A partially observed non-zero sum differential game of forward-backward stochastic differential equations and its application in finance

0.23 MB·

by  Jie Xiong

A partially observed non-zero sum differential game of
forward-backward stochastic differential equations and
6
its application in finance
1
0
2
Jie Xiong a, Shuaiqi Zhang b, Yi Zhuang c
n
a
a Department of Mathematics, University of Macau, Macau, PR China
J b School of Economics and Commerce, Guangdong University of Technology, Guangzhou 510520, PR China

4 c
School of Mathematics, ShandongUniversity,Jinan 250100, PR China

]
C January 5, 2016
O
.
h Abstract
t
a In this article, we concern a kind of partially observed non-zero sum stochastic differential game
m
based on forward and backward stochastic differential equations (FBSDEs). It is required that each
[ player has his own observation equation, and the corresponding open-loop Nash equilibrium control is

requiredtoadaptedtothefiltrationthattheobservationprocessgenerated. Tofindthisopen-loopNash
1
equilibriumpoint,weprovethemaximumprincipleasanecessaryconditionoftheexistenceofthispoint,
v
andgiveaverificationtheoremasasufficientconditiontoverifyitistherealopen-loopNashequilibrium
8
point. Combined this with reality, a financial investment problem is raised. We can obtain the explicit
3
5 observable investmentstrategy by using stochastic filtering theory and theresults above.
0
Keywords. forward-backwardstochastic equation, differential game, maximum principle, partial
0
information, stochastic filtering
.
1
0
6 1 Introduction
1
:
v 1.1 Historical contribution
i
X
Thegeneraltheoryofbackwardstochasticdifferentialequation(BSDE)wasfirstintroducedbyPardouxand
r Peng [18]. For the BSDE coupled with a forward stochastic differential equation (SDE), it is so-called the
a
forwardand backwardstochastic differential equation (FBSDE), which has important applications in many
areasinoursociety. Instochasticcontrolarea,the Hamiltoniansystemisoneofthe formofFBSDEs. More
essentiallyinfinancialmarket,the famousBlack-Scholesoptionpricing formulacanbe deducedby acertain
FBSDE. Some research based on FBSDE is surveyed by Ma and Yong [11].
In stochastic control theory, one can use control to reach a maximum or minimum objection based on
stochasticdifferentialsystem. Peng[19]firstlyconsideredthemaximumprincipleofconvexdomainforward-
backwardstochastic control system. In the following, Xu [34] dealt with a case that controldomain doesn’t
need to be convex and there is no control variable in diffusion coefficient in the forward equation. In more
general case, Tang and Li [22] considered that the control domain is non-convex and diffusion coefficient
contains control variable. Moreover, Shi and Wu [20], [21] solved the corresponding fully-coupled case,
etc. All these previous work were based on the “complete information” case, meaning that the control
variable is adapted to the truth complete filtration. In reality, there are many cases the controller can only
obtains “partial information”, reflecting in mathematics that the controlvariable is adapted to the filtraion
generated by an observable process. Based on this phenomenon, Xiong and Zhou [33] dealt with a Mean-
Variance problem in financial marcket that the investor’s optimal portfolio is only based on the stock and
1
bond process he observed. This assumption of partial information is indeed natural in financing market.
What’s more, Wang and Wu [25] consideredthe Kalman-Bucyfiltering equationof FBSDE system. Huang,
Wang and Xiong [7] dealt with the backward stochastic control system under partial information. Wang
and Wu [26], Wu [31], Wang Wu and Xiong [27] solvedthe partially observedcase of forwardand backward
stochastic control system.
The game theory was firstly constructed by Von Neumann since 1928. Nash [12] - [15] made the fun-
damental contribution in the Non-cooperate Games, considered there are N-players acting independently
to maximize their own objective conducted. He gave the notion of equilibrium point. Then, Isaacs [9],
Basar and Olsder [2] conducted the game research on differential equation system. Varaiya [24], Eliott and
Davis [3] considered the stochastic case. Next, many articles of forward stochastic differential games which
is based on SDEs appeared, like Hamadene [4] - [6], Karoui and Hamadene [10], Wu [30], Øksendal [1], etc.
For the backward case, Yu and Ji [36] studied the Linear Quadratic (LQ) system, Wang and Yu [28] gave
the maximum principle of backward system. Øksendal and Sulem [16], Hui and Xiao [8] had a research
on the maximum principle of forward-backward system. Recently, Tang and Meng [23] solved the partial
informationcaseofzero-sumforwardandbackwardsystem. WangandYu[29]solvedthepartialinformation
case of non-zero sum backwardsystem, etc.
Inourarticle,wegenerategametheorytothepartiallyobservednon-zerosumforward-backwardsystem.
The main difference here is that, we suppose every player has his own observation equation, not just as
partial information that focusing only on a smaller sub-filtration. In section 1, we introduce some historical
contributions and make notions we need. In section 2, we establish the necessary condition of maximum
principle for Nash equilibrium point and give a sufficient condition (verification theorem) to help us check
if the candidate equilibrium points are real. In section 3, we consider a reasonable financial investment
problem and use the theorems in section 2 to obtain the open-loop Nash equilibrium point and give the
explicit observable solution of investment strategy.
1.2 Basic Notions
Throughout our article, we denote (Ω,F,{F } ,P) the complete probability space, on which (W(·),Y (·),
t t≥0 1
Y (·)) be astandard3-dimensionalF Brownianmotion. LetFW,F1,F2 be thenaturalfiltrationgenerated
2 t t t t
by W(·),Y (·),Y (·) respectively. We set F =FW ⊗F1⊗F2. For fixed terminal time T, F =F . What’s
1 2 t t t t T
more, we denote the 1-dimensional Euclidean space by R, the Euclidean norm by |·|, and the transpose
of matrix A by Aτ, the partial derivative of function f(·) with respect to x by f (·). We also denote the
x
L2(0,T;S)representingthe setofS-valued, F -adaptedsquareintegrableprocess(i.e. E T |x(t)|2dt<∞),
F t 0
and the L2(Ω;S) representing the set of S-valued, F-measured square integrable random variable. In the
F R
following discussion, we only consider 1-dimensional case if there is no specific illustration.
1.3 Problem formulation
We consider a partially observed stochastic differential game problem of forward-backward stochastic sys-
tems, focusing on necessity and sufficiency of the existence of open-loop Nash equilibrium point.
We formulate the controlled forward and backward stochastic differential equation (FBSDE) as
dx(t)=b(t,x(t),v (t),v (t))dt+σ(t,x(t),v (t),v (t))dW(t)
1 2 1 2
+σ (t,x(t),v (t),v (t))dWv1,v2(t)+σ (t,x(t),v (t),v (t))dWv1,v2(t),
 1 1 2 1 2 1 2 2
−dxy((0t))==xf0(t,,x(t),y(t),z(t),z1(t),z2(t),v1(t),v2(t))dt−z(t)dW(t)−z1(t)dY1(t)−z2(t)dY2(t), (1.1)
y(t)=g(x(T)),
wherev (·),v (·)aretwocontrolprocessestakingvaluesinconvexsetsU ⊂R,U ⊂Rrespectively,Wv1,v2(·)
1 2 1 2 1
andWv1,v2(·)arecontrolledstochasticprocesstakingvaluesinR, b,σ,σ ,σ :Ω×[0,T]×R×U ×U 7→R,
1 1 2 1 2
2
f :Ω×[0,T]×R×R×R×R×R×U ×U 7→R, g :Ω×R7→R are continuous maps, x ∈R, g(x(T)) is
1 2 0
a F measurable square integrable random variable. Here for simplicity, we omit the notation of ω in each
T
process.
We regard v (·),v (·) as two strategies of player 1 and 2. For both of them, they cannot observe the
1 2
process x(·),y(·),z (·),z (·) directly. However, they can observe their own related processes Y (·), Y (·),
1 2 1 2
which satisfy the following equations 1
dY (t)=h (t,x(t),v (t),v (t))dt+dWv1,v2(t),
i i 1 2 i (1.2)
( Yi(0)=0 (i=1,2),
where h : Ω×[0,T]×R ×U ×U 7→ R,i = 1,2 is a continuous map. We define the filtration Fi =
i 1 2 t
σ{Y (s)|0 ≤ s ≤ t},i = 1,2 as the information for player i obtained at time t, and the admissible control
i
v (·) as
i
v (t)∈U ={v (·)∈U |v (t)∈Fi and sup E|v (t)|8 <∞,a.e} (i=1,2), (1.3)
i i i i i t i
0≤t≤T
where U ,i=1,2 is called the open-loop admissible control set for player i.
i
Hypothesis(H1). Supposefunctionsb,σ,σ ,σ ,h ,h ,f,garecontinuouslydifferentiablein(x,y,z,z ,z ,v ,v ).
1 2 1 2 1 2 1 2
The partial derivatives b ,b ,σ ,σ ,σ ,σ ,h ,h ,f ,
x vi x vi jx jvi jx jvi x
f ,f ,f ,f ,g ,i,j = 1,2 are uniformly bounded. Further, we assume there is constant C such that
y z zj vi x
|h(t,x,v ,v )| +|σ (t,x,v ,v )|+|σ (t,x,v ,v )|≤C for ∀(t,x,v ,v )∈[0,T]×R×U ×U .
1 2 1 1 2 2 1 2 1 2 1 2
From the Hypothesis(H1), we can defined a new probability measure Pv1,v2 by
dPv1,v2
=Zv1,v2(t), (1.4)
dP
(cid:12)Ft
(cid:12)
where Zv1,v2(·) is a F -martingale (cid:12)
t (cid:12)
2 t 1 2 t
Zv1,v2(t)=exp h (s,x(s),v (s),v (s))dY (s)− h2(s,x(s),v (s),v (s))ds . (1.5)
j 1 2 j 2 j 1 2
j=1Z0 j=1Z0
(cid:8)X X (cid:9)
Equivalently, it can be written in the SDE form
dZv1,v2(t)=h (t,x(t),v (t),v (t))Zv1,v2(t)dY (t)+h (t,x(t),v (t),v (t))Zv1,v2(t)dY (t),
1 1 2 1 2 1 2 2
(1.6)
( Zv1,v2(0)=1.
ByusingtheGirsanovtheorem,(W(·),Wv1,v2(·),Wv1,v2(·))becomesa3-dimensionalstandardBrownian
1 2
motiondefinedon(Ω,F,{F } ,Pv1,v2),where(Wv1,v2(·),Wv1,v2(·))isa2-dimensionalcontrolledBrownian
t t≥0 1 2
motion and Y (·),j =1,2 turn out to be a stochastic observation process.
j
Based on the construction above, we define two cost functional under space (Ω,F,{F } ,Pv1,v2).
t t≥0
T
J (v (·),v (·))=Ev1,v2[ l (t,x(t),y(t),z(t),z (t),z (t),v (t),v (t))dt+Φ (x(T))+γ (y(0))]
i 1 2 i 1 2 1 2 i i
Z0 (1.7)
T
=E[ Zv1,v2(t)l (t,x(t),y(t),z(t),z (t),z (t),v (t),v (t))dt+Zv1,v2(T)Φ (x(T))+γ (y(0))],
i 1 2 1 2 i i
Z0
for two players i = 1,2, where Ev1,v2 is the corresponding expectation. l : Ω×[0,T]×R×R×R×R×
i
R×U ×U 7→R, Φ :Ω×R7→R, γ :R7→R are continuous maps. In this cost functional, it contains the
1 2 i i
running cost part representing an utility in duration, and the terminal and initial representing the restrict
on the endpoints.
1Hereweassumethatthecontrolvariablesv1(·),v2(·)explicitlyappearedintheobservationfunctionhi(·),whichiscommon
incontrol problemsunderpartialobservation(see,e.g.,[27]).
3
Hypothesis(H2). Suppose functions l , Φ , γ ,i = 1,2 are continuously differentiable in (x,y,z,z ,z ,
i i i 1 2
v ,v ), x, y respectively,thepartialderivativesl ,l ,l ,l ,l ,i,j =1,2areboundedbyC(1+|y|+|z|+
1 2 ix iy iz izj ivj
|z |+|z |+|v |+|v |) where C is a constant.
1 2 1 2
For each of the player, his goal is to minimise his own cost. Here we set (u ,u )∈U ×U such that
1 2 1 2
J (u (·),u (·))= min J (v (·),u (·)),
1 1 2 1 1 2
v1(·)∈U1
(1.8)

J (u (·),u (·))= min J (u (·),v (·)).
 2 1 2 v2(·)∈U2 2 1 2
In this definition, (u ,u ) is the well-known open-loop Nash equilibrium point of our partially-observed
1 2
forward-backwardnon-zerosumsystem,and(x,y,z,z ,z ,Z)isthecorrespondingequilibriumstateprocess.
1 2
Similar to the optimal control,what we want to do is to find this equilibrium control. We denote the whole
problem above as Problem(NEP).
In particular, If we set J(v (·),v (·))=J (v (·),v (·))=−J (v (·),v (·)), then (1.8) is equivalent to
1 2 1 1 2 2 1 2
J(u (·),v (·))≤J(u (·),u (·))≤J(v (·),u (·)), (1.9)
1 2 1 2 1 2
for ∀(v (·),v (·))∈U ×U .
1 2 1 2
In that case, the reward of player 1 is actually the cost of player 2, and the sum is always zero. We can
regard it as a special case of non-zero sum game. We define this problem of our system as Problem(EP).
Remark 1.1. If we at first suppose W (·) = Wv1,v2(·),j = 1,2 to be a F -Brownian motion under P,
j j t
then the distribution of observation process Y(·) will be depending on the control process. In that way, our
admission control is adapted to a controlled filtration, which appears a circulation. Here, we break through
the circulation by Girsanov theorem, making observation process to be an uncontrolled stochastic process and
depict the controlled Brownian motion under related equivalent probability measure.
2 Maximum principle
Inthissection,wewillestablishthenecessarycondition(maximumprinciple)ofexistenceofopen-loopNash
equilibrium point in problem (NEP), and give a sufficient condition (verification theorem) of a special class
of system.
2.1 Variational equation
Let (v (·),v (·))∈L8 (0,T;R)×L8 (0,T;R) such that (u (·)+v (·),u (·)+v (·))∈U ×U .
1 2 F1 F2 1 1 2 2 1 2
For any ǫ∈[0,1], we make the variational controls as
uǫ(·)=u (·)+ǫv (·),
1 1 1 (2.1)
uǫ(·)=u (·)+ǫv (·).
2 2 2
Because U ,U are convex sets, we have (uǫ(·),uǫ(·))∈U ×U . We denote
1 2 1 2 1 2
φuǫi(·), φ=x,y,z,z1,z2,Z (i=1,2),
as the corresponding state processes of variation (uǫ,u ) or (u ,uǫ).
1 2 1 2
It is noteworthy that when using the variational technique, we had better require the Brownian motion
do not affected by the control process. Then our state equation can be written as
4
2
dx(t)= b(t,x(t),v (t),v (t))− σ (t,x(t),v (t),v (t))h (t,x(t),v (t),v (t)) dt
1 2 j 1 2 j 1 2
 +σh(t,x(t),v1(t),v2(t))dWXj(=t)1+Xj=21σj(t,x(t),v1(t),v2(t))dYj(t)2, i (2.2)
−dy(t)=f(t,x(t),y(t),z(t),z (t),z (t),v (t),v (t))dt−z(t)dW(t)− z (t)dY (t),
1 2 1 2 j j
where (W(·),Y1(xy·()(0,t))Y2==(·xg)(0)x,i(sTF))t,-Brownian motion under P. Xj=1
Then we have the following estimates under Hypothesis (H1).
Lemma 2.1.
sup E|x(t)|8 ≤C(1+ sup E|v(t)|8), (2.3)
0≤t≤T 0≤t≤T
sup E|y(t)|2 ≤C(1+ sup E|v(t)|2), (2.4)
0≤t≤T 0≤t≤T
T T T
E |z(t)|2dt+ |z (t)|2dt+ |z (t)|2dt ≤C(1+ sup E|v(t)|2), (2.5)
1 2
Z0 Z0 Z0 0≤t≤T
(cid:0) (cid:1)
sup E|Zv1,v2(t)|≤K, (2.6)
0≤t≤T
where C,K is constant independent of ǫ.
Lemma 2.2.
sup E|xuǫi(t)−x(t)|8 ≤Cǫ8, (2.7)
0≤t≤T
sup E|yuǫi(t)−y(t)|2 ≤Cǫ2, (2.8)
0≤t≤T
T
E |zuǫi(t)−z(t)|2dt≤Cǫ2, (2.9)
Z0
T
E |zuǫi(t)−z (t)|2dt≤Cǫ2, (2.10)
1 1
Z0
T
E |zuǫi(t)−z (t)|2dt≤Cǫ2, (2.11)
2 2
Z0
sup E|Zuǫi(t)−Z(t)|2 ≤Cǫ2, (2.12)
0≤t≤T
for i=1,2., where C is constant independent of ǫ.
For notation simplicity, we set
ζ(t)=ζ(t,x(t),u (t),u (t)), for ζ =b,σ,σ ,h (i=1,2),
1 2 i i
ψ(t)=ψ(t,x(t),y(t),z(t),z (t),z (t),u (t),u (t)), for ψ =f,l (i=1,2).
1 2 1 2 i
5
We introduce the following variational equations
2 2
dx1(t)= [b (t)− (σ (t)h (t)+σ (t)h (t))]x1(t)+[b (t)− (σ (t)h (t)+σ (t)h (t))]v (t) dt
i x jx j j jx i vi jvi j j jvi i
 j=1 j=1
−dy1(t)=+[[(cid:8)fσx((tt))xx11i((ttX))++fσv(it()ty)v1i((tt))+]dWf ((tt))z+1(Xjt=)21+[σj2x(tf)x1i((tt))z1+(tσ)jv+i(ftX)v(it()tv)]d(tY)j]d(tt)−, z1(t)dW(t)− 2 z1(t(cid:9))dY (t),
i x i y i z i zj ji vi i i ji j
 yxi11i((T0))==g0x,(x(T))x1i(T) (i=1,2), Xj=1 Xj=1(2.13)
and
2
dZ1(t)= Z1(t)h (t)+Z(t)(h (t)x1(t)+h (t)v (t)) dY (t),
 i i j jx i jvi i j (2.14)
j=1
 Z1(0)=0X((cid:2)i=1,2). (cid:3)
i
From Hypothesis(H1), we know that (2.13) and (2.14) exist a unique solution respectively.
Next, we make the notation
φǫ(t)= φuǫi(t)−φ(t) −φ1(t), for φ=x,y,z,z ,z ,Z (i=1,2),
i ǫ i 1 2
and
φ¯(t)=φuǫi(t)−φ(t), for φ=x,y,z,z1,z2,Z,b,σ,σ1,σ2,h1,h2,Z (i=1,2).
Then we have
Lemma 2.3. For i,j =1,2,
lim sup E|xǫ(t)|4 =0, (2.15)
i
ǫ→00≤t≤T
lim sup E|Zǫ(t)|2 =0, (2.16)
i
ǫ→00≤t≤T
lim sup E|yǫ(t)|2 =0, (2.17)
i
ǫ→00≤t≤T
T
limE |zǫ(t)|2dt=0, (2.18)
i
ǫ→0 Z0
T
limE |zǫ (t)|2dt=0. (2.19)
ji
ǫ→0 Z0
Proof. We only consider the first two case for i=1. The rest are similar and are well-known results.
6
(i).
¯b(t) 2 σ¯ (t)
dxǫ(t)= −b (t)x1(t)−b (t)v (t) − j −σ (t)x1(t)−σ (t)v (t) h (t)
1 ǫ x 1 vi i ǫ jx 1 jv1 1 j
(cid:26) j=1(cid:20)
(cid:0) (cid:1) X (cid:0) (cid:1)
h¯ (t) σ¯ (t)
+ j −h (t)x1(t)−h (t)v (t) σ (t)+ j h¯ (t) dt
ǫ jx 1 jv1 1 j ǫ j
(cid:21)(cid:27)
(cid:0) (cid:1) 2
σ¯(t) σ¯ (t)
+ −σ (t)x1(t)−σ (t)v (t) dW(t)+ j −σ (t)x1(t)−σ (t)v (t) dY (t)
ǫ x 1 v1 1 ǫ jx 1 jv1 1 j
j=1
(cid:0) (cid:1) X(cid:0) (cid:1)
x¯(t) 1 1
= xǫ(t)b (t)+ b (Θ)dλ−b (t) +v (t) b (Θ)dλ−b (t) dt
1 x ǫ x x 1 v1 v1
(cid:20) Z0 Z0 (cid:21)
x¯(t)(cid:0) 1 (cid:1) (cid:0) 1 (cid:1)
+ xǫ(t)σ (t)+ σ (Θ)dλ−σ (t) +v (t) σ (Θ)dλ−σ (t) dW(t)
1 x ǫ x x 1 v1 v1
(cid:20) Z0 Z0 (cid:21)
2 (cid:0) x¯(t) 1 (cid:1) (cid:0) 1 (cid:1)
+ xǫ(t)σ (t)+ σ (Θ)dλ−σ (t) +v (t) σ (Θ)dλ−σ (t) h (t)dt
1 jx ǫ jx jx 1 jv1 jv1 j
j=1(cid:26)(cid:20) Z0 Z0 (cid:21)
X (cid:0) (cid:1) (cid:0) (cid:1)
x¯(t) 1 1
+ xǫ(t)h (t)+ h (Θ)dλ−h (t) +v (t) h (Θ)dλ−h (t) σ (t)dt
1 jx ǫ jx jx 1 jv1 jv1 j
(cid:20) Z0 Z0 (cid:21)
x¯(t) 1 (cid:0) 1 (cid:1) (cid:0) (cid:1)
+ σ (Θ)dλ+v (t) σ (Θ)dλ h¯ (t) dt,
ǫ jx 1 jv1 j
(cid:20) Z0 Z0 (cid:21) (cid:27)
(2.20)
where
(Θ)=(t,x(t)+λx¯(t),u (t)+λǫv(t),u (t)).
1 2
Thus we have
E|xǫ(t)|4 ≤CE t|xǫ(s)|4ds+C E t|x¯(s)|8ds E t 1b (Θ)dλ−b (s) 8ds
1 Z0 1 s Z0 ǫ (cid:20)s Z0 Z0 x x
(cid:0) (cid:1)
t 1 2 t 1
+ E σ (Θ)dλ−σ (s) 8ds+ E σ (Θ)dλ−σ (s) 8ds
x x jx jx
s s
Z0 Z0 j=1 Z0 Z0
(cid:0) (cid:1) X (cid:0) (cid:1)
2 t 1
+ E h (Θ)dλ−h (s) 8ds
jx jx
s
j=1 Z0 Z0 (cid:21)
X (cid:0) (cid:1)
t t 1
+C E |v (s)|8ds E b (Θ)dλ−b (s) 8ds (2.21)
1 v1 v1
s Z0 (cid:20)s Z0 Z0
(cid:0) (cid:1)
t 1 2 t 1
+ E σ (Θ)dλ−σ (s) 8ds+ E σ (Θ)dλ−σ (s) 8ds
v1 v1 jv1 jv1
s s
Z0 Z0 j=1 Z0 Z0
(cid:0) (cid:1) X (cid:0) (cid:1)
2 t 1
+ E h (Θ)dλ−h (s) 8ds
jv1 jv1
s
j=1 Z0 Z0 (cid:21)
X (cid:0) (cid:1)
t x¯(s) t t
+C E | |8ds+ E |v (s)|8ds E x¯(s)ds,
s Z0 ǫ s Z0 1 s Z0
(cid:0) (cid:1)
where C is a constant. From Hypothesis (H1), we know that the right side of the inequality converges to 0
when ǫ7→0. The case of i=2 is similar.
7
(ii).
dZǫ(t)= 2 hujǫ1(t)Zuǫ1(t)−hj(t)Z(t) −Z1(t)h (t)−Z(t)h (t)x1(t)−Z(t)h (t)v (t) dY (t)
1 ǫ 1 j jx 1 jv1 1 j
j=1(cid:20) (cid:21)
X
= 2 Zuǫ1(t)ǫ−Z(t)hj(t)+Zuǫ1(t)hujǫ1(t)ǫ−hj(t) −Z11(t)hj(t)−Z(t)hjx(t)x11(t)
j=1(cid:20)
X
−Z(t)h (t)v (t) dY (t)
jv1 1 j
(cid:21)
2 h¯ (t) Z¯(t)
= Zǫ(t)h (t)+Z(t) j −Z(t)h (t)x1(t)−Z(t)h (t)v (t)+ h¯ (t) dY (t)
1 j ǫ jx 1 jv1 1 ǫ j j
Xj=1(cid:20) (cid:21) (2.22)
2 x¯(t) 1
= Zǫ(t)h (t)+Z(t) xǫ(t)h (t)+ h (Θ)dλ−h (t)
1 j 1 jx ǫ jx jx
j=1(cid:20) (cid:18) Z0
X (cid:0) (cid:1)
1
+v (t) h (Θ)dλ−h (t) + Zǫ(t)+Z1(t) h¯ (t) dY (t)
1 jv1 jv1 1 1 j j
Z0 (cid:19) (cid:21)
= 2 Zǫ(t)hu(cid:0)ǫ1(t)+Z(t) xǫ(t)h (t)+(cid:1) x¯(t)(cid:0) 1h (Θ)dλ(cid:1)−h (t)
1 j 1 jx ǫ jx jx
j=1(cid:20) (cid:18) Z0
X (cid:0) (cid:1)
1
+v (t) h (Θ)dλ−h (t) +Z1(t)h¯ (t) dY (t).
1 jv1 jv1 1 j j
Z0 (cid:19) (cid:21)
(cid:0) (cid:1)
Thus we have
t t
E|Zǫ(t)|2 ≤C E |Zǫ(s)|2ds+E |Z(s)xǫ(s)|2ds
1 1 1
(cid:20) Z0 Z0
t x¯(s) 1 2
+E |Z(s) |2 h (Θ)dλ−h (s) ds
jx jx
ǫ
Z0 (cid:16)Z0 (cid:17)
t 1 2 t
+E |v (s)|2 h (Θ)dλ−h (s) ds+E |Z1(s)h¯ (s)|2ds
1 jv1 jv1 1 j
Z0 (cid:16)Z0 (cid:17) Z0
t t
≤C E |Zǫ(s)|2ds+ E |xǫ(s)|4ds
(cid:20) Z0 1 s Z0 1 (2.23)
+4 E t|x¯(s)|8ds E t 1h (Θ)dλ−h (s) 4ds
jx jx
s Z0 ǫ s Z0 Z0
(cid:0) (cid:1)
t t 1
+ E |v (s)|4ds E h (Θ)dλ−h (s) 4ds
1 jv1 jv1
s Z0 s Z0 Z0
(cid:0) (cid:1)
t t
+ E |Z1(s)|4ds E |x¯(s)|4ds,
1
s Z0 s Z0
where C is a constant. From Hypothesis (H1), we know that the right side of the inequality converges to 0
when ǫ7→0. The case of i=2 is similar.
2.2 Variational inequality
From the definition of open-loop Nash equilibrium point (u (·),u (·)) in Problem (NEP), it is clear that
1 2
8
ǫ−1[J (uǫ(·),u (·))−J (u (·),u (·))]≥0,
1 1 2 1 1 2
(2.24)
ǫ−1[J (u (·),uǫ(·))−J (u (·),u (·))]≥0.
2 1 2 2 1 2
Let Γ (·)=Z1(·)Z−1(·),i=1,2., From Itoˆ’s formula, we have
i i i
2
dΓ (t)= h (t)x1(t)+h (t)v (t) dWu1,u2(t),
 i jx i jvi i j (2.25)
j=1
 Γ (0)=0X((cid:2)i=1,2). (cid:3)
i
From (2.24), we can derive the variational inequality.
Lemma 2.4.
T
Eu1,u2 Φ (x(T))x1(T)+γ (y(0))y1(0)+Φ (x(T))Γ (T)+ Γ (t)l (t)dt
ix i iy i i i i i
(cid:20) Z0
(2.26)
T 2 T
+ [l (t)x1(t)+l (t)y1(t)+l (t)z1(t)+ l (t)z1(t)]dt+ l (t)v (t)dt ≥0.
ix i iy i iz i izj ji ivi i
Z0 j=1 Z0 (cid:21)
X
Proof. we only consider the case i=1. From (1.7), we have
T
ǫ−1[J (uǫ(·),u (·))−J (u (·),u (·))]=ǫ−1E Zuǫ1(t)luǫ1(t)−Z(t)l (t) dt
1 1 2 1 1 2 1 1
(cid:20)Z0 (cid:16) (cid:17) (2.27)
+ Zuǫ1(T)Φuǫ1(x(T))−Z(T)Φ (x(T)) + γuǫ1(y(0))−γ (y(0)) ≥0.
1 1 1 1
(cid:16) (cid:17) (cid:16) (cid:17)(cid:21)
According to Lemma 2.3 and Hypothesis (H2),
ǫ−1[γ1uǫ1(y(0))−γ1(y(0))]= 1γ1y y(0)+λ yuǫ1(0)−y(0) dλ(yuǫ1(0)ǫ−y(0)) →γiy(y(0))y11(0), (2.28)
Z0 (cid:16) (cid:0) (cid:1)(cid:17)
and
ǫ−1E[Zuǫ1(T)Φuǫ1(x(T))−Z(T)Φ (x(T))]
1 1
=ǫ−1E[Zuǫ1(t)(Φuǫ1(x(T))−Φ (x(T)))+Φ (x(T))(Zuǫ1(T)−Z(T))]
1 1 1
=E[Zuǫ1(T) 1Φ1x x(T)+λ xuǫ1(T)−x(T) dλ(xuǫ1(T)ǫ−x(T)) +Φ1(x(T))(Zuǫ1(T)ǫ−Z(T))]
Z0 (cid:16) (cid:0) (cid:1)(cid:17) !
→E[Z(T)Φ (x(T))x1(T)+Φ (x(T))Z1(T)].
1x 1 1 1
(2.29)
Similarly, we have
T T 2
ǫ−1E[ (Zuǫ1(t)l1uǫ1(t)−Z(t)l1(t))dt]→E Z(t) l1x(t)x11(t)+l1y(t)y11(t)+l1z(t)z11(t)+ l1zj(t)zj11(t)
Z0 (cid:20)Z0 j=1
(cid:0) X
T
+l (t)v (t) dt+ l (t)Z1(t)dt .
1v1 1 1 1
Z0 (cid:21)
(cid:1) (2.30)
From the definition of Γ(·) and (2.28)-(2.30), we derive the variational inequality.
9
2.3 A necessary condition (maximum principle)
Inthefollowing,weignorethesuperscriptofWu1,u2(·),j =1,2fornotationsimplicity. Weformulateadjoint
j
equaions under probability measure Pu1,u2.
2
−dP (t)=l (t)dt−Q (t)dW(t)− Q dW (t),
i i i ji j
 (2.31)
j=1
 P (T)=Φ (x(T)) (i=1,2). X
i i

2
dp (t)=[f (t)p (t)−l (t)]dt+[f (t)p (t)−l (t)]dW(t)+ f (t)−h (t) p (t)−l (t) dW (t),
i y i iy z i iz zj j i izj j
 j=1
−dqi(t)=n(cid:2)bx(t)−Xj=21σj(t)hjx(t)(cid:3)qi(t)+σx(t)k2i(t)+Xj=21(cid:2)σXjx(t(cid:2))(cid:0)kji(t)+hjx(t)(cid:1)Qji(t)(cid:3) (cid:3)
−f (t)p (t)+l (t) dt−k (t)dW(t)− k (t)dW (t),
x i ix i ji j
It iqpsii(n(T0o))te==wo−−rtgγhxyy((xyh((e0Tr)e))),tphia(Tt)Wo+(Φ·)i,xj(x=(T1),)2 is(ith=e1BXj,=r21o)w. nianmotionunder probability measure Pu1,u2(.2D.3u2e)
j
totheobservationequation(1.2)andtheappearanceofWv1,v2(·)andY (·),j =1,2intheforward-backward
j j
stateequation(1.1),theequation(2.32)isnottheclassicformanymore. Alsoweshouldintroduceequation
(3.25) to deal with the controlled probability measure Pv1,v2 or expectation Ev1,v2 related to observation
process when using the variational method.
Now we give the necessary condition.
Theorem 2.1. Suppose (H1) and (H2) hold, (u (·),u (·)) is an open-loop Nash equilibrium point of problem
1 2
(NEP), and (x,y,z,z ,z ) is the corresponding state process, then we have
1 2
Eu1,u2[H˜ (t,x,y,z,z ,z ,u ,u ;q ,k ,k ,k ,p ,Q ,Q )(v −u (t))|F1]≥0,
1v1 1 2 1 2 1 1 11 21 1 11 21 1 1 t
(2.33)
Eu1,u2[H˜ (t,x,y,z,z ,z ,u ,u ;q ,k ,k ,k ,p ,Q ,Q )(v −u (t))|F2]≥0,
2v2 1 2 1 2 2 2 12 22 2 12 22 2 2 t
for ∀(v ,v )∈U ×U , a.e.. Here we set
1 2 1 2
H˜ (t)=H˜ (t,x,y,z,z ,z ,u ,u ;q ,k ,k ,k ,p ,Q ,Q )
ivi ivi 1 2 1 2 i i 1i 1i i 1i 2i
2 (2.34)
=H (t)− q (t)σ (t,x(t),u (t),u (t))h (t,x(t),u (t),u (t)),
ivi i j 1 2 jvi 1 2
j=1
X
where
2
H (·),b(t,x,u ,u )q (t)+σ(t,x,u ,u )k (t)+ [σ (t,x,u ,u )k (t)+h (t,x,u ,u )Q (t)]
i 1 2 i 1 2 i j 1 2 ji j 1 2 ji
j=1
X (2.35)
2
−[f(t,x,y,z,z ,z ,u ,u )− h (t,x,u ,u )z (t)]p (t)+l (t,x,y,z,z ,z ,u ,u ) (i=1,2).
1 2 1 2 j 1 2 j i i 1 2 1 2
j=1
X
Proof. We only consider the i = 1 case. Applying Itoˆ’s formula to q (·)x1(·), p (·)y1(·), P (·)Γ (·) respec-
1 1 1 1 1 1
tively, we have
10