.34
17.642
0.23
17.632
17.610
0.11
17.621
17.599
0.00
0.00
0.11
0.23
0.34
0.45
0.57
0.68
0.79
0.91
1.02
Fig. 5. Value function W for = 0:3, = 10%, r = 7%, = (11%; 15%),
= (30%; 35%), = (1%; 2%).
1.13
28
M. AKIAN, J.L. MENALDI AND A. SULEM
0.0946
0.0933
1.13
0.0920
1.13
0.57
0.57
y1
y2
0.00
0.00
Fig. 6. Optimal consumption C for = 0:3, = 10%, r = 7%, = (11%; 15%),
= (30%; 35%), = (1%; 2%).
1.13
0.092257
1.02
0.092488
0.092720
0.092952
0.093184
0.093416
0.093648
0.093880
0.91
0.79
0.092257
0.092488
0.094343
0.68
0.094111
0.57
0.45
0.34
0.093880
0.23
0.093648
0.093184
0.092952
0.11
0.093184
0.093416
0.092720
0.092952
0.00
0.00
0.11
0.23
0.34
0.45
0.57
0.68
0.79
0.91
1.02
1.13
Fig. 7. Optimal consumption C for = 0:3, = 10%, r = 7%, = (11%; 15%),
= (30%; 35%), = (1%; 2%).
INVESTMENT-CONSUMPTION MODEL WITH TRANSACTION FEES
1.56
1.40
1.25
B1 \ S2
NT1 \ S2
S1 \ S2
B1 \ NT2
NT
S1 \ NT2
B1 \ B2
NT1 \ B2
S1 \ B2
1.09
0.94
0.78
0.62
0.47
0.31
0.16
0.00
0.00
0.16
0.31
0.47
0.62
0.78
0.94
1.09
1.25
1.40
1.56
Fig. 8. Boundaries of the regions Bi , Si and NTi for = 0:3, = 10%, r = 7%,
= (11%; 15%), = (30%; 35%), = (1%; 2%).
1.56
1.40
1.25
B1 \ S2
NT1 \ S2
S1 \ S2
B1 \ NT2
NT
S1 \ NT2
B1 \ B2
NT1 \ B2
S1 \ B2
1.09
0.94
0.78
0.62
0.47
0.31
0.16
0.00
0.00
0.16
0.31
0.47
0.62
0.78
0.94
1.09
1.25
1.40
1.56
Fig. 9. Boundaries of the regions Bi , Si and NTi for = 0:3, = 10%, r = 7%,
= (11%; 15%), = (30%; 35%), = (1%; 0:5%).
29
30
M. AKIAN, J.L. MENALDI AND A. SULEM
1.56
1.40
1.25
B1 \ S2
NT1 \ S2
S1 \ S2
B1 \ NT2
NT
S1 \ NT2
B1 \ B2
NT1 \ B2
S1 \ B2
1.09
0.94
0.78
0.62
0.47
0.31
0.16
0.00
0.00
0.16
0.31
0.47
0.62
0.78
0.94
1.09
1.25
1.40
1.56
Fig. 10. Boundaries of the regions Bi , Si and NTi for = 0:3, = 10%, r = 7%,
= (11%; 15%), = (30%; 35%), = (1%; 1%).
1.56
1.40
1.25
1.09
B1 \ S2
NT1 \ S2
S1 \ S2
B1 \ NT2
NT
S1 \ NT2
B1 \ B2
NT1 \ B2
S1 \ B2
0.94
0.78
0.62
0.47
0.31
0.16
0.00
0.00
0.16
0.31
0.47
0.62
0.78
0.94
1.09
1.25
1.40
1.56
Fig. 11. Boundaries of the regions Bi , Si and NTi for = 0:3, = 10%, r = 7%,
= (11%; 20%), = (30%; 50%), = (1%; 1%).
INVESTMENT-CONSUMPTION MODEL WITH TRANSACTION FEES
31
7. Theoretical analysis of the optimal strategy.
7.1. No transaction costs : the Merton problem. When the transaction
costs are equal to zero, the optimal investment strategy is to keep a constant fraction
of total wealth in each risky asset (see Merton [24], Sethi and Taksar [27], Karatzas,
Lehoczky, Sethi and Shreve [17], Davis and Norman [10]). Indeed, set = = 0 in
equation (71). We obtain :
~ + u(BW); max @W ; max (? @W )) = 0 in (R+)n
(87) max(AW
1in @yi 1in; yi >0 @yi
which is equivalent to
(88)
W = constant,
?(y)W + u(W) 0 8y 2 (R+)n ;
with (y) dened in (68). Uniqueness of the solution of (88) is not guaranteed since
assumption (A.2) is not satised, but the function W dened in (58) is the minimal
solution of VI (87). Hence, we have
(89)
R
max f?(y)W
y2( +)n
+ u (W)g = 0:
Equation (89) coincides with the Bellman equation of the problem where the
proportion yi is considered as a control variable (see [10]). Under assumption (A.1),
the optimal proportion denoted by i and called the \Merton proportion" is given by
(90)
i = 2(1i ??r) :
i
The optimal fraction of wealth dedicated to consumption is
n ? r 2 !!
X
1
1
i
C = 1 ? ? r + 2(1 ? )
;
i
i=1
and the value function W is equal to
( ?1)
W=C :
The regions \sell i" and \buy i" are characterized by
Bi = fy 2 (R+)n; yi i g;
Si = fy 2 (R+)n; yi i g:
Note that these regions are not obtained by merely setting = = 0 in (86) but
by taking the limit of these expressions when and tend to 0.
7.2. A general shape of the transaction regions. In this section, we derive formally from VI (71), without numerical computation, the general shape of the
transaction regions, given in Figure 12. To that purpose, we assume the function W
to be C 2 in the interior of (R+)n . Although this is not true in general, what is done
below can be adapted by using the theory of viscosity solutions. This approach is
32
M. AKIAN, J.L. MENALDI AND A. SULEM
y2
B1 \ S2
d02
c02
NT1 \ S2
D
S1 \ S2
A
B1 \ NT2
C NT
B S1 \ NT2
B1 \ B2
NT1 \ B2
S1 \ B2
c01
b01
y1
Fig. 12. General shape of the transaction regions
used for example in Fleming and Soner [13] to obtain regularity results for the value
function V and general properties of the transaction regions for n = 1.
From (71), we have M~ i W 0; besides, the concavity of W implies that M~ i W =
@W
? @yi is non decreasing with respect to yi . Consequently, the region Si dened in (86)
can be written as
Si = fy 2 (R+)n; yi i+ (^y )g
where i+ is some mapping of
y^ = (y1 ; : : :; yi?1 ; yi+1; : : :; yn):
To obtain a similar characterization for Bi , we consider another change of variables (0 ; y0 ) obtained by substituting ?i for i in (55) for some xed i 2 f1; : : :; ng.
Proceeding as above, and using Remark 6.1, we obtain :
Bi = fy 2 (R+)n ; yi0 i0 ? (^y0 )g
(91)
with
(92)
1
i)yi
0
yi0 = (11 +
+ y and y^ = 1 + y y^:
i i
i i
Since yi0 is non decreasing with respect to yi , we get
Bi = fy 2 (R+)n; yi i? ( 1 +1 y y^)g:
i i
Suppose i+ < +1 and i? > 0. This implies that i+ and i? are continuous
functions and that the regions Si and Bi are connected.
We restrict now to the case n = 2, but what is done below can easily be generalized
to n > 2.
In S1 , M~ 1 W = ? @W
@y1 = 0. The function W is thus constant with respect to y1
in S1 . Consequently the parts of the boundaries @B2 and @S2 included in S1 are
straight lines of equation y2 = constant. Similarly, using the change of variables (92)
INVESTMENT-CONSUMPTION MODEL WITH TRANSACTION FEES
33
with i = 1, we infer that the parts of the boundaries @B2 and @S2 included in B1 are
straight lines of equation
y20 = 1 +y2 y = constant:
1 1
By symmetry, we get similar properties for the boundaries @B1 and @S1 as displayed in Figure 12. No other property has been obtained for the boundary of NT.
A question which arises now is how is located the \Merton proportion" . In
general, is not necessarily in the region NT. Nevertheless, we have
Proposition 7.1. We use the notation of Figure 12 :
A = (a1; a2) = @S1 \ @S2 ;
B = (b1; b2) = @S1 \ @B2 ;
C = (c1 ; c2) = @B1 \ @B2 ;
D = (d1; d2) = @B1 \ @S2 ;
c01 = 1 +c1 c ; b01 = 1 +b1 b ; c02 = 1 +c2 c ; d02 = 1 +d2 d :
22
22
11
1 1
and set
8
i
1
<
~i = : 1 + i ? i i if i < 1 + i ;
+1
otherwise.
Then,
1 a1 ; b01; 2 a2; d02
and
d1; c01 ~1 ; b2 ; c02 ~2 :
Proof. We prove i ai , i = 1; 2. The other inequalities are obtained similarly
by using the change of variables (92). In S1 \ S2 , the function W is equal to a constant
W0 and satises (71) which reduces to :
?(y)W0 + u(W0 ) 0
with (y) given in (68).
Hence,
1
?(y) + (1 ? ) ?1 1 W0?1 0 8y 2 S1 \ S2 :
On the other hand, the point A is in S1 \ S2 \ NT. Assuming that W is C 2 at point
A, we obtain
(93)
and
~ + u (BW) = 0
AW
1
?(A) + (1 ? ) 1= ?1 W0?1 = 0:
34
M. AKIAN, J.L. MENALDI AND A. SULEM
Consequently
(y) (A) 8y 2 S1 \ S2 = [a1; +1) [a2; +1):
As the function (y) is of the form 1 (y1 ) + 2 (y2 ) with quadratic functions i , we
get
i (yi ) i (ai ) 8yi ai :
Consequently, ai Argmini = i .
7.3. Special case of no transaction cost for one of the risky assets. We
suppose here n = 2, 1 = 0, 2 > 0. The VI (71) then reduces to
~ + u (BW); @W ; @W ? 2BW; max (? @W ) = 0
(94)
max(AW
i=1;2; yi >0 @yi
@y1 @y2
which implies that the function W is independent of y1 . Consequently the boundaries
of B2 and S2 are horizontal straight lines of equation y2 = 2? and y2 = 2+ respectively. Since equation (94) holds for all y1 0, and W is the minimal solution of (94),
we have
~ + u (BW)); @W ? 2 BW; ? @W ) = 0 for y2 > 0:
max(max
(AW
y1 0
@y2
@y2
The regions B1 and S1 are delimited by the curve of equation y1 = 1(y2 ) where
~ + u (BW))
1(y2 ) = Argmax(AW
y1 0
is the solution of
2
2
y1 12 y22 @@yW2 ? (2( ? 1)12y1 + (1 ? r))y2 @W
@y2 + ((1 ? r) + ( ? 1)1 y1 )W = 0:
2
Consequently
1 BW
1(y2 ) =
:
y
@BW
2
BW + 1 ? @y
2
In particular 1(0) = 1 and 1(y2 ) = (1 + 2y2 )1 in B2 . In S2 , W is constant
and 1(y2 ) = 1 (see Figure 13). Moreover, by using the concavity of W, we obtain
the estimate
0 < 1(y2 ) 1 ?1 y :
2 2
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pp 113{122.
INVESTMENT-CONSUMPTION MODEL WITH TRANSACTION FEES
35
y2
2+
2?
S2 \ B1
S2 \ S1
NT2 \ B1
NT2 \ S1
B2 \ B1
B2 \ S1
1
y1
Fig. 13. Boundaries of the transaction regions in the case of no transaction cost for the rst
risky asset.
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