#### Non-zero sum differential games of anticipated forward-backward stochastic differential delayed equations under partial information and application

Zhuang Advances in Difference Equations
Non-zero sum differential games of anticipated forward-backward stochastic differential delayed equations under partial information and application
Yi Zhuang
This paper is concerned with a non-zero sum differential game problem of an anticipated forward-backward stochastic differential delayed equation under partial information. We establish a maximum principle and a verification theorem for the Nash equilibrium point by virtue of the duality and convex variation approach. We study a linear-quadratic system under partial information and present an explicit form of the Nash equilibrium point. We derive the filtering equations and prove the existence and uniqueness for the Nash equilibrium point. As an application, we solve a time-delayed pension fund management problem with nonlinear expectation to measure the risk and obtain the explicit solution. MSC: Primary 93E20; 34K50; secondary 93A23; 93E11
stochastic differential game; maximum principle; stochastic differential delayed equation; linear-quadratic problem; partial information; g-expectation; convex risk measure
1 Introduction
The general nonlinear backward stochastic differential equations (BSDEs) were first
developed by Pardoux and Peng [
1
] and have been widely applied in the optimal control,
mathematical finance, and related fields (see Peng [
2, 3
], Karoui et al. [4]). The classical
Black-Scholes option pricing formula in the financial market can also be deduced by virtue
of the BSDE theory. If a BSDE is coupled with a forward stochastic differential equation
(SDE), it is called the forward-backward stochastic differential equation (FBSDE). The
fundamental research based on FBSDEs has been surveyed by Ma and Yong [
5
]. FBSDEs
are widely encountered in the application of the stochastic recursive utility (see, e.g., Wang
and Wu [
6
]), financial optimization problem with large investors (see Cvitanic and Ma [
7
]),
asset pricing problem with differential utilities (see Antonellia et al. [
8
]), etc. The classical
Hamiltonian system is also one of the forms of FBSDEs in the stochastic control field (see,
e.g., Yong and Zhou [
9
]).
In classical cases, there are many phenomena that have the nature of past-dependence,
i.e., their behavior not only depends on the situation at the present time, but also on their
past history. Such models were identified as stochastic differential delayed equations
(SDDEs), which are a natural generalization of the classical SDEs and have been widely
studied in engineering, life science, finance, and other fields (see, for example, the population
growth model in Mohammed [
10
], Arriojas et al. [
11
]). Chen and Wu [
12
] first studied a
stochastic control problem based on SDDE. When introducing the adjoint equation, they
need some new types of BSDEs, which have been introduced by Peng and Yang [
13
] for
the general nonlinear case and called anticipated BSDEs (ABSDEs). The anticipated term
defined by the conditional expectation can be regarded as a predicted value of the future
state. It can be applied to the insider trading market, describing the asset price influenced
by insiders (see, e.g., Øksendal and Sulem [
14
], Kyle [
15
]). Moreover, a class of BSDEs with
time-delayed generators (BSDDEs) has also been studied in the stochastic control field
(see Wu and Wang [
16
], Shi and Wang [
17
], Wu and Shu [
18
]). Recently, Chen and Wu
[
19
], Huang et al. [
20
] studied a linear-quadratic (LQ) case based on a coupled SDDE and
ABSDE called the anticipated forward-backward stochastic differential delayed equation
(AFBSDDE).
Game theory has been pervading the economic theory, and it attracts more and more
research attention. It was firstly introduced by Von Neumann and Morgenstern [
21
]. Nash
[
22
] made the fundamental contribution in non-cooperate games and gave the classical
notion of Nash equilibrium point. In recent years, many articles on stochastic
differential game problems driven by stochastic differential equations have appeared. Researchers
try to consider the strategy on multiple players rather than one player and try to find an
equilibrium point rather than an optimal control. These problems are more complicated
than the classical control problems but much closer to social and behavior science. Yu [
23
]
solved the LQ game problem on a forward and backward system. Øksendal and Sulem [
24
],
Hui and Xiao [
25
] made a research on the maximum principle of a forward-backward
system. Chen and Yu [
26
] studied the maximum principle of an SDDE case, Shi and Wang
[
17
], Wu and Shu [
18
] discussed a BSDDE case.
In reality, instead of complete information, there are many cases where the controller
can only obtain partial information, reflecting in mathematics that the control variable
is adapted to a smaller filtration. Based on this phenomenon, Xiong and Zhou [
27
] dealt
with a mean-variance problem in the financial market that the investor’s optimal portfolio
is only based on the stock process he observed. This assumption of partial information is
indeed natural in the financial market. Recently, Wu and Wang [
16
], Wu and Shu [
18
] also
considered the partial information case.
To our best knowledge, the research on general AFBSDDEs and its wide applications
in mathematical finance are quite lacking in the literature. Recently, Huang and Shi [
28
]
discussed the optimal control problem based on the AFBSDDE system. Our work
distinguished itself from the above-mentioned ones in the following aspects. First, we study the
stochastic differential game problem with multiple players rather than the control
problem with only one controller. We aim to find the equilibrium point rather than the optimal
control. Second, we consider the case that the diffusion coefficient can contain control
variables and the control field is convex. Third, we deal with the system under partial
information that the available information to the players is partial, which can be seen as a
generalization of the complete information case. Fourth, we derive the filtering equations
of an LQ system and get worthwhile results about the existence and uniqueness of the
equilibrium point. Finally, as an example, we solve a financial problem by virtue of the
theoretical results we obtain.
The rest of this paper is organized as follows. In Section 2, we give some necessary
notions and state some preliminary results. In Section 3, we establish a necessary condition
(maximum principle) and a sufficient condition (verification theorem) for the Nash
equilibrium point. In Section 4, we study a linear-quadratic game problem under partial
information. We derive the filtering equations and prove the existence and uniqueness for the
Nash equilibrium point. In Section 5, we solve a pension fund management problem with
nonlinear expectation and obtain the explicit solution.
2 Preliminary results
Throughout this article, we denote by Rk the k-dimensional Euclidean space; by Rk×l the
collection of k × l matrices. For a given Euclidean space, we denote by ·, · (resp. | · |)
the scalar product (resp. norm). The superscript τ denotes the transpose of vectors or
matrices.
Let ( , F , {Ft}t≥0, P) be a complete filtered probability space equipped with a d + d¯
dimensional, Ft -adapted standard Brownian motion (W (·), W¯ (·)), where F = FT . EFt [·] =
E[·|Ft] denotes the conditional expectation under natural filtration Ft , and fx(·) denotes
the partial derivative of function f (·) with respect to x. Let T > 0 be the finite time duration
and 0 < δ < T be the constant time delay. Moreover, we denote by C([–δ, 0]; Rk) the space
of a uniformly bounded continuous function on [–δ, 0], by Lp ( ; Rk) the space of an F
F
measurable random variable ξ satisfying E|ξ |p < ∞ for any p ≥ 1, and by LpF (r, s; Rk) the
space of Rk -valued Ft-adapted processes ϕ(·) satisfying E rs |ϕ(t)|p dt < ∞ for any p ≥ 1.
We consider the following AFBSDDE:
⎧⎪ dxv(t) = b(t, xv(t), xδv(t), v1(t), v2(t)) dt + σ (t, xv(t), xδv(t), v1(t), v2(t)) dW (t)
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪ –dyv(t) =+fσ¯(t(,tx,xv(vt()t,)y,vx(δvt()t,)z,vv(1t()t,)z¯,vv(2t()t,)y)δvd+(Wt¯),(vt)1,(t), v2(t)) dt
⎪⎪⎪⎪⎪⎪⎩⎪⎪⎪⎪⎪ yxvv((Tt))==ξG(t(–)x, vz(vT(tt))∈)d,[W–δ(,t0)y]–v,(tz¯)v(=t)ϕd(Wt¯),(t),t ∈ t(T∈,[T0,+Tδ],].
(1)
Here, (xv(·), yv(·), zv(·), z¯v(·)) : × [–δ, T ] × [0, T + δ] × [0, T ] × [0, T ], b : × [0, T ] × Rn ×
Rn × Rk1 × Rk2 → Rn, σ : × [0, T ] × Rn × Rn × Rk1 × Rk2 → Rn×d, σ¯ : × [0, T ] × Rn ×
Rn × Rk1 × Rk2 → Rn×d¯ , f : × [0, T ] × Rn × Rm × Rm×d × Rm×d¯ × Rm × Rk1 × Rk2 →
Rm, G : × Rn → Rn are given continuous maps, xδv(t) = xv(t – δ), yδv+ (t) = EFt [yv(t + δ)],
ξ (·) ∈ C([–δ, 0]; Rn) is the initial path of xv(·), ϕ(·) ∈ L2F (T , T + δ; Rm) is the terminal path
of yv(·). Here, for simplicity, we omit the notation of ω in each process.
Let Ui be a nonempty convex subset of Rki , Gti ⊆ Ft be a given sub-filtration which
represents the information available to the player i, and vi(·) be the control process of
player i (i = 1, 2). We denote by Uaid the set of Ui-valued Gti-adapted control processes
vi(·) ∈ L2 i (0, T ; Rki ), and it is called the admissible control set for player i (i = 1, 2).
G
Uad = Ua1d × Ua2d is called the set of admissible controls for the two players. We also
introduce the following assumption:
H1. Functions b, σ , σ¯ are continuously differentiable in (x, xδ, v1, v2), f is continuously
differentiable in (x, y, z, z¯, yδ+ , v1, v2), G is continuously differentiable in x. All the
partial derivatives of b, σ , σ¯ , f , G are uniformly bounded.
Then we have the following existence and uniqueness result which can be found in [
12,
13
].
Theorem 2.1 If v1(·) and v2(·) are admissible controls and assumption H1 holds,
AFBSDDE (1) admits a unique solution (x(·), y(·), z(·), z¯(·)) ∈ L2F (–δ, T ; Rn) × L2F (0, T + δ; Rm) ×
L2F (0, T ; Rm×d) × L2F (0, T ; Rm×d¯ ).
The players have their own preferences which are described by the following cost
functionals:
Ji v1(·), v2(·) = E
0
T
li t, xv(t), yv(t), zv(t), z¯v(t), v1(t), v2(t) dt + i xv(T )
+ γi yv(0) .
Here, li : × [0, T ] × Rn × Rm × Rm×d × Rm×d¯ × Rk1 × Rk2 → R, i : × Rn → R,
γi : × Rm → R (i = 1, 2) are given continuous maps. li, i, and γi satisfy the following
condition:
H2. Functions li, i, and γi are continuously differentiable with respect to
(x, y, z, z¯, v1, v2), x, and y, respectively. Moreover, there exists a positive constant C
such that the partial derivatives of li, i, and γi are bounded by
C(1 + |x| + |y| + |z| + |z¯| + |v1| + |v2|), C(1 + |x|), and C(1 + |y|), respectively.
Now we suppose that each player hopes to maximize his cost functional Ji(v1(·), v2(·)) by
selecting a suitable admissible control vi(·) (i = 1, 2). The problem is to find an admissible
control (u1(·), u2(·)) ∈ Uad such that
⎧
⎨ J1(u1(·), u2(·)) = supv1(·)∈U1 J1(v1(·), u2(·)),
⎩ J2(u1(·), u2(·)) = supv2(·)∈U2 J2(u1(·), v2(·)).
(2)
If we can find an admissible control (u1(·), u2(·)) satisfying (2), then we call it a Nash
equilibrium point. In what follows, we aim to establish the necessary and sufficient condition
for the Nash equilibrium point subject to this game problem.
3 Maximum principle
In this section, we will establish a necessary condition (maximum principle) and a
sufficient condition (verification theorem) for problem (2).
Let (u1(·), u2(·)) be an equilibrium point of the game problem, (v1(·), v2(·)) ∈ L2 1 (0, T ;
G
Rk1 ) × L2 2 (0, T ; Rk2 ) be such that (u1(·) + v1(·), u1(·) + v2(·)) ∈ Uad. Then, for any 0 ≤
G
≤ 1, we take the variational control u1(·) = u1(·) + v1(·) and u2(·) = u2(·) + v2(·).
Because both U1 and U2 are convex, (u1(·), u2(·)) is also in Uad. For simplicity, we
denote by (xu1 (·), yu1 (·), zu1 (·), z¯u1 (·)), (xu2 (·), yu2 (·), zu2 (·), z¯u2 (·)), and (x(·), y(·), z(·), z¯(·)) the
corresponding state trajectories of system (1) with control (u1(·), u2(·)), (u1(·), u2(·)), and
(u1(·), u2(·)).
The following lemma gives an estimation of (x(·), y(·), z(·), z¯(·)).
Lemma 3.1 Let H1 hold. For i = 1, 2,
E
0
sup E xui (t) – x(t) 2 ≤ C 2,
0≤t≤T
T
zui (t) – z(t) 2 dt ≤ C 2,
sup E yui (t) – y(t) 2 ≤ C 2,
0≤t≤T
E
0
T
z¯ui (t) – z¯(t) 2 dt ≤ C 2.
Proof Using Itô’s formula to |xui (t) – x(t)|2 and Gronwall’s inequality, we draw the
conclusion.
(3)
(4)
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎧⎪⎪⎪⎪⎪⎨⎪ d–xdi1y(i1t()t)==[b{xfx(t(t)x)xi1i1(t(t))++bfxyδ((tt))yxi1i1((tt) –+ δfz)(t+)zbi1v(it()t)+vif(z¯t()t])dz¯i1t(t)
+ [σx(t)xi1(t) + σxδ (t)xi1(t – δ) + σvi (t)vi(t)] dW (t)
+ [σ¯x(t)xi1(t) + σ¯xδ (t)xi1(t – δ) + σ¯vi (t)vi(t)] dW¯ (t),
+ EFt [fyδ+ (t)yi1(t + δ)] + fvi (t)vi(t)} dt
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ yxi1i1((Tt))==0G,x(x(T ))xi1(T ),
– zi1(t) dW (t) – z¯i1(t) dW¯ (t),
t ∈ [–δ, 0],
iτx x(T ) xi1(T ) + γiτy y(0) yi1(0) ≤ 0.
We introduce the adjoint equation as
⎪⎪⎪⎪⎪⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ d–pdiq(ti()t=)=+[f+{y[τbf(Ezτxtτ)((Fptt)t)iq[p(btii()τx(tδt+))(t+–fy+τδσl+iδxzτ()(t(qtt–)i)](ktδdi()+Wtp)iδ(+()tt+)σ–¯+xστδ(x[)τtδf)–z¯(τk¯t(ilt+(iy)t(p)δti)–)(k]tif)d(xτt–t(+tl)izpδ¯()it()t])dW¯ (t),
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩⎪⎪⎪⎪⎪ qqpiii(((tT0)))===k––i(γGt+y)(xτσ=y¯((xxτ0kδ¯((i)T(t)t,)+))p=δi)(0k¯T,ip()ti+(+tt)∈δ=)i(x]0T(+x,,(lTTixt()+t)∈),δ}[]d–(tiδ–,=0k1)i,,(2t)).dW (t) – k¯i(t) dW¯ (t),
This equation is also an AFBSDDE. By the existence and uniqueness result in [
12, 13
],
we know that (5) admits a unique solution (pi(t), qi(t), ki(t), k¯i(t)) (i = 1, 2).
Define the Hamiltonian function Hi (i = 1, 2) by
Hi(t, x, y, z, z¯, xδ , yδ+ , v1, v2; pi, qi, ki, k¯i)
= qi, b(t, x, xδ , v1, v2) + ki, σ (t, x, xδ, v1, v2) + k¯i, σ¯ (t, x, xδ, v1, v2)
– pi, f (t, x, y, z, z¯, yδ+ , v1, v2) + li(t, x, y, z, z¯, v1, v2).
Then (5) can be rewritten as a stochastic Hamiltonian system of the following type:
⎪⎪⎪⎪⎪⎧⎪⎪⎪⎨ pd–ipd(i0q()ti()=t=)–=[γ–{yHH(yii(yx0((tt)))),–+ HEFiyδtp+[Hi((tti)–xδ=(δt)0]+,dδt)–]t}∈Hdt[iz––(δtk),0id()Wt,) d(Wt) –(tH)–iz¯ (k¯ti)(td)Wd¯ W(¯t)(,t),
⎪⎪⎪⎪⎩⎪⎪⎪⎪ qqii((tT))==k–i(Gt)xτ=(xk¯(iT(t)))p=i(0T, ) +t ∈i(xT(x,(TT )+),δ],
(5)
(6)
where Hi(t) = Hi(t, x(t), y(t), z(t), z¯(t), xδ (t), yδ+ (t), u1(t), u2(t); pi(t), qi(t), ki(t), k¯i(t)).
Theorem 3.1 Let H1 and H2 hold. Suppose that (u1(·), u2(·)) is an equilibrium point of
our problem and (x(·), y(·), z(·), z¯(·)) is the corresponding state trajectory. Then we have
E
Hivi (t), vi – ui(t) |Gti ≤ 0
(i = 1, 2)
for any vi ∈ Ui a.e., where (pi(·), qi(·), ki(·), k¯i(·)) (i = 1, 2) is the solution of the adjoint
equation (5).
Proof Applying Itô’s formula to q1(·), x11(·) , we get
E –Gxτ x(T ) p1(T ) +
1x x(T ) , x11(T )
= E
0
T
fxτ (t)p1(t), x11(t) + bτxδ (t)q1(t), x11(t – δ)
– EFt bτxδ (t + δ)q1(t + δ) , x11(t) + σxτδ (t)k1(t) + σ¯xδ (t)k¯1(t), x11(t – δ)
τ
– EFt σxτδ (t + δ)k1(t + δ) + σ¯xτδ (t + δ)k¯1(t + δ) , x11(t)
+ q1(t), bv1 (t)v1(t) + k1(t), σv1 (t)v1(t)
+ k¯1(t), σ¯v1 (t)v1(t) – l1x(t), x11(t) dt.
Noticing the initial and terminal conditions, we have
bxδ (t)q1(t), x11(t – δ) – EFt bxδ (t + δ)q1(t + δ) , x11(t) dt
τ τ
0
0
T
δ
bxδ (t)q1(t), x11(t – δ) dt – E
τ
bxδ (t)q1(t), x11(t – δ) dt – E
τ
T+δ
T+δ
δ
T
τ
bxδ (t)q1(t), x11(t – δ) dt
τ
bxδ (t)q1(t), x11(t – δ) dt
Similarly, we also have
τ τ
σxδ (t)k1(t) + σ¯xδ (t)k¯1(t), x11(t – δ)
– EFt σxτδ (t + δ)k1(t + δ) + σ¯xτδ (t + δ)k¯1(t + δ) , x11(t) dt = 0.
E
E
E
T
0
Applying Itô’s formula to p1(·), y11(·) , we obtain
1 1
p1(T ), Gx x(T ) x1(T ) + γy y(0) , y1(0)
– p1(t), fx(t)x11(t) + fv1 (t)v1(t) – l1y(t), y11(t)
– l1z(t), z11(t) – l1z¯ (t), z¯11(t) dt.
Noticing the initial and terminal conditions, we have
fyδ+ (t – δ)p1(t – δ), y11(t) – p1(t), EFt fyδ+ (t)y11(t + δ)
τ
fyδ+ (t – δ)p1(t – δ), y11(t) – p1(t), EFt fyδ+ (t)y11(t + δ)
τ
dt
fyδ+ (t – δ)p1(t – δ), y11(t) dt – E
τ
τ
fyδ+ (t – δ)p1(t – δ), y11(t) dt
fyδ+ (t – δ)p1(t – δ), y11(t) dt – E
τ
τ
fyδ+ (t – δ)p1(t – δ), y11(t) dt
T+δ
T+δ
δ
T
From (7) and (8), we have
E
1x x(T ) , x1(T ) + γy y(0) , y11(0)
1
q1(t), bv1 (t)v1(t) + k1(t), σv1 (t)v1(t) + k¯1(t), σ¯v1 (t)v1(t)
(7)
(8)
– p1(t), fv1 (t)v1(t) – l1y(t), y11(t)
– l1z(t), z11(t) – l1z¯(t), z¯11(t) – l1x(t), x11(t) dt.
Substituting (9) into (4) leads to
E
0
T
for any v1(·) such that u1(·) + v1(·) ∈ Ua1d.
We set
for 0 ≤ t ≤ T , and v¯1(·) ∈ Ua1d. Then we have
1 E
ε
t
t+
H1v1 (s), v¯1(s) – u1(s) ds ≤ 0.
Letting
→ 0, we get
E H1v1 (t), v¯1(t) – u1(t) ≤ 0
(9)
for any admissible control v¯1(·) ∈ Ua1d.
Furthermore, we set v¯1(t) = v11A + u1(t)1 –A for any v1 ∈ U1 and A ∈ Gt1, then it is
obvious that v¯1(·) defined above is an admissible control.
So E H1v1 (t), v¯1(t) – u1(t) = E[1A H1v1 (t), v1 – u1(t) ] ≤ 0 for any A ∈ Gt1. This implies
E H1v1 (t), v1 – u1(t) |Gt1 ≤ 0, a.e.
for any v1 ∈ U1. Repeating the same process to deal with the case i = 2, we can show that
the other equality also holds for any v2 ∈ U2.
Remark 3.1 If (u1(·), u2(·)) is an equilibrium point of non-zero sum differential game and
(u1(·), u2(·)) is an interior point of U1 × U2 for all t ∈ [0, T ], then the inequality in
Theorem 3.1 is equivalent to the following equation:
E Hivi (t)|Gti = 0,
∀vi ∈ Ui a.e. (i = 1, 2).
Proof It is obvious that the “⇐” part holds. For the “⇒” part, we assume that a Nash
equilibrium point (u1(·), u2(·)) takes values in the interior of U1 × U2, a.e. for all t ∈ [0, T ].
Then, for (ω, t) ∈ × [0, T ] and i = 1, 2, there exists a closed ball B¯ui(t)(k) ⊂ Ui, where ui(t)
is the center and k > 0 denotes the radius. For any η ∈ Rki with |η| = 1, both vi = ui(t) + kη
and vi = ui(t) – kη belong to B¯u1(t)(k). Then, by (6), we have E[Hivi (t)|Gti]kη = 0. From the
arbitrariness of η, we get E[Hivi (t)|Gti] = 0, a.e. and finish the proof.
On the other hand, we will aim to build a sufficient maximum principle called the
verification theorem for the equilibrium point under some concavity assumptions of Hi. At
this moment, assumption H2 can be relaxed to the following:
H3. Functions li, i, and γi are differentiable with respect to (x, y, z, z¯, v1, v2), x, and y,
respectively, satisfying the condition that for each (v1(·), v2(·)) ∈ Uad,
li(·, xv(t), yv(t), zv(t), z¯v(t), v1(t), v2(t)) ∈ L1F (0, T ; R), and
liφ (·, xv(·), yv(·), zv(·), z¯v(·), v1(·), v2(·)) ∈ L2F (0, T ; R) for φ = x, y, z, z¯, vi (i = 1, 2).
Theorem 3.2 Let H1 and H3 hold. Let (u1(·), u2(·)) ∈ Ua1d × Ua2d be given and (x(·), y(·),
z(·), z¯(·)) be the corresponding trajectory. Setting
H1v1 (t) = H1 t, x(t), y(t), z(t), z¯(t), xδ (t), yδ+ (t), v1(t), u2(t); p1(t), q1(t), k1(t), k¯1(t) ,
H2v2 (t) = H2 t, x(t), y(t), z(t), z¯(t), xδ (t), yδ+ (t), u1(t), v2(t); p2(t), q2(t), k2(t), k¯2(t) .
Suppose
(x, y, z, z¯, xδ , yδ+ , vi) → Hivi (t) (i = 1, 2),
x →
i(x) (i = 1, 2),
y → γi(y) (i = 1, 2)
are concave functions respectively, and G(x) = MT x, MT ∈ Rm×n, ∀x ∈ Rn. If condition (6)
holds, then (u1(·), u2(·)) is an equilibrium point.
Proof For any v1(·) ∈ Ua1d, let (xv1 (·), yv1 (·), zv1 (·), z¯v1 (·)) be the trajectory corresponding to
the control (v1(·), u2(·)) ∈ Uad. We consider
J1 v1(·), u2(·) – J1 u1(·), u2(·) = A + B + C,
l1 t, v1 (t), v1(t), u2(t) – l1 t, (t), u1(t), u2(t)
dt,
v1 (t) = (xv1 (t), yv1 (t), zv1 (t), z¯v1 (t)).
with
Applying Itô’s formula to p1(·), yv1 (·) – y(·) and taking expectation, we get
C ≤ E
0
T
– p1(t), f v1 (t) – f (t) – H1y(t) + H1yδ+ (t – δ), yv1 (t) – y(t)
– H1z(t), zv1 (t) – z(t) – H1z¯ (t), z¯v1 (t) – z¯(t) dt
– E p1(T ), MT xv1 (T ) – x(T ) ,
(10)
where f (t) = f (t, (t), EFt [yδ+ (t)], u1(t), u2(t)), and f v1 (t) = f (t, v1 (t), EFt [yδv1+ (t)], v1(t),
u2(t)).
Due to 1 being concave on x,
B ≤ E τ1x x(T ) xv1 (T ) – x(T ) .
Applying Itô’s formula to q1(·), xv1 (·) – x(·) and taking expectation, we get
B ≤ E
q1(t), bv1 (t) – b(t) + k1(t), σ v1 (t) – σ (t)
+ k¯1(t), σ¯ v1 (t) – σ¯ (t) – H1x(t) + EFt H1xδ (t + δ) , xv1 (t) – x(t) dt
+ E MT p1(T ), xv1 (T ) – x(T ) ,
τ
where b(t) = b(t, x(t), xδ(t), u1(t), u2(t)) and bv1 (t) = b(t, xv1 (t), xδv1 (t), v1(t), u2(t)), etc.
Moreover, we have
0
T
T
T–δ
0
δ
due to the fact that xv1 (t) = x(t) = ξ (t) for any t ∈ [–δ, 0) and H1xδ (t) = 0 for any t ∈
(T , T + δ].
Similarly, we have
H1yδ+ (t – δ), yv1 (t) – y(t) dt – E
H1yδ+ (t – δ), yv1 (t) – y(t) dt
A = E
H1v1 (t) – H1(t) dt – E
q1(t), bv1 (t) – b(t)
+ k1(t), σ v1 (t) – σ (t) + k¯1(t), σ¯ v1 (t) – σ¯ (t)
– p1(t), f v1 (t) – f (t) dt.
0
T
From (10)-(12), we can obtain
J1 v1(·), u2(·) – J1 u1(·), u2(·)
= A + B + C
≤ E
0
T
H1v1 (t) – H1(t) – H1x(t) + EFt H1xδ (t + δ) , xv1 (t) – x(t)
– H1y(t) + H1yδ+ (t – δ), yv1 (t) – y(t) – H1z(t), zv1 (t) – z(t)
– H1z¯ (t), z¯v1 (t) – z¯(t) dt.
H1xδ (t), xv1 (t – δ) – x(t – δ) dt –
EFt H1xδ (t + δ) , xv1 (t) – x(t) dt
H1xδ (t + δ), xv1 (t) – x(t) dt – E
H1xδ (t + δ), xv1 (t) – x(t) dt
Note that
E
E
T
0
due to the fact that yv1 (t) = y(t) = ϕ(t) for any t ∈ (T , T + δ] and H1yδ+ (t) = 0 for any t ∈
[–δ, 0).
By the concavity of H1, we derive that
In conclusion, with the help of Theorems 3.1 and 3.2, we can formally solve the Nash
equilibrium point (u1(·), u2(·)). We can first use the necessary principle to get the candidate
equilibrium point and then use the verification theorem to check whether the candidate
point is the equilibrium one. Let us discuss a linear-quadratic case.
4 A linear-quadratic case
In this section, we study a linear-quadratic case, which can be seen as a special case of the
general system discussed in Section 3, and aim to give a unique Nash equilibrium point
explicitly. For notational simplification, we suppose the dimension of Brownian motion
d = d¯ = 1 and notations are the same as in the above sections if there is no specific
illustration.
Consider a linear game system with delayed and anticipated states:
⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎧⎪ d–xdvy(vt()t)==+[A[[E(Ct()t(xt)x)vx(vtv()t(t)+)+A+¯F(C¯t()t(x)tyδv)vx(t(δvt)()t+)+B+G1D((tt1))(vzt1v)((vtt1))(++t)BG+¯2(D(tt)2)z¯v(vt2()(tvt)2)](td)t] dW (t),
+ F¯ (t)yδv+ (t) + H1(t)v1(t) + H2(t)v2(t)] dt – zv(t) dW (t)
⎪⎪⎩⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪ yxvv((Tt))==ξ M(t–)T,xz¯vv((tTt)∈)d,[W–¯δ(,t0)y,]v,(tt) ∈= ϕ[0(,tT), ], t ∈ (T , T + δ],
(13)
where all the coefficients are bounded, deterministic matrices defining on [0, T ], ξ (·) ∈
C([–δ, 0]; Rn), ϕ(·) ∈ L2F (T , T + δ; Rm). For any given (v1(·), v2(·)) ∈ Uad, it is easy to know
that (13) admits a unique solution (xv(·), yv(·), zv(·), z¯v(·)). Here, we only consider the case
that xv(·) is driven by one Brownian motion W (·) just for notational simplicity. All the
techniques and proof are similar.
In addition, two players aim to maximize their index functionals for i = 1, 2:
Oi(t)xv(t), xv(t) + Pi(t)yv(t), yv(t)
+ Mixv(t), xv(t) + Niyv(0), yv(0) ,
where Oi(·), Pi(·), Qi(·), Q¯ i(·) are bounded deterministic non-positive symmetric matrices,
Ri(·) is a bounded deterministic negative symmetric matrix, Ri–1(·) is bounded, Mi, Ni are
deterministic non-positive symmetric matrices for i = 1, 2.
According to Theorem 3.1, the Hamiltonian function is given by
Hi(t, x, y, z, z¯, xδ, yδ+ , v1, v2; pi, qi, ki)
= qi, A(t)x + A¯ (t)xδ + B1(t)v1 + B2(t)v2
+ ki, C(t)x + C¯ (t)xδ + D1(t)v1 + D2(t)v2 – pi, E(t)x + F(t)y + G(t)z + G¯ (t)z¯
If (u1(·), u2(·)) is the Nash equilibrium point, then
Oi(t)x, x + Pi(t)y, y + Qi(t)z, z
ui(t) = –Ri–1(t) Biτ (t)qˆ i(t) + Diτ (t)kˆi(t) – Hiτ (t)pˆ i(t) ,
where qˆ i(t) = E[qi(t)|Gt] for i = 1, 2, etc., and (pi(·), qi(·), ki(·)) is the solution of the following
adjoint equation:
(15)
(16)
⎪⎪⎪⎪⎪⎧⎪⎪⎪⎪⎪⎨⎪ d–pdiq(ti()t=)=–[F{QτA(iτt()(tpt))ziq((tti())t]+)d+FW¯τC((tτt()–t+)δk[)iGp(¯tiτ)(t(–t–)Epδτi()(tt–))p–Pii(Q(¯tt)i)(y+t()tEz¯)(F]td)t][tAd¯+Wτ¯([tG(+tτ)(,δt))qpii((tt)+ δ)
⎪⎪⎪⎪⎪⎪⎪⎪⎪ pi(0) = –N+iyC¯(0τ )(,t + δ)kpii((tt+)=δ)0],+ Oti∈(t)[x–(δt,)}0d),t – ki(t) dW (t),
⎪⎩⎪ qi(T ) = –MT pi(T ) + Mix(T ), qi(t) = 0,
We note that the setting Gt ⊆ Ft is very general. In order to get an explicit expression of
the equilibrium point, we suppose Gt = σ {W (s); 0 ≤ s ≤ t} in the rest of this section.
We denote the filtering of state process x(t) by xˆ(t) = E[x(t)|Gt], etc., and note that
E[y(t + δ)|Gt ] = E{[y(t + δ)|Gt+δ ]|Gt} = E[yˆ(t + δ)|Gt ]. By Theorem 8.1 in Lipster and
Shiryayev [
29
] and Theorem 5.7 (Kushner-FKK equation) in Xiong [
30
], we can get the state
filtering equation for (13):
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎧⎪ d–xdˆ(yˆt()t=)=+[A{[E(Ct()(txˆt))(xˆxtˆ()(tt+))+A+¯F(C¯t()t(xˆ)tyδˆ)x((ˆttδ))(t–+) G–(i2t=)1zˆi2=B(1ti)D(t+)i(RGt¯i–)(R1t()i–tzˆ¯1)((Btt))i(B+t)iF(]¯td()t]t)dEWGt [(ytˆ)(,t + δ)]
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ yxˆˆ((Tt))==ξ M(t–)T,xˆ(Tti2=)∈1, H[–i(δt,)y0ˆR(]i–t,1)(=t)ϕBˆ(it()t,)} dtt ∈– (zˆT(t,)TdW+ δ(t]),, t ∈ [0, T ],
where Bi(t) = Biτ (t)qˆi(t) + Diτ (t)kˆi(t) – Hiτ (t)pˆ i(t), and the adjoint filtering equation for (15)
satisfying
⎪⎪⎪⎪⎪⎪⎧⎪⎪⎪⎪⎪⎨ d–pdˆiqˆ(ti()t=)=+[F{[τAG(τtτ)((ptˆt)i)q(ˆpˆtii()(t+t))+F–¯τCQ(tτi((–tt))δkzˆˆ)i(p(ˆtt)i)(]t–d–WEδτ()(tt–)),pˆPii((tt))+yˆ(tE)G]dt[tA¯τ (t + δ)qˆi(t + δ)
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ qpˆˆii((T0))==––NM+iyTC¯(p0ˆτi)((,tT+) δ+)kMˆpˆii((itxtˆ+)(T=δ))0,],+ Oti∈q(ˆti)([xt–ˆ)(δt=,)}00d),,t –tkˆ∈i(t()Td, WT +(t)δ,] (it=∈1[,02,)T. ],
(17)
From Theorems 3.1 and 3.2, it is easy to know that (u1(·), u2(·)) is an equilibrium point
for the above linear-quadratic game problem if and only if (u1(·), u2(·)) satisfies the
expression of (14) with (xˆ, yˆ, zˆ, pˆ i, qˆi, kˆi) (i = 1, 2) being the solution of the coupled triple
dimensions filtering AFBSDDE (16)-(17) (TFBSDDE). Then the existence and uniqueness of the
equilibrium point is equivalent to the existence and uniqueness of the TFBSDDE.
However, we note that TFBSDDE (16)-(17) is so complicated. Fortunately, in some
particular cases, we can make some transactions to link it with a double dimensions filtering
AFBSDDE, called DFBSDDE. Now we present our result in the following.
H4. The dimension of x is equal to that of y: n = m, G¯ (t) ≡ 0 and coefficients Bi(t) = Bi,
Di(t) = Di, Hi(t) = Hi are independent of time t for any i = 1, 2.
Theorem 4.1 Under H4, we assume that one of the following conditions holds true:
(a) D1 = D2 = H1 = H2 ≡ 0 and BiRi–1Biτ S = SBiRi–1Biτ (i = 1, 2);
(b) B1 = B2 = H1 = H2 ≡ 0 and DiRi–1Diτ S = SDiRi–1Diτ (i = 1, 2);
(c) B1 = B2 = D1 = D2 ≡ 0 and HiRi–1Hiτ S = SHiRi–1Hiτ (i = 1, 2),
where Sτ = A(·), A¯(·), C(·), C¯ (·), E(·), F(·), F¯ (·), G(·), MT , Oi(·), Pi(·), Qi(·), Mi, Ni. Then (u1(·),
u2(·)) given by (14) is a unique Nash equilibrium point.
Proof We only prove (a). The same method can be used to get (b) and (c). From the above
discussion, we need to prove only that there exists a unique solution of the coupled
TFBSDDE (16)-(17). In the case that D1 = D2 = H1 = H2 ≡ 0, it becomes
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪ dd–pxdˆˆ(iyˆ(t(t)t)=)==+[+[A{F[E([CτtG(()(ttxˆτt)))(x(ˆpˆxttˆ()i)(t(ptˆ+)t)i)+(A+¯+tF)(CF¯t–¯()t(τxˆ)tQ(yδˆ)tx((iˆ(t–tδt))()δt–+zˆ))(]pˆGtd)i((]Wti2td=)–1zWˆ(B(tδt)i()),Rt–+)i–,1PF¯Bi((iτtt)qˆ)Eyiˆ((Gtt)t)[]]yˆdd(ttt + δ)]} dt – zˆ(t) dW (t),
⎪⎪ –dqˆi(t) = {Aτ (t)qˆi(t) + Cτ (t)kˆi(t) – Eτ (t)pˆ i(t) + EGt [ A¯τ (t + δ)qˆi(t + δ)
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪ xqpˆˆˆ(ii((tT0))=)==ξ –(–tN)M+,iyTC¯(p0ˆtτi)(∈(,tT+[)–δ+δ),kMˆpˆ0ii(](it;xtˆ+)(T=δ))0,]y,ˆ+(TOt)i∈q(=ˆti)([Mxt–ˆ)(δTt=,)xˆ}0k(ˆd)Ti,(tt)–,)=kˆi0(t,)yˆd(tWt)∈=(t(ϕT)ˆ,(,tT),t+∈δt[]0∈., (TT],, T + δ],
(18)
Now we consider another DFBSDDE:
dx(t) = [A(t)x(t) + A¯ (t)x (t) – q(t)] dt + [C(t)x(t) + C¯ (t)x (t)] dW (t),
˜ ˜ ˜δ ˜ ˜ ˜δ
–dy(t) = E(t)x(t) + F(t)y(t) + G(t)z(t) + F¯ (t)
˜ { ˜ ˜ ˜
E
Gt
δ
[y(t + )] dt – z(t) dW (t),
˜ } ˜
τ τ
δ δ
⎪ dp(t) = [F (t)p(t) + F¯ (t – )p(t – ) –
˜ ˜ ˜
⎪
2
i=1
B R
i
–1
i
τ
B P (t)y(t)] dt
i ˜
i
τ
+ [G (t)p(t) –
˜
2
i=1
B R
i
–1
i
τ
B Q (t)z(t)] dW (t),
i ˜
i
τ τ τ τ
¯ δ
–dq(t) = A (t)q(t) + C (t)k˜ (t) – E (t)p(t) + A (t + )
˜ { ˜ ˜
E
Gt
δ
[q(t + )]
˜
(19)
τ
δ
+ C¯ (t + )
E
Gt
δ
[k˜ (t + )] +
2
i=1
B R
i
–1
i
τ
B O (t)x(t) dt – k˜ (t) dW (t),
i ˜ }
i
⎪
ξ
⎪ x(t) = (t),
˜
⎪
⎪ ˜
p(0) = –
⎩ q(T ) =
˜
t
∈
From the commutation relation between matrices, we notice that, if (x, y, z, p , q , kˆ ) (i =
ˆ ˆ ˆ ˆ i ˆ i i
1, 2) is a solution of (18), then (x, y, z, p, q, k˜ ) solves (19), where
˜ ˜ ˜ ˜ ˜
⎪
⎪
⎧
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎧
⎪
⎩
⎪
⎪
⎪
⎨
⎧
⎪
⎩
x(t) = x(t),
˜ ˆ
⎪
p(t) = B R
⎨ ˜ 1
q(t) = B R
˜ 1
⎪ k˜ (t) = B R
1
y(t) = y(t),
˜ ˆ
z(t) = z(t),
˜ ˆ
–1
1
–1
1
–1
1
τ
τ
B p (t) + B R
ˆ 1 2
1
B q (t) + B R
ˆ 1 2
1
–1
2
–1
2
τ
τ
B p (t),
ˆ 2
2
B q (t),
ˆ 2
2
τ
ˆ
B k (t) + B R
1 2
1
–1
2
τ
ˆ
B k (t).
2
2
We let
p(t) = B R
¯ 1
q(t) = B R
¯ 1
k¯ (t) = B R
1
–1
1
–1
1
–1
1
τ
τ
B p (t) + B R
ˆ 1 2
1
B q (t) + B R
ˆ 1 2
1
–1
2
–1
2
τ
τ
B p (t),
ˆ 2
2
B q (t),
ˆ 2
2
τ
ˆ
B k (t) + B R
1 2
1
–1
2
τ
ˆ
B k (t).
2
2
p(t) = p(t) = B R
˜ ¯ 1
q(t) = q(t) = B R
˜ ¯ 1
k˜ (t) = k¯ (t) = B R
1
–1
1
–1
1
–1
On the other hand, if (x, y, z, p, q, k˜ ) is a solution of (19), we can let x(t) = x(t), y(t) = y(t),
˜ ˜ ˜ ˜ ˜ ˆ ˜ ˆ ˜
z(t) = z(t). From the existence and uniqueness result of SDDE and ABSDE (see [
12, 13
]),
ˆ ˜
we can get (p (t), q (t), kˆ (t)) from the following filtering AFBSDDE:
ˆ i ˆ i i
τ τ τ
δ δ
dp (t) = [F (t)p (t) + F¯ (t – )p (t – ) – P (t)y(t)] dt + [G (t)p (t)
ˆ i ˆ i ˜ i i ˆ ˆ i
τ τ τ τ
¯ δ
–dq (t) = A (t)q (t) + C (t)kˆ (t) – E (t)p (t) + A (t + )
ˆ i { ˆ i i ˆ i
E
Ft
δ
[q (t + )]
ˆ i
– Q (t)z(t)] dW (t),
i ˆ
τ
δ
+ C¯ (t + )
E
Ft
ˆ
– k (t) dW (t),
i
δ
[kˆ (t + )] + O (t)x(t) dt
i i ˆ }
t
∈
[0, T ],
⎪ p (0) = –N y(0),
ˆ i i ˆ
p (t) = 0,
ˆ i
t
∈
δ
[– , 0),
⎩ ˆ i
q (T ) = –M p (T ) + M x(T ),
T ˆ i i ˆ
ˆ
q (t) = k (t) = 0,
ˆ i i
t
∈
δ
(T , T + ].
By Itô’s formula and the uniqueness result of the solution of the SDDE and ABSDE for
fixed (x( ), y( ), z( )), we have
ˆ · ˆ · ˆ ·
(20)
(21)
Then (xˆ, yˆ, zˆ, pˆ i, qˆi, kˆi) (i = 1, 2) is a solution of (18). Moreover, the existence and
uniqueness of (19) is equivalent to the existence and uniqueness of (18). According to the
monotonicity condition in [
19, 28
], it is easy to check that DFBSDDE (19) satisfies the condition
and it has a unique solution. So TFBSDDE (18) admits a unique solution. We complete
the proof.
5 An example in finance
This section is devoted to studying a pension fund management problem under partial
information with time-delayed surplus arising from the financial market, which
naturally motivates the above theoretical research. The financial market is the Black-Scholes
market, while the pension fund management framework comes from Federico [
31
]. To
get close to reality, we study this problem in the case when the performance criterion
Ji(v1(·), v2(·)) is measured by a criterion involving risk. If we interpret risk in the sense of a
convex risk measure, it can be performed by a nonlinear expectation called g-expectation,
which can also be used to represent a nonlinear human preference in behavioral
economics (see [
3, 32–35
] and recent articles [
25, 36, 37
]). Now we introduce it in
detail.
In the following, we only consider the one-dimensional case just for simplicity of
notations. First, we give the definition of convex risk measure and its connection with
gexpectation.
Definition 5.1 ([
32
]) Let F be the family of all lower bounded FT -measurable random
variables. A convex risk measure on F is a functional ρ : F → R such that
(a) (convexity) ρ(λX1 + (1 – λ)X2) ≤ λρ(X1) + (1 – λ)ρ(X2), X1, X2 ∈ F, λ ∈ (0, 1),
(b) (monotonicity) if X1 ≤ X2 a.e., then ρ(X1) ≥ ρ(X2), X1, X2 ∈ F,
(c) (translation invariance) ρ(X + m) = ρ(X) – m, X ∈ F, m ∈ .
R
The convex risk measure is a useful tool widely applied in the measurement of financial
positions. For the financial interpretation (see, e.g., [
34
]), the property (a) in Definition 5.1
means that the risk of a diversified position is not more than the weighted average of the
individual risks; (b) means that if portfolio X2 is better than X1 under almost all scenarios,
then the risk of X2 should be less than the risk of X1; (c) implies that the addition of a sure
amount of capital reduces the risk by the same amount. It is also a generalization of the
concept of coherent risk measure in [
38
]. Here, if ρ(X) ≤ 0, then position X is called
acceptable, and ρ(X) represents the maximal amount that investors can withdraw without
changing the acceptability of X. If ρ(X) ≥ 0, then X is called unacceptable and ρ(X)
represents the minimal extra wealth that investors have to add into the position X to make it
acceptable.
Consider the following BSDE:
⎧⎨ –dy(t) = g(t, y(t), z(t)) dt – z(t) dW (t),
⎩ y(T ) = ξ .
(22)
Under certain assumptions, (22) has a unique solution (y(·), z(·)). If we also set g(t, y,
z)|z=0 ≡ 0, we can make the definition as follows.
Definition 5.2 ([
3, 35
]) For each ξ ∈ FT , we call
Eg (ξ )
y(0)
the generalized expectation (g-expectation) of ξ related to g.
The well-known Allais [
39
] and Ellsberg [
40
] paradox indicates that the classical
vonNeumann-Morgenstern linear expected utility theory (here we mean that the linear
expectation E is used) cannot exactly express people’s subjective preferences or criterion
involving risk. Then one naturally tries to replace E by some kind of nonlinear expectation.
From Definition 5.2, the g-expectation Eg (·) based on the BSDE possesses all the properties
that E has, except the linearity (see [
3
]). It can be seen as a (subject) nonlinear preference
and is closely related to the stochastic differential utility (see, e.g., [
4
]). It is obvious that
when g(·) = 0, Eg is reduced to the classical expectation E.
Here, we present the g-expectation as a nonlinear measurement of risk and give the
connection between the convex risk measure and the g-expectation as follows (see [
33,
41
] for more details).
Definition 5.3 The risk ρ(ξ ) of the random variable ξ ∈ L2F ( ; R) (ξ can be regarded as
a financial position in the financial market) is defined by
ρ(ξ )
Eg [–ξ ] = y(0),
where Eg [·] is defined in Definition 5.2 with ξ replaced by –ξ . Here, g is independent of y
and is convex with respect to z.
Assuming that there are two assets in the financial market for the pension fund managers
to invest:
⎧⎪ dS0(t) = r(t)S0(t) dt,
⎪
⎨ dS1(t) = μ(t)S1(t) dt + σ (t)S1(t) dW (t),
⎪⎪⎩ S0(0) = 1, S1(0) > 0,
where S1(·) is a risky finance asset price and S0(·) is one risk-free asset price. μ(·) is an
appreciation rate of the asset process, and σ (·) is the volatility coefficient. We assume that
μ(·), r(·) and σ (·) are deterministic bounded coefficients, and σ –1(·) is bounded.
Suppose that there are two pension fund managers (players) working together to invest
the risk-free and risky assets. In the real financial market, it is reasonable for the investors
to make decisions based on the historical price of the risky asset S1(·). So the observable
filtration can be set as Gt = σ {S1(s)|0 ≤ s ≤ t}, and it is clear that Gt = FtW = σ {W (s)|0 ≤
s ≤ t}. The pension fund wealth x(·) can be modeled by
⎧⎪ dx(t) = (r(t)x(t) + (μ(t) – r(t))π (t) – α(x(t) – x(t – δ)) – c1(t) – c2(t)) dt
⎨⎪ + π (t)σ (t) dW (t) + σ¯ (t) dW¯ (t),
⎪⎪⎩ x(0) = x0 > 0, x(t) = 0, t ∈ [–δ, 0).
(23)
(24)
(25)
(26)
Here, we denote by π (t) the amount of portfolio invested in the risky asset at time t, and
α(x(t) – x(t – δ)) represents the surplus premium to fund members or their capital
transfusions depending on the performance of fund growth during the past period with
parameter α > 0 (see, e.g., [
16, 17
]). Meanwhile, there is an instantaneous consumption rate
ci(t) for manager i (i = 1, 2). We assume that the value of x(·) is not only affected by the
risky asset, but also by some practical phenomena like the physical inaccessibility of some
economic parameters, inaccuracies in measurement, insider trading or the information
asymmetry, etc. (see, e.g., [
42, 43
]). Here, σ¯ (·) represents the instantaneous volatility
affected by these unobservable factors, FtW¯ represents the unobservable filtration
generated by W¯ (·). We set x(t) be adapted to the filtration Ft generated by Brownian motion
(W (·), W¯ (·)), and the control processes ci(t) (i = 1, 2) be adapted to the observation
filtration Gt ⊆ Ft .
The controlled process ci(·) (i = 1, 2) is called admissible for manager i if ci(t) > 0 is
adapted to the filtration Gt at time t, ci(t) ∈ L2(0, T ; R), and the family of admissible control
(c1(·), c2(·)) is denoted by C1 × C2.
We assume that the insurance company hopes for more terminal capital with less risks
and more consumption ci(·). According to Definitions 5.1 and 5.3, we can define the cost
functional as
Jig c1(·), c2(·) = –KiEg –x(T ) + E
0
T
e–βtLi ci(γt)γ dt, i = 1, 2,
where Ki, Li are positive constants representing the different extent of preferences of two
managers. β is a discount factor and 1 – γ ∈ (0, 1) is a constant called the Arrow-Pratt
index of risk aversion. Here, we set g be a linear form as g(·, y(·), z(·)) = g(·)z(·), where g(·)
is a deterministic bounded coefficient.
Then our problem is naturally to find an equilibrium point (c1∗(·), c2∗(·)) ∈ C1 × C2 such
that
⎨⎧ J1g (c1∗(·), c2∗(·)) = supc1∈C1 J1 (c1(·), c2∗(·)),
g
g g
⎩ J2 (c1∗(·), c2∗(·)) = supc2∈C2 J2 (c1∗(·), c2(·)).
Then our problem can be reformulated as
and
⎧⎪ dx(t) = (r(t)x(t) + (μ(t) – r(t))π (t) – α(x(t) – x(t – δ))
⎪⎪⎪⎪⎪⎪⎪ – c1(t) – c2(t)) dt + π (t)σ (t) dW (t) + σ¯ (t) dW¯ (t),
⎨ –dy(t) = g(t)z(t) dt – z(t) dW (t), t ∈ [0, T ],
⎪⎪⎪⎪⎪⎪⎪⎩⎪ yx((T0))==x–0x,(T ), x(t) = 0, t ∈ [–δ, 0),
Jig c1(·), c2(·) = E
e–βtLi ci(γt)γ dt – Kiy(0), i = 1, 2.
Now we will apply the theoretical results obtained in Section 3 to solve the above game
problem. The Hamiltonian function is in the form of
Hi t, x(t), y(t), z(t), xδ (t), c1(t), c2(t); pi(t), qi(t), ki(t), k¯i(t)
= qi(t) r(t)x(t) + μ(t) – r(t) π (t) – α x(t) – x(t – δ) – c1(t) – c2(t)
+ ki(t)π (t)σ (t) + k¯i(t)σ¯ (t) – pi(t)g(t)z(t) + e–βt Li ci(γt)γ ,
where the adjoint process (pi(·), qi(·), ki(·), k¯i(·)) satisfies
⎧⎪ dpi(t) = g(t)pi(t) dW (t),
⎪⎪⎪⎨⎪ –dqi(t) = {(r(t) – α)qi(t) + αEFt [qi(t + δ)]} dt – ki(t) dW (t) – k¯i(t) dW¯ (t),
qi(t) = ki(t) = k¯i(t) = 0,
Then we use the necessary maximum principle (Theorem 3.1) to find a candidate
equilibrium point:
where qˆ i(t) = E[qi(t)|Gt ] (i = 1, 2).
Now we have to deal with qˆ i(t), the optimal filtering of qi(t) on the observation Gt . We
also set pˆ i(t) = E[pi(t)|Gt]. Note that
E E qi(t + δ)|Ft |Gt = E qi(t + δ)|Gt = E E qi(t + δ)|Gt+δ |Gt = E qˆ i(t + δ)|Gt .
(27)
⎪⎪⎪⎪⎪⎩ qpii((T0))==Kpii,(T ),
1
c1∗(t) = L1–1eβt qˆ 1(t) γ –1 ,
1
c2∗(t) = L2–1eβt qˆ 2(t) γ –1 ,
Then, by Theorem 8.1 in [
30
], we have
⎧⎪ dpˆ i(t) = g(t)pˆ i(t) dW (t),
⎪⎪⎨⎪⎪ –dqˆ i(t) = {(r(t) – α)qˆ i(t) + αEGt [qˆ i(t + δ)]} dt – kˆi(t) dW (t),
⎪⎪ pˆi(0) = Ki,
⎪⎪⎪⎩ qˆ i(T ) = pˆ i(T ),
qˆ i(t) = kˆi(t) = 0,
From (28), we can derive the explicit expression of pˆ i(t) as
pˆ i(t) = Ki exp
0
t
g(s) dW (s) – 1
2 0
t
g2(s) ds > 0,
which is a Gt -exponential martingale.
By Theorem 5.1 in [
13
], we can prove qˆ i(t) ≥ 0, t ∈ [0, T ]. Thus ci∗(t) > 0 for all t ∈ [0, T ].
Next, we will solve the anticipated BSDE of qˆ i(t) recursively. This method can also be
found in [
44, 45
].
qˆi(t) = exp
r(s) – α ds EGt pˆ i(T ) = exp
r(s) – α ds pˆ i(t), t ∈ [T – δ, T ].
t
T
From Proposition 5.3 in [
4
], (qˆi(t), kˆi(t)) is Malliavin differentiable and {Dtqˆi(t); T – δ ≤
t ≤ T } provides a version of {kˆi(t); T – δ ≤ t ≤ T }, i.e.,
kˆi(t) = Dtqˆi(t) = exp
r(s) – α ds Dtpˆ i(t), t ∈ [T – δ, T ].
(2) If we have solved ABSDE (28) on the interval [T – nδ, T – (n – 1)δ] (n = 1, 2, . . .), and
the solution {(qˆi(t), kˆi(t)); T – nδ ≤ t ≤ T – (n – 1)δ} is Malliavin differentiable, then we
continue to consider the solvability on the next interval [T – (n + 1)δ, T – nδ], where we
can rewrite ABSDE (28) as follows:
qˆi(t) = qˆi(T – nδ) +
r(s) – α qˆi(s) + αEGs qˆi(s + δ) ds –
kˆi(s) dW (s).
t
T–nδ
(1) When t ∈ [T – δ, T ], the ABSDE in (28) becomes a standard BSDE (without
anticipation):
t
T
r(s) – α qˆi(s) ds –
kˆi(s) dW (s), t ∈ [T – δ, T ].
qˆi(t) = pˆ i(T ) +
Obviously, we have
t
T
t
T
t
T
t
T–nδ
t
t
s
s
t
t
T–nδ
t
T–nδ
T–nδ
t
T–nδ
exp
exp
qˆi(t) = exp
+ α
kˆi(t) = exp
r(s) – α ds EGt Dtqˆi(T – nδ)
r(s) – α ds EGt qˆi(T – nδ)
r(η) – α dη EGt qˆi(s + δ) ds,
We note that {(qˆi(s + δ), kˆi(s + δ)); t ≤ s ≤ T – nδ} has been solved and is Malliavin
differentiable. So the same discussion leads to {(qˆi(t), kˆi(t)); T – (n + 1)δ ≤ t ≤ T – nδ} is Malliavin
differentiable, and
+ α
r(η) – α dη EGt Dtqˆi(s + δ) ds
for any t ∈ [T – (n + 1)δ, T – nδ], i = 1, 2.
We notice that all the conditions in the verification theorem (Theorem 3.2) are satisfied,
then Theorem 3.2 implies that (c1∗(·), c2∗(·)) given by (27) is an equilibrium point.
Proposition 5.1 The investment problem (23)-(24) admits an equilibrium point (c1∗(·),
c2∗(·)) which is defined by (27).
6 Conclusions
To the author’s best knowledge, this article is the first attempt to study the non-zero sum
differential game problem of AFBSDDE under partial information. Throughout this paper,
there are four distinguishing features worthy of being highlighted. First, we considered the
time-delayed system, which has wide applications and can explain various past-dependent
situations. Second, we studied the game problem with multiple players and aimed to find
the Nash equilibrium point other than the optimal control. We established a necessary
condition (maximum principle) and a sufficient condition (verification theorem) by virtue
of the duality and convex variation approach. Third, we discussed an LQ system under the
partial information condition. Applying the stochastic filtering formula, we derived the
filtering equation and proved the existence and uniqueness of the filtering equation and the
corresponding Nash equilibrium point. Fourth, we solved a pension fund management
problem with a nonlinear expectation to measure the risk (convex risk measure) and
obtained the explicit solution.
Acknowledgements
This work was supported by the Natural Science Foundation of China (No. 61573217, No. 11601285), the Natural Science
Foundation of Shandong Province (No. ZR2016AQ13), the National High-level Personnel of Special Support Program of
China, and the Chang Jiang Scholar Program of Chinese Education Ministry. The author would like to thank Prof. Zhen Wu
for his valuable suggestions.
Competing interests
The author declares that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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