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中国科技论文在线
Exact controllability of backward stochastic control system#
Wang Xiangrong, Huang Hong*
(Shandong University of Science and Technology,College of Information Science and
Engineering, Shandong Qingdao,266590) 5
Foundations: This work is supported by NSFC()and The Doctoral Found of the Ministry of
Education of China()
Brief author introduction:王向荣(1966-),男，山东科技大学，教授、博士研究生导师，主要从事金融数学，金
融工程与风险管理方面的研究
Abstract: Based on the study of Peng[1], this paper gives a necessary condition of exact controllability
for backward stochastic control systems, and introduces three equivalent conditions of necessary and
sufficient for the system to be exactly controllable when the system is linear. In the end, two examples
on the optimal portfolio problem with consumption will be given.
Key words: control theory;backward stochastic control systems; exact controllability; portfolio 10
0 Introductions
In the view of backward stochastic differential equation, Peng[1] has firstly defined
the exactly terminal-controllable and exactly controllable for a forward stochastic control
system, and then he gave a necessary condition to the exactly terminal-controllable for
the stochastic control system with deterministic random coefficients, when the random 15
coefficient is linear, he got necessary and sufficient conditions for the system to be
exactly terminal-controllable, finally he gave a necessary and sufficient condition for
stochastic exact controllability and algebraic criterion for the linear system. . Liu,
Peng[2,3] discussed exact controllability of stochastic control system when the control
energy constraint is concerned, then obtained the necessary and sufficient condition 20
which determines linear stochastic systems to be exactly controllable with control
constraint by using the method of moment theory. F. Liu [4] has discussed the nonlinear
stochastic control systems which the coefficient is time-dependent, and gave a necessary
and sufficient condition for the system to be exactly controllable. Based on the study of
Peng and Liu, . Li and . Yao [5] given a necessary and sufficient conditions for 25
linear stochastic control system whose coefficient is time-dependent.
Along with the rapid progress of financial theory, backward stochastic differential
equations play a more and more important role in financial research. Z. Wu and .[6]
have studied the problem of the European option pricing from the view of backward
stochastic differential equation and got a European option price formula when dividend 30
was considered and asset and portfolio were satisfied some conditions. H. Y. Wang, X.
R. Wang [7] discussed an optimal control problem on portfolio and consumption choice
in international security market and they have obtained the explicit optimal portfolio and
consumption rates.
Based on those outstanding researches in above, we will discuss the exact 35
controllable of following stochastic optimal control problem in this paper,
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中国科技论文在线
( , , )t t t t t t
T
dy f y z v dt z dB
y ξ
− = −
= [0, ]t T∈
Where ( , , )F PΩ is a probability space, { , 0}tB t ≥ is a 1-dimensional standard
Wiener process in this space, and let { ,0 }t sB s tσ= ≤ ≤F denote its natural
filtration, [0, ] KU T R⊂ is the set of admissible controls. All the processes in this paper 40
are t −F adapted and square integrals, and the set of all mR −valued square integral
processes is denoted by 2 (0, ; )mFL T R .
In section 2, we will introduce the definitions of initial-controllable and exact control
for the backward stochastic control system above; at the same time some conditions
will be given for each of them. In section 3, a linear backward stochastic control system 45
will be introduced, and we will give three equivalent conditions for this linear system to
be exact control. In section 4, we study the applications of the introduced linear control
system; two examples on the optimal portfolio problem with consumption will be
given.
1 Exact Controllable Backward Stochastic Control Systems 50
In this section, we will study the following backward stochastic control system
( , , )t t t t t t
T
dy f y z v dt z dB
y ξ
− = −
= [0, ]t T∈ (1)
with initial point 0(0)
my y R= ∈ (2)
Here
( , , ) : m m k mf y z v R R R R× × → , 2 ( , , , )mF tL F P Rξ ∈ Ω 55
Our problem is to find a pair of processes ( , , ) m m ky z v R × ×∈ which is satisfied (1) and (2)
in some conditions in brief.
In Ref. [1], the definitions of exactly terminal-controllable and exactly controllable
have be given by Peng, on the basis of these, we will give analogous definitions about
control system (1). 60
Definition : A stochastic control system (1) is called initial-controllable, if for
any 0
my y R= ∈ , there exists at least one admissible control [ ]0,v U T∈ and a
process 2 ( , , , )mF tL F P Rξ ∈ Ω such that the corresponding trajectory ty satisfies the
initial condition (2).
For any 2 (0, ; )mFL T Rξ ∈ and 0 my R∈ , a stochastic control system (1) is called 65
exact control, if there exists at least one admissible control [ ]0,v U T∈ , such that the
corresponding ty satisfies the initial condition(2).
Let us consider the following stochastic differential equation fristly
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中国科技论文在线
0
( , , )
(0)
t t t t t tdy f y z v dt z dB
y y
= +
= [0, ]t T∈ (3)
For the initial- controllable of (1), if we regard a trajectory and the related admissible 70
control ( , , )t t ty z v as a pair of processes ( , , )t t ty z v in (3), then the initial-
controllability is just equivalent to the well-posed of the SDE (3) when tz and tv are given.
And as we all know for the well-posed of the SDE (3), we have the following existence
and uniqueness theorem.
Theorem [8] (existence and uniqueness of solution to SDE ) 75
For a SDE
0
( , ) ( , )
(0)
t t t tdx b t x dt t x dB
x x
σ= +
= [0, ]t T∈ (4)
Let ( , ) :[0, ] , ( , ) :[0, ]n n n n mb T R R T R Rσ ×⋅ ⋅ × → ⋅ ⋅ × → be measurable functions, and
2
0[ ]E x < ∞ , for some constant 0C > satisfying
( , ) ( , ) (1 )b t x t x C xσ+ ≤ + , [0, ]nx R t T∈ ∈ 80
and for some constant 0D > satisfying
( , ) ( , ) ( , ) ( , )b t x b t y t x t y D x yσ σ− + − ≤ − , , [0, ]nx y R t T∈ ∈
Then SDE (4) exists unique solution.
For SDE (3), it is easy to verify that, when 2 (0, ; )mt Fz L T R∈ and [0, ]tv U T∈ are
given, assume 20[ ]E y < ∞ , if for some constant 0C > satisfying 85
( , , , ) (1 )f t y z v C y≤ + , [0, ]my R t T∈ ∈
and for some constant 0D > satisfying
( , , , ) ( , , , )f t x z v f t y z v D x y− ≤ − , , [0, ]mx y R t T∈ ∈
Then SDE (3) exists unique solution. It is easy for us to get the following theorem which
is about necessary condition for the system to be initial- controllable. 90
: A necessary condition for a backward Stochastic Control Systems (1)
to be initial- controllable is that for some constant 0C > satisfying
( , , , ) (1 )f t y z v C y≤ + , [0, ]my R t T∈ ∈
and for some constant 0D > satisfying
( , , , ) ( , , , )f t x z v f t y z v D x y− ≤ − , , [0, ]mx y R t T∈ ∈ 95
The proof of this theorem is obviously.
2 Exact control criterions of linear backward stochastic control system
For this part, we will discuss exact control of the following linear stochastic
control system
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中国科技论文在线
1( )t t t t t t
T
dy Ay A z Bv dt z dB
y ξ
− = + + −
= (5) 100
Here, A and 1A are m m× matrices,B ism k× matrices, and 2 ( , , , )mtL F P Rξ ∈ Ω .
For the following BSDE:
1( )
0
t t t t t t
T
dy Ay A z Bu dt z dB
y
− = + + −
= (6)
Set { }20 ; (0, ; )u kFS y u L T R= ∈ .
: Exact control criterions of stochastic control system (5) 105
( )i
{ }20 ; (0, ; )u kFS y u L T R= ∈ mR= ;
( )ii 1 1 1( , , , , , )Rank B AB AB AAB A AB m=L ;
( )iii Matrix
0
( )( )
T
T
t tG E x B x B dt= ∫ is positive.
For tx in ( )iii which is satisfied the following SDE:
1
0
( )t t t
m
dx x Adt A dB
x I
= +
= [0. ]t T∈ (7) 110
mI is m-dimensional unit vector.
Actually, optimum control is 1 0( )
T
t tu x B G y
−= .
Proof: Firstly, from the definition of set S , it follows immediately that
dissertation ( )i is equivalent to exact controllability of system (5).
( )i ⇒ ( )ii : In Ref. [2], we know that 115
{ }20 1 1 1; (0, ; ) [ , , , , , ]u kFy u L T R Span B AB AB AAB A AB∈ = L
Consequently
1 1 1( , , , , , )Rank B AB AB AAB A AB m=L
( )ii ⇒ ( )iii Suppose that Matrix
0
( )( )
T
T
t tG E x B x B dt= ∫ is not positive, then there
exists a non-zero mRα ∈ such that 120
0
( )( ) 0
T
T T T T
t tG E x B x B dtα α α α= =∫ . [ ]0,t T∀ ∈
That is
0T tx Bα =
This is impossible, soG is positive.
( )iii ⇒ ( )i : Using ˆIto s′ formula applied to t tx y⋅ and integral fromt toT yields that 125
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中国科技论文在线
( ) ( ) ( ) ( ) ( ) ( )
T
t
y T x T y t x t E x s Bu s ds= − ∫
Let 0t = , then we get
0
0
( ) ( )
T
y E x t Bu t dt= ∫ (8)
That is to say, for any admissible control [0, ]tu U T∈ will satisfies (8).
For any given 0y , since ( )r G m= , we can find a control 130
1
0( )
T
t tu x B G y
−= (9)
Easy to verify that (9) is satisfied (8).
That means for any given 0
my R∈ , we can find an admissible control 2 (0, ; )kt Fu L T R∈
which is satisfied (8), so
{ }20 ; (0, ; )u k mFS y u L T R R= ∈ = 135
Next we will proof that the admissible control tu defined by (9) is optimum control.
Suppose that, , [0, ]t tu u U T′ ∈ are two different admissible control, so they all satisfied
(8)
0
0
( )
T
ty E x t Bu dt= ∫ (10)
0
0
( )
T
ty E x t Bu dt′= ∫ (11) 140
Let (10) (11)−
0
0 ( ) ( )
T
t tE x t B u u dt′= −∫ (12)
multiplied 10 ( )
T Ty G − on both sides of (12)
1
0
0 0
0 ( ) ( ) ( ) ( )
T T
T T T
t t t t tE y G x t B u u dt E u B u u dt
− ′ ′= − = −∫ ∫
Thus
145
2
0 0
T T
T
t t tE u dt E u u dt′=∫ ∫
So
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中国科技论文在线
2
0
22
0
2 2
0
0
( 2 )
( )
T
t t
T
T
t t t t
T
t t
E u u dt
E u u u u dt
E u u dt
′≤ −
′ ′= − +
′= −
∫
∫
∫
That is
2 2
0 0
T T
t tE u dt E u dt′ ≥∫ ∫ 150
So 1 0( )
T
t tu x B G y
−= is the optimum control.
The proof is completed.
3 Applications
In the field of finance, portfolio problem has been the focus of the researchers
for a long time. In the deepening research of finance market, portfolio problem has 155
not only confined to the income, risk research, more need to consider the influence of
utility, form of investment and other factors. Backward stochastic differential equation
theory, the optimal control theory has been proved to be an effective tool in the
research of the portfolio problem. In the following we will give two examples of the
applications to the backward stochastic control system, different from the general 160
portfolio problem, we consider the portfolio problem with consumption.
Example Consider a simple stock market, there is only one kind of stocks
and a bond, assume that stock market is complete and transaction cost is not in the
considered range.
0 ( )p t is the pricing process of the bond which satisfied
165
0 0( ) ( )dp t rp t dt= (13)
( )p t is the pricing process of the stock which satisfied
( ) ( ) ( ) tdp t p t dt p t dBα σ= + (14)
The investment values in moment t of bond is
0 0( ) [ ( ) ( )]dS t rS t C t dt= − (15) 170
and the stock is
( ) ( ) ( ) tdS t S t dt S t dBα σ= + (16)
where ( )C t is consumer flow in moment t ,and assume ( )C t is take value in risk free
assets.
Let 0( ) ( ) ( )y t S t S t= + is the total investments of investors at moment t , and 175
assume that ( )y T ξ= , then from (15) to (16) we can obtain
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中国科技论文在线
( ) [ ( ) ( ) ( ) ( )] ( )
( )
tdy t ry t r S t C t dt S t dB
y T
α σ
ξ
= + − − +
= (17)
As Rσ ∈ is a constant, if we let ( ) ( )S t z tσ = , then (17) can be rewritten
( ) [ ( ) ( ) ( )] ( )
( )
t
rdy t ry t z t C t dt z t dB
y T
α
σ
ξ
−− = − + + −
=
(18)
It is clearly that (16) is a backward stochastic control system in the form of system (5) 180
when 3( , , )y z C R∈ , then from , system (16) can be verified exact control
easily, and optimum control is
2
02
2( ) exp{ }
2 22exp{ } 2
2
t
a b b aC t B t y
b T aT
+= +
+ −
(19)
Where 0y R∈ is the initial assets and
2 2
2
2 ( )r ra σ ασ
+ −= −
2( )rb ασ
−= (20) 185
That is to say, the optimal proportion of consumption ( )C t is defined as (19).
Example : Different from example , this time we assume that there is a
bond and m kinds of stocks in the stock market. The prices of the bond and each kind
of stocks are defined as same as (13) and (14). The investment values in moment t of
190
the bond is also satisfied (15), but to the stocks it satisfied
( ) ( ) ( )i i i i i tdS t S t dt S t dBα σ= + 1, 2i m= ⋅⋅⋅ (21)
The total investments of investors at moment t has changed
0
1
( ) ( ) ( )
m
i i
i
y t S t S tπ
=
= +∑ (22)
Here iπ is the ratio of assets invested in the i stock accounted for all assets invested 195
risk, and
1
1
m
i
i
π
=
=∑
Then (22) can be rewritten in form of
0
1
( ) [ ( ) ( )]
m
i i
i
y t S t S tπ
=
= +∑
Set
200
0( ) [ ( ) ( )]i i iy t S t S tπ= +
We can get
( ) [ ( ) ( ) ( ) ( )] ( )
( )
i i i i i i i i i t
i i
dy t ry t r S t C t dt S t dB
y T
α π π π σ
ξ
= + − − +
=
1, 2i m= ⋅⋅⋅ (23)
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中国科技论文在线
Let ( ) ( )i i i iz t S tπ σ= , ( ) ( )i iC t C tπ= then (23) can be transformed
( )( ) [ ( ) ( ) ( )] ( )
( )
i
i i i i i t
i
i i
rdy t ry t z t C t dt z t dB
y T
α
σ
ξ
−− = − + − −
=
1, 2i m= ⋅⋅⋅ (24) 205
From example , we know that (24) is exact control, and
2
02
2( ) exp{ }
2 22exp{ } 2
2
i i i i
i t
i
i
a b b aC t B t y
b T a T
+= +
+ − 1, 2i m= ⋅⋅⋅ (25)
Where
2 2
2
2 ( )i i
i
i
r ra σ ασ
+ −= − ,
2( )i
i
i
rb ασ
−= , 1, 2i m= ⋅⋅⋅
Then the optimum consumption is
210
1
( ) ( )
m
i
i
C t C t
=
= ∑ 1, 2i m= ⋅⋅⋅ (26)
4 Conclusions
This paper gives a necessary condition of exact controllability for backward
stochastic control systems, and introduces three equivalent conditions of necessary
and sufficient for the system to be exactly controllable when the system is linear. 215
These conclusions are very valuable for research on the exact controllability of
forward-backward stochastic control system.
Acknowledgements
This work is supported by the Doctoral Fund of the Ministry of China
()220
References
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[2] ,Application of backward stochastic differential equations in control system[D], Shandong University,
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[3] ,,Exact controllability of stochastic control system with control energy constraint[J],Journal of
Shandong University,1999, 34[4]:361-366.
[4] ，, The algebraic criterion for nonlinear stochastic control systems which the cofficient is
time-dependent[J],Proceedins of 26th Chinese control conference,2007,754-756.
[5] ,,Exact controllability of linear stochastic control system[J],Journal of Jiaxing 230
College,2003,15(6):11-14.
[6] ,,The backward stochastic differential equation and control theory applied to the problem of
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倒向随机控制系统的精确能控性
王向荣，黄虹 240
（山东科技大学，信息科学与工程学院，山东青岛，266590）
摘要：给出了倒向随机控制系统的精确能控性的一个必要条件，并获得了线性倒向随机控制
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中国科技论文在线
系统精确能控的三个等价条件。最后，为体现该系统的实际应用价值，讨论了两个带有消费
的最优投资组合问题的例子。
关键词：控制理论，倒向随机控制系统;精确能控性；投资组合 245
中图分类号：
