MATHEMATICAL CONTROL AND
RELATED FIELDS
Volume 7, Number 2, June 2017
doi:10.3934/mcrf.2017010
pp. 289–304
NASH EQUILIBRIUM POINTS OF RECURSIVE NONZERO-SUM
STOCHASTIC DIFFERENTIAL GAMES WITH UNBOUNDED
COEFFICIENTS AND RELATED MULTIPLE
DIMENSIONAL BSDES
Rui Mu
Center for Financial Engineering
Soochow University
Suzhou 215006, China
Zhen Wu∗
School of Mathematics
Shandong University
Jinan 250100, China
(Communicated by Qi Lü)
Abstract. This paper is concerned with recursive nonzero-sum stochastic differential game problem in Markovian framework when the drift of the state
process is no longer bounded but only satisfies the linear growth condition. The
costs of players are given by the initial values of related backward stochastic
differential equations which, in our case, are multidimensional with continuous
coefficients, whose generators are of linear growth on the volatility processes
and stochastic monotonic on the value processes. We finally show the wellposedness of the costs and the existence of a Nash equilibrium point for the
game under the generalized Isaacs assumption.
1. Introduction. In this article, we discuss a recursive nonzero-sum stochastic differential game (NZSDG for short) under Markovian framework. Generally speaking,
stochastic differential game theory deals with conflict or cooperate problems in a
dynamic system which is influenced by multiple players. Let us introduce the setting of the problem briefly. Assume that we have a system which is described as
follows:
dxt = σ(t, xt )dBt for t ≤ T and x0 = x,
(1)
where B is a Brownian motion. This system can also be controlled by two players
which we represent by weak formulation of a stochastic differential equation (SDE
for short):
dxt = f (t, xt , ut , vt )dt + σ(t, xt )dBtu,v for t ≤ T and x0 = x.
(2)
2010 Mathematics Subject Classification. Primary: 49N70, 60H10; Secondary: 91A15.
Key words and phrases. Recursive utility, nonzero-sum stochastic differential games, Nash
equilibrium point, backward stochastic differential equations, Isaacs condition.
The first author is supported by the Natural Science Foundation for Young Scientists of Jiangsu
Province, P.R. China (No. BK20160300); The second author is supported by National Natural
Science Foundation of China (61573217, 11626247, 11125102), the National High-level personnel
of special support program and the Chang Jiang Scholar Program of Chinese Education Ministry.
∗ Corresponding author: Zhen Wu.
289
290
RUI MU AND ZHEN WU
The process B u,v is a new Brownian motion generated from B by applying Girsanov’s transformation. The precise analysis will be introduced in the following
text. Processes u = (ut )t≤T and v = (vt )t≤T represent the control actions of the
two players imposed on this system. Indeed, the controls are not free, which bring
some costs for players. What we discussed is a recursive type of cost functional,
which is defined by the initial value of the following backward stochastic differential
equation (BSDE for short): for i = 1, 2,
RT yti,u,v = g i (xT ) + t zsi,u,v σ −1 (s, xs )f (s, xs , us , vs ) + hi (s, xs , ysi,u,v , us , vs ) ds
R T i,u,v
− t zs dBs .
(3)
The costs are defined by J i (u, v) = y0i,u,v for players i = 1, 2, respectively. The
objective of this game model is to find a Nash equilibrium point (u∗ , v ∗ ) such that,
J 1 (u∗ , v ∗ ) ≤ J 1 (u, v ∗ ) and J 2 (u∗ , v ∗ ) ≤ J 2 (u∗ , v)
for any admissible control (u, v). This is actually to say that both of the two players
would like to minimize their costs and no one can cut more by unilaterally changing
her own control.
In the following, let us discuss the main contribution of our work, as well as
the main difference between recursive cost and the classical one. The concept of
stochastic differential recursive utility has been considered by Duffie and Epstein
in [3] which extends the classical utility. The recursive one involves instantaneous
utility depending not only on instantaneous consumption rate but also on the future utility. The manner of using solutions of BSDEs to describe cost functionals
of stochastic differential game is initially inspired by [5], where some formulations
of recursive utilities and their properties are also discussed. In BSDE (3), If the
functions hi are independent on parameters y i , then by applying Girsanov’s transRT
formation, the costs J i will be reduced to E u,v [g i (xT ) + 0 hi (s, xs , us , vs )], which
is the accumulation of the instantaneous cost hi and the terminal cost g i . This is
the classical structure of non-recursive cost functions as studied in [9], [11]. Some
recursive optimal control problems are studied by [19]. There are also works study
the zero-sum case of recursive game, such as [20]. Readers are referred to a series of
works by Hamadène for research on classical NZSDGs without the recursive part,
say [7, 8, 9] and the references therein. Our main contribution is that we study a
nonzero-sum game with recursive cost which is defined by initial value of BSDE (3).
Besides, assumptions on the coefficients are irregular. We finally show the existence
of a Nash equilibrium point.
The method of BSDEs has been shown as an efficient tool to deal with the
recursive nonzero-sum stochastic differential game, see the works by [13], [20] for
example. A complete review on BSDEs theory as well as some applications are
introduced in a survey paper by [18]. The connection of BSDE with NZSDG and
some other popular methods to deal with game problem, such as partial differential
equation, are presented in a celebrated survey paper by [2].
In the present paper, we study the recursive NZSDG through BSDE technique in
the same line as [20]. However in [20], the drift function f of the state process in (2)
is bounded or almost equivalent to bounded one. This boundedness is important
when we consider the related BSDEs since this guarantees the good Liptsitz property
of the generator of the corresponding BSDE with respect to z component. However,
this restriction is too strict to some extent. Therefore, the motivation of our work
RECURSIVE SDG AND RELATED BSDE
291
is to relax this limitation on f . To instead, we consider a drift f which is of linear
growth on the state process x. This has already been considered in some classical
game problems without recursive part by [9] and [10]. To our knowledge, this general
recursive case has not been studied in literatures. This is our main innovation.
Besides, under appropriate assumptions on function hi (see Assumption 3), we find
that the generator of BSDE (3) is not regular, which is of stochastic linear growth
on z and stochastic monotonic on y. This BSDE is new and has not been studied
before. We give the existence of solutions for BSDE (3) which provides the wellposedness of the cost function. This result is summerized as Theorem 3.1 which
we mainly deal with in this article. Then, with the help of the generalized Isaacs
condition and a kind of multiple dimensional BSDE (22) below (whose existence of
solutions has been show in [16]), we show the existence of NEP for this recursive
NZSDG by applying comparison properties between BSDEs.
Finally, we point out that this work establishes a model involving only two players, however, it can be generalized to multiple players case following the same way
without any difficulty.
The rest of this work is organized as follows:
In Section 2, we give the precise statement of the recursive game problem and
some assumptions on coefficients. Section 3 is devoted to the well-posedness of
cost functionals, i.e., the existence of solutions of BSDE (3). The idea is to take a
partition of interval [0, T ] and firstly solve this BSDE in a small interval [T − δ, T ],
then, extend it backwardly to the whole interval. The existence of Nash equilibria is
shown in Section 4. Finally, in Section 5, we give a simple one-dimensional example.
We can clearly see that Nash equilibrium point exists for the recursive game under
our assumptions.
2. Statement of the problem. In this section, we will give some basic notations,
the preliminary assumptions throughout this paper, as well as the statement of the
recursive nonzero-sum stochastic differential game. Let T be fixed and let (Ω, F, P)
be a probability space on which we define a d-dimensional Brownian motion B =
(Bt )0≤t≤T with integer d ≥ 1. Let us denote by F = {Ft , 0 ≤ t ≤ T }, the natural
filtration generated by the process B and augmented by NP the P-null sets, i.e.
Ft = σ{Bs , s ≤ t} ∨ NP .
Let P be the σ-algebra on [0, T ] × Ω of Ft -progressively measurable sets. Let
p ∈ [1, ∞) be a real constant and t ∈ [0, T ] be fixed. We then define the following
spaces: Lp = {ξ : Ft -measurable and Rm -valued random variable s.t. E[|ξ|p ] < ∞};
p
St,T
= {ϕ = (ϕs )t≤s≤T : P-measurable and Rm -valued s.t. E[sups∈[t,T ] |ϕs |p ] < ∞}
RT
p
p
and Ht,T
= {ϕ = (ϕs )t≤s≤T : P-measurable and Rm -valued s.t. E[( t |ϕs |2 ds) 2 ] <
p
p
∞}. Hereafter, S0,T
and H0,T
are simply denoted by STp and HTp .
The following assumptions are in force throughout this paper. Let σ be the function defined by: σ : [0, T ]×Rm −→ Rm×m which satisfies the following assumption.
Assumption 1.
(i): σ is uniformly Lipschitz w.r.t x. i.e. there exists a constant C1 such that, ∀t ∈ [0, T ], ∀x, x0 ∈ Rm , |σ(t, x) − σ(t, x0 )| ≤ C1 |x − x0 |.
(ii): σ is invertible and bounded and its inverse is bounded,
i.e., there
exits a
constant Cσ such that ∀(t, x) ∈ [0, T ] × Rm , |σ(t, x)| + σ −1 (t, x) ≤ Cσ .
Remark 1 (Uniform elliptic condition). Under Assumption 1, we can verify that,
there exists a real constant > 0 such that for any (t, x) ∈ [0, T ] × Rm , .I ≤
σ(t, x).σ > (t, x) ≤ −1 .I where I is the identity matrix of dimension m.
292
RUI MU AND ZHEN WU
Suppose that we have a system whose dynamic is described by a SDE as follows:
for (t, x) ∈ [0, T ] × Rm ,
Z s

 X t,x = x +
σ(u, Xut,x )dBu , s ∈ [t, T ] ;
s
(4)
t
 t,x
Xs = x, s ∈ [0, t].
The solution X = (Xst,x )s≤T exists and is unique under Assumption 1. (cf. [17],
p.289). We recall here two well-known results associate to the integrability of the
solution. For any fixed (t, x) ∈ [0, T ] × Rm , p ≥ 2, it holds that, P-a.s.
t,x p
E sup |Xs | ≤ C(1 + |x|p ),
(5)
0≤s≤T
where the constant C is only depend on the Lipschitz coefficient and the bound of
σ. In addition, for a constant α ∈ (0, 2), we also have, P-a.s.
E exp sup |Xst,x |α
< ∞.
(6)
0≤s≤T
We consider a two players game model in this article for simplicity. The general
multiple players case is a straightforward adaptation. Each of the two players
imposes a control type strategy to this system. Let us now denote by U1 and U2
two compact metric spaces and let M1 (resp. M2 ) be the set of P-measurable
processes u = (ut )t≤T (resp. v = (vt )t≤T ) with values on U1 (resp. U2 ). We denote
by M the set M1 × M2 . Hereafter M is called the set of admissible control.
We then introduce the following Borelian function f : [0, T ] × Rm × U1 × U2 −→
m
R which satisfies:
Assumption 2. For any (t, x) ∈ [0, T ] × Rm , (u, v) 7→ f (t, x, u, v) is continuous
on U1 × U2 . Moreover f is of linear growth w.r.t x, i.e. there exists a constant Cf
such that |f (t, x, u, v)| ≤ Cf (1 + |x|), ∀(t, x, u, v) ∈ [0, T ] × Rm × U1 × U2 .
For (u, v) ∈ M, let Pu,v
t,x be the measure on (Ω, F) defined as follows:
Z .
u,v
−1
t,x
t,x
dPt,x = ET
σ (s, Xs )f (s, Xs , us , vs )dBs dP
(7)
0
where for any (Ft , P)-continuous local martingale M = (Mt )t≤T ,
E(M ) := (exp{Mt − hM it /2})t≤T .
(8)
The notation h i. denotes the quadratic variation process. By Assumptions 1 and
2, we know Pu,v
t,x is a new probability on (Ω, F) (see Appendix A, [4] or [17]
p.200).
From
Girsanov’s theorem ([6], pp.285-301), the process B u,v := (Bs −
R s −1
t,x
σ
(r,
X
)f
(r, Xrt,x , ur , vr )dr)s≤T is a (Fs , Pu,v
t,x )-Brownian motion and the pror
0
t,x
cess (Xs )s≤T satisfies the following SDE in weak formulation:
(
dXst,x = f (s, Xst,x , us , vs )ds + σ(s, Xst,x )dBsu,v , s ∈ [t, T ] ;
(9)
Xst,x = x, s ∈ [0, t].
Actually, the process (Xst,x )s≤T is not adapted with respect to the filtration
generated by the Brownian motion (Bsu,v )s≤T . Therefore, it is known as the weak
solution of (9). Besides, properties (5) and (6) hold true, as well, for the expectation
under the probability Pu,v
t,x (see [11], Lemma 3.3-(ii)).
Now, this system is controlled by two players through dynamic function f . The
control actions are not free, which bring the players corresponding costs, or payoffs
RECURSIVE SDG AND RELATED BSDE
293
in some circumstances. Before introducing the costs, we first present the following
Borelian functions hi (resp. g i ) : [0, T ] × Rm × R × U1 × U2 (resp. Rm ) −→ R, i =
1, 2 which satisfy
Assumption 3.
(i): For any (t, x, y) ∈ [0, T ]×Rm ×R, (u, v) 7→ hi (t, x, y, u, v)
is continuous on U1 × U2 , i = 1, 2. Moreover hi is of polynomial growth w.r.t.
x and of linear growth w.r.t. y and satisfies the stochastic monotonic property
on y, for i = 1, 2. i.e. there exist constants Ch , γ ≥ 0 and α ∈ (0, 2) such that
∀(t, x, y, y0, u, v) ∈ [0, T ] × Rm × Rm × R2 × U1 × U2 ,
(a) |hi (t, x, y, u, v)| ≤ Ch (1 + |x|γ + |y|);
(b) (y − y0)(hi (t, x, y, u, v) − hi (t, x, y0, u, v)) ≤ Ch (1 + |x|α )|y − y0|2 ;
(ii): The function g i , i = 1, 2, is of polynomial growth with respect to x, i.e.
there exist constants Cg and γ ≥ 0 such that |g i (x)| ≤ Cg (1 + |x|γ ), ∀x ∈
Rm , for i = 1, 2.
Now, let x0 ∈ Rm be fixed. The costs (or payoffs) of the players for their controls
(u, v) ∈ M are given by the initial values of related BSDEs. More precisely, we
define:
i,(u,v)
J i (u, v) = Y0
,
i = 1, 2
(10)
i,(u,v)
and the process (Yt
i,(u,v)
Yt
)t≤T satisfies the following BSDE: for i = 1, 2,
Z Th
0,x0
i
= g (XT ) +
Zsi,(u,v) σ −1 (s, Xs0,x0 )f (s, Xs0,x0 , us , vs )
t
+
i
hi (s, Xs0,x0 , Ysi,(u,v) , us , vs )
Z
ds −
(11)
T
Zsi,(u,v) dBs .
t
Actually, BSDE (11) has solution in some appropriate space which will be shown
in the next section (see Theorem 3.1). Therefore, the costs (10) are well-defined.
However, for the integrity of the statement of the problem, we would like to go
ahead and put the proof of the existence later. This manner of definition of the cost
functional has already been considered in [20], [13]. Indeed, if hi is independent of
y component, which is the unrecursive case, we can express the value process of the
RT
i,(u,v)
0,x0
i
BSDE (11) by Yt
= Eu,v
) + t hi (s, Xs0,x0 , us , vs )ds|Ft ] where Eu,v
0,x0 [g (XT
0,x0
u,v
is the expectation under the probability P0,x
.
It
is
easy
to
check
this
conditional
0
expectation is well-defined from the assumptions on g i and hi . Apparently, in this
RT
i,(u,v)
0,x0
i
case cost function is J i (u, v) = Y0
= Eu,v
) + t hi (s, Xs0,x0 , us , vs )ds]
0,x0 [g (XT
since F0 is nothing but some null measure sets. Then functions hi and g i can
be viewed as the instantaneous cost and the terminal cost respectively for player
i = 1, 2. This situation is coincident with the classical nonzero-sum stochastic
differential game model as in the work by Hamadène (see [8]).
u,v
u,v
Hereafter Eu,v
(resp.Pu,v ).
0,x0 (resp. P0,x0 ) will be simply denoted by E
What we concerned in this article is to find an admissible control (u∗ , v ∗ ) such
that
J 1 (u∗ , v ∗ ) ≤ J 1 (u, v ∗ ) and J 2 (u∗ , v ∗ ) ≤ J 2 (u∗ , v), ∀(u, v) ∈ M.
The control (u∗ , v ∗ ) is called a Nash equilibrium point for the recursive NZSDG.
It reads that each player chooses her best control, while, an equilibrium is a pair
of controls, such that, when applied, no player will lower any cost by unilaterally
changing her own control.
294
RUI MU AND ZHEN WU
Now, we define the Hamiltonian functions Hi , i = 1, 2, of the game from [0, T ] ×
Rm × R × Rm × U1 × U2 into R by: Hi (t, x, y, p, u, v) = pσ −1 (t, x)f (t, x, u, v) +
hi (t, x, y, u, v).
Obviously, under Assumptions 2 and 3, Hi satisfies the following hypothesis, for
each (t, x, y, y 0 , z, z 0 , u, v) ∈ [0, T ] × Rm × R2 × R2m × U1 × U2 ,

|H (t, x, y, z, u, v)| ≤ Cσ Cf (1 + |x|)|z| + Ch (1 + |x|γ + |y|), γ ≥ 0;

 i
(y − y 0 )(Hi (t, x, y, z, u, v) − Hi (t, x, y 0 , z, u, v)) ≤ Ch (1 + |x|α )|y − y 0 |2 , α ∈ (0, 2);


|Hi (t, x, y, z, u, v) − Hi (t, x, y, z 0 , u, v)| ≤ Cσ Cf (1 + |x|)|z − z 0 |.
(12)
For the existence of Nash equilibria, we also need the following assumption.
Assumption 4 (Generalized Isaacs condition). (i) There exist two Borelian applications u∗ , v ∗ defined on [0, T ] × Rm × R2 × R2m , with values in U1 and U2 , respectively, such that for any (t, x, y 1 , y 2 , p, q, u, v) ∈ [0, T ] × Rm × R2 × R2m × U1 × U2 ,
we have:
H1∗ (t, x, y 1 , y 2 , p, q) = H1 (t, x, y 1 , p, u∗ (t, x, y 1 , y 2 , p, q), v ∗ (t, x, y 1 , y 2 , p, q))
≤ H1 (t, x, y 1 , p, u, v ∗ (t, x, y 1 , y 2 , p, q))
and
H2∗ (t, x, y 1 , y 2 , p, q) = H2 (t, x, y 2 , q, u∗ (t, x, y 1 , y 2 , p, q), v ∗ (t, x, y 1 , y 2 , p, q))
≤ H2 (t, x, y 2 , q, u∗ (t, x, y 1 , y 2 , p, q), v).
(ii) the mapping (y 1 , y 2 , p, q) ∈ R2 × R2m 7−→ (H1∗ , H2∗ )(t, x, y 1 , y 2 , p, q) ∈ R is
continuous for any fixed (t, x) ∈ [0, T ] × Rm .
3. Well-posedness of costs. In this section, we focus on the well-posedness of
costs J i (u, v) for admissible control (u, v) ∈ M and i = 1, 2. Precisely speaking, we
need to show the existence of the solutions for BSDE (11) which we summarized as
the following theorem.
Theorem 3.1. Under Assumptions 1,2 and 3, BSDE (11) has solution (Y i,(u,v) ,
Z i,(u,v) ) which belongs to STq̄ × HTq̄ for any q̄ > 1 and any admissible control (u, v) ∈
M, i = 1, 2.
For simplicity, in this section, we omit the subscript (u, v) and denote the pair
(Y i,(u,v) , Z i,(u,v) ) by (Y i , Z i ). We first provide an uniform priori estimate of the
solution for BSDE (11). For this, we need the following result by [12], which related
to the integrability of the Doléans-Dade exponential of X t,x . Actually, we only
need the following Lemma 3.2, Lemma 3.3 and Lemma 3.7, readers who are not
interested in the technique proofs can skip the other lemmas and the proof process
in the following subsection.
3.1. Integrability of Doléans-Dade exponential.
Lemma 3.2 ([12]). Under Assumption 1, let ϕ be a P ⊗B(Rm )-measurable application from [0, T ]×Ω×Rm to Rm which is uniformly of linear growth, that is, P-a.s.,
∀(s, x) ∈ [0, T ] × Rm , |ϕ(s, ω, x)| ≤ Cϕ (1 + |x|). Then, there exists some p0 ∈ (1, 2)
and a constant C, where p0 depends only on Cσ , Cϕ , m while the constant C,
depends only on m and p0 , but not on ϕ, such that:
h
p0 i
R·
E ET 0 ϕ(s, Xst,x )dBs ≤ C,
where the process (Et )t≤T is the density function defined in (8).
RECURSIVE SDG AND RELATED BSDE
295
For the same function ϕ in Lemma 3.2 and a fixed t ∈ [0, T ], let us now define a
process (Γt,s )t≤s≤T as follows:
dΓt,s = Γt,s · ϕ(s, Xst,x )dBs ,
∀s ∈ [t, T ) and Γt,t = 1, Γt,T = ET /Et .
Actually,
Rs 2
Rs
t,x
t,x
1
Γt,s := Γt,s ϕ(r, Xrt,x ) = e t ϕ(r,Xr )dBr − 2 t ϕ (r,Xr )dr ,
∀s ∈ [t, T ].
(13)
Then, the following lemma holds true by using the same mind as Lemma 3.2. We
provide the proof here for readers’ better understanding.
Lemma 3.3. Under Assumption 1, for the same ϕ as in Lemma 3.2, there exists
some δ ∈ (0, T ) small enough, such that
−p
E
sup |ΓT −δ, t |
< ∞ for any p > 1.
T −δ≤t≤T
To prove Lemma 3.3, we need the following lemmas.
Rt
Lemma 3.4. Under Assumption 1, let Mt = 0 σ(s, Xst,x )dBs for each t ≤ T , then
for any p > 1, there exists a constant C0 depending on Cσ , T and p, such that,
|Xst,x |p ≤ C0 (1 + |x|p + |Ms |p ) a.s.
Lemma 3.5. If B := (Bt )t≤T is a Rm -valued
is
h R Brownian
i motion and (σt )t≤T
T
m
2
a R -valued stochastic process such that E 0 |σt | dt < ∞, then I S(t) is a
Rt
standard Brownian motion on [0, R(T )] where R(t) = 0 |σr |2 dr < ∞; S(t) =
Rt
inf {s > 0, R(s) = t} and I(t) = 0 σs dBs .
Proof. See [14] p.29.
Lemma 3.6. Let B = (Bt )t≥0 be a standard one dimensional Brownian motion.
The law of |B| has density
E[e
λ|Bt |2
x2
√ 2 e− 2t
2πt
,
t ≥ 0. If 2λt < 1 for a constant λ, then,
] < ∞.
We are now ready to provide the proof of Lemma 3.3.
Proof of Lemma 3.3. In this proof, the process X t,x is denoted simply by X. For a
constant δ ∈ (0, T ), let us define a stopping time
Rt
Rt
τN := inf{t ≥ T − δ, | T −δ ϕ(s, Xs )dBs | ≥ N or T −δ |ϕ(s, Xs )|2 ds ≥ N }.
For p > 1, since
hZ T
i
2
E
1τN (t)ΓT −δ, t (−pϕ(s, Xs )) |ϕ(t, Xt )|2 dt
T −δ
hZ
≤E
T ∧τN
T −δ
2pN
≤ Ne
Rt
e2{
T −δ
−pϕ(s,Xs )dBs − 12
Rt
T −δ
p2 |ϕ(s,Xs )|2 ds}
i
· |ϕ(t, Xt )|2 dt
,
R t∧τN
then the process ( T −δ ΓT −δ, s (−pϕ(r, Xr ))·ϕ(s, Xs )dBs )T −δ≤t≤T is a Ft -martingale.
296
RUI MU AND ZHEN WU
Therefore, by Itô’s formula,
h
i
E ΓT −δ, T ∧τN (−pϕ(s, Xs ))
h Z T ∧τN
i
=1−E
ΓT −δ, t∧τN (−pϕ(s, Xs )) · pϕ(t, Xt )dBt
T −δ
= 1.
Rt
We now define Mt := T −δ σ(s, Xs )dBs for each t ∈ [T − δ, T ]. Then we obtain
from the linear growth of ϕ and Lemma 3.4 that
R T ∧τ
ΓT −δ, T ∧τ (ϕ(s, Xs )) −p = ΓT −δ, T ∧τ (−pϕ(s, Xs )) · e 12 (p2 +p) T −δ N |ϕ(s,Xs )|2 dt
N
N
2
2
2
1
≤Γ
(−pϕ(s, X )) · e 2 (p +p)C(1+|x| +|MT ∧τN | )
T −δ, T ∧τN
s
(14)
where the constant C depends on T, C0 and Cϕ .
R t∧τ
Let the process B N = (BtN )T −δ≤t≤T := (Bt − BT −δ − 0 N −pϕ(s, Xs )ds)t≤T .
Hence the process B N is a Brownian motion under the probability PRN which satist
fies that dPN = ΓT −δ, T ∧τN (−pϕ(s, Xs ))dP. Let us denote MtN := T −δ σ(s, Xs )d
BsN for each t ∈ [T − δ, T ]. Then
Z t∧τN
Mt = MtN +
−pσ(s, Xs )ϕ(s, Xs )ds
T −δ
and from Assumption 1, the linear growth of ϕ and Lemma 3.4, we know,
Z t∧τN
|Mt |2 ≤ 2|MtN |2 + C̄p2 Cσ Cϕ2
1 + |Xs |2 ds
T −δ
Z t
≤ C̄ |MtN |2 + 1 + |x|2 +
|Ms |2 ds , for t ∈ [T − δ, T ∧ τN ],
T −δ
2
Cσ2 Cϕ2 C0 δ).
Thanks to Gronwall’s inequality, we
where the constant C̄ = 2 ∨ (2p
have,
|MT ∧τN |2 ≤ C̄ 1 + |x|2 + |MTN∧τN |2 eC̄δ .
(15)
Back to (14) and take expectation on both sides, we obtain, there exists a constant
which we still denoted by C̄ depending on C0 , Cσ , Cϕ , p, m, T , such that,
h
h 1 2
2 C̄δ i
−p i
2
N
E ΓT −δ, T ∧τN (ϕ(s, Xs ))
≤ EN e 2 (p +p)C̄(1+|x| +|MT ∧τN | )e
h 1 2
2i
N
C̄δ
2
C̄δ
2
1
≤ e 2 (p +p)C̄e (1+|x| ) · EN e 2 (p +p)C̄e |MT ∧τN |
(16)
where EN is the expectation under the probability PN . If σi (t) is the ith row
(i = 1, 2, ..., m) of the matrix σ(t, Xt ), then by a technique of splitting the stochastic
integral into the integrals on random intervals, we get the following inequality,
m Z T
m
2 X
X
N
βi Ri (T ) 2 ,
|MTN |2 ≤
σi (t)dBtN ≤
i=1
R Si (t)
0
i=1
where βiN (t) = T −δ σi (s)dBsN and Si (t) = inf{s ≥ T − δ : Ri (s) = t} with
Rs
Ri (s) = T −δ |σi (t)|2 dt.
RECURSIVE SDG AND RELATED BSDE
297
It follows from Lemma 3.5 that βiN is a Brownian motion on the random interval
[T − δ, Ri (T )]. Now Hölder’s inequality implies for a constant λ,
1
h
i
hY
h
Y
2i
2m i
N 2
N
N
EN eλ|MT | ≤ EN
eλ|βi (Ri (T ))| ≤
EN emλ|βi (Ri (T ))|
i
i
=
Y
EN
h
m1 i
2
emλ|β(Ri (T ))|
i
where β is a scalar Brownian motion on (Ω, F, PN ). Since Ri (T ) ≤ δ(Cσ )2 , then
by Lemma 3.6, if 2λmδ(Cσ )2 < 1, we have,
h
i
h
i
2 2
N 2
EN eλ|MT | ≤ EN emλ|β(δ|Cσ | )| ≡ e0 < ∞.
Now let λ = 21 (p2 + p)C̄eC̄δ , the same C̄ as in (16). Considering inequality (16) and
the fact that δ < T , we can conclude that if (p2 + p)C̄eC̄T < ε = (mδ|Cσ |2 )−1 then
1
2
1
−p E |ΓT −δ,T ∧τN (ϕ(s, Xs ))|
≤ e 2 ε(1+|x| ) e02 ≡ C.
We can choose δ small enough, such that,
0<δ<
(p2
1
, for any fixed p > 1.
+ p)C̄eC̄T m|Cσ |2
(17)
Then Fatou’s lemma yields that there exists a δ ∈ (0, T ) small enough such that
E [|ΓT −δ, T (ϕ(s, Xs ))|−p ] ≤ C holds true for any p > 1, with constant C depending
only on p, m but not on ϕ.
Finally, BDG inequality yields that Lemma 3.3 is true.
By exactly examining the proof of Lemma 3.3, we can also have the following
result.
Lemma 3.7. Under Assumption 1, let ϕ satisfies the assumption in Lemma 3.2,
then, there exists some δ ∈ (0, T ) small enough, such that
p
E
sup |ΓT −δ, t | < ∞ for any p > 1.
T −δ≤t≤T
Now, we are ready to introduce the proof of Theorem 3.1.
3.2. Proof of Theorem 3.1. We begin with the integrability of Y i , i = 1, 2
(the superscription (u, v) is omitted for simplicity). Since the technique restriction
as shown in Lemma 3.3, we will divide [0, T ] into small intervals, i.e., [0, T ] =
[T − δ, T ] ∪ [T − 2δ, T − δ] ∪ ... ∪ [0, T − (n − 1)δ] with n = T /δ for small δ ∈ (0, T )
and solve BSDE (11) in small time intervals backwardly. If the choice of δ is
independent with the terminal value of the BSDE, then, with this time partition
method, we can find the global solution on the whole time space. Next, we start
with some t ∈ [T − δ, T ]. The following technique is inspired by [1]. We first take a
linearization and take i = 1 for example. Let
H1 (s, Xs0,x0 , Ys1 , Zs1 , us , vs ) − H1 (s, Xs0,x0 , 0, Zs1 , us , vs )
;
Ys1
H1 (s, Xs0,x0 , 0, Zs1 , us , vs ) − H1 (s, Xs0,x0 , 0, 0, us , vs )
bs =
· Zs1 .
2
|Zs1 |
as =
298
RUI MU AND ZHEN WU
Then, from BSDE (11), (Y 1 , Z 1 ) solves the following linear BSDE: ∀t ∈ [T − δ, T ],
Z T
Z T
0,x0
1
1
0,x0
1
1
Yt = g (XT ) +
[h1 (s, Xs , 0, us , vs ) + as · Ys + bs · Zs ]ds −
Zs1 dBs .
t
Rt
t
as ds
, by Itô’s formula, we have, ∀t ∈ [T − δ, T ],
Z T
Z
et Yt1 = eT g 1 (XT0,x0 ) +
[es · h1 (s, Xs0,x0 , 0, us , vs ) + es bs Zs1 ]ds −
Let et = e
0
t
T
es Zs1 dBs .
t
It follows from Assumptions 1-(ii) and 2 that, |bs | ≤ Cσ Cf (1+|Xs0,x0 |). Once again,
by Girsanov’s transformation, we have,
Z T
Z T
0,x0
0,x0
1
1
es Zs1 dBs∗ , t ∈ [T −δ, T ],
es ·h1 (s, Xs , 0, us , vs )ds−
et Yt = eT g (XT )+
t
t
(18)
where Bs∗ = Bs − BT −δ − T −δ br dr, s ∈ [T − δ, T ] is a Brownian motion under the
probability P∗ which is defined by dP∗ = ΓT −δ, T (br )dP, where ΓT −δ,· is defined
in (13). Considering additionally that, as ≤ Ch (1 + |Xs0,x0 |α ) for α ∈ (0, 2) which
is obtained by Assumption 3-(i) on h1 , therefore, we know (et )t≤T has moments of
any order by (6), the same with g 1 and h1 following the estimate (5).
Besides, if we assume that BSDE (11) has solution and Z 1 belongs to HTp −δ,T
for some δ ∈ (0, T ) and any p > 1, then for any q ∈ (1, p),
q2 i
q2 i
h
R
h R
T
T
1 2
2
q
1 2
|
ds
E
|Z
e
|
ds
≤
E
sup
e
|Z
s
s
s
s
T −δ
T −δ
T −δ≤s≤T
h R
p2 i
h
pq i
T
1 2
|
ds
≤ C{E
sup esp−q + E
|Z
} < ∞.
s
T −δ
Rs
T −δ≤s≤T
This enable us to take the conditional expectation of (18) under the probability P∗ ,
RT
i.e., et Yt1 = E∗ [eT g 1 (XT0,x0 ) + t es · h1 (s, Xs0,x0 , 0, us , vs )ds|Ft ], t ∈ [T − δ, T ]. As
we analyzed above, the conditional expectation is well-posed. More precisely,
"
#
Z T
es
0,x0
∗ eT 1
1
0,x0
Yt = E
g (XT ) +
· h1 (s, Xs , 0, us , vs )dsFt , t ∈ [T − δ, T ].
et
et
t
(19)
let us now denote by Γt,s the process (Γt,s (br ))t≤s≤T for fixed t ∈ [0, T ] as in (13).
Equation (19) can be rewrite as: for t ∈ [T − δ, T ],
"
!
#
Z T
eT 1 0,x0
es
1
0,x0
1
Yt =
E ΓT −δ, T
g (XT ) +
· h1 (s, Xs , 0, us , vs )ds Ft .
ΓT −δ, t
et
et
t
For any p > 1, by applying Young’s inequality and conditional Jensen’s inequality, and considering the fact that functions g 1 and h1 are of polynomial growth on
x and Lemma 3.3, 3.7, we have
p i
sup Yt1 T −δ≤t≤T
n h
io p
q̂
≤C E
sup |ΓT −δ, t |−q̂
×
T −δ≤t≤T
n h
RT
× E
sup ΓT −δ,T eeTt g 1 (XT0,x0 ) + t
T −δ≤t≤T
p
≤ C E |ΓT −δ,T |p̃ p̃ ×
E
h
es
et
p̄ io pp̄
· h1 s, Xs0,x0 , 0, us , vs ds RECURSIVE SDG AND RELATED BSDE
299
p̄p̃ io p(p̃−p̄)
n h
p̃−
RT
p̄
p̄p̃
× E eeTt g 1 (XT0,x0 ) + t eest · h1 s, Xs0,x0 , 0, us , vs ds
p̄)
io p(p̃−p̄)
n h
io p(2p̃−
n h
2p̄p̃T Ch
0,x
2p̄p̃
p̄p̃
2p̄p̃
1+|Xt 0 |α
≤C E
sup e p̃−p̄
×
E |g 1 (XT0,x0 )| p̃−p̄ )
+
T −δ≤t≤T
2p̄p̃ io p(p̃−p̄) n h R
p̃−p̄
2p̄p̃
T
0,x0
h
(s,
X
+ E
,
0,
u
,
v
)ds
1
s
s
s
T −δ
≤C
(20)
pq̂
with p < q̂ and p̄ = q̂−p
< p̃.
The uniform integrability of process Z 1 follows from Itô’s formula and the facts
that Y 1 ∈ STp −δ,T for any p > 1 from (20). Indeed, we have, for 1 < q < p,
h
h Z T
q2 i
≤ C̄E
sup (|Yr1 |q + |Yr1 |p )
E
|Zr1 |2 dr
T −δ≤r≤T
T −δ
+
sup
T −δ≤r≤T
i
pq
(1 + |Xr0,x0 | p−q + |Xr0,x0 |qγ )
(21)
which is finite with the constant C̄ depending only on Ch , Cσ , Cf , T , p and q. The
details are omitted here.
Let us summarize the estimates (20) and (21) as the following lemma.
Lemma 3.8. Under Assumptions 1 and (12), if for any admissible control (u, v) ∈
M, i = 1, 2, (Y i,(u,v) , Z i,(u,v) ) are solutions to BSDE (11), such that, Y i,(u,v) ∈
STp −δ,T for some δ ∈ (0, T ) and any p > 1. Then, for any q ∈ (1, p), (Y i,(u,v) ,
Z i,(u,v) ) ∈ STq −δ,T × HTq −δ,T and there exists q̄ > q such that,
h
R
q2 i
T
i,(u,v) q
i,(u,v) 2
E
sup |Yt
| + T −δ |Zr
| dr
T −δ≤t≤T
h
i qq̄ h R
q̄ i qq̄
T
0,x0
+ E
}
≤ C{ E |g 1 (XT0,x0 )|q̄
h
(s,
X
,
0,
u
,
v
)ds
i
s
s
s
T −δ
which is obviously finite by considering g i (x) and hi (s, x, 0, u, v) are of polynomial
growth on x and the fact that X 0,x0 has moments of any order. The constant C
depends only on Ch , Cσ , Cf , T, q, q̄.
We come back to
Proof of Theorem 3.1. For each n ≥ 1, let us define the following stopping time:
Z t
τn = inf{t ≥ T −δ,
Cσ2 Cf2 (1+|Xs0,x0 |2 )+Ch (1+|Xs0,x0 |γ )ds ≥ n}∧T, δ ∈ (0, T ).
T −δ
This stopping time is of stationary type and it will converge P-a.s. to T as n tends
to infinity. Let us set g 1n (x) = g 1 (x)1g1 (x)≤n . Then by the result of [15], we know
that there exists a bounded process Y 1n and a process Z 1n ∈ HT2 −δ,T , which solve
the following BSDE:∀t ∈ [T − δ, T ],
Z T
Z T
Yt1n = g 1n (XT0,x0 ) +
1s≤τn H1 (s, Xs0,x0 , Ys1n , Zs1n , us , vs )ds −
Zs1n dBs .
t
t
Indeed, for any (s, y, z, u, v) ∈ [T − δ, T ] × R × Rm × U1 × U2 , we know,
1s≤τn H1 (s, Xs0,x0 , y, z, u, v) ≤ 1s≤τn Ch (1 + |Xs0,x0 |γ ) + 1 Cσ2 Cf2 (1 + |Xs0,x0 |2 )
2
1 2
+ |z|
2
300
RUI MU AND ZHEN WU
and
T
1
3n
1s≤τn Ch (1 + |Xs0,x0 |γ ) + Cσ2 Cf2 (1 + |Xs0,x0 |2 ) ds ≤
.
2
2
T −δ
Z
Then, Lemma 3.8 yields that, (Y 1n , Z 1n ) ∈ STq −δ,T ×HTq −δ,T uniformly with respect
to n for any q > 1.
Let us show that (Y 1n , Z 1n )n≥1 is a Cauthy sequence in STq −δ,T × HTq −δ,T for all
q > 1. Let m, n be integers such that m > n > 1 and let us set δY = Y 1m − Y 1n ,
δZ = Z 1m − Z 1n . Then (δY, δZ) solves the BSDE: ∀t ∈ [T − δ, T ],
Z T
Z T
δYt = g 1m (XT0,x0 ) − g 1n (XT0,x0 ) +
F (s, Xs0,x0 , δYs , δZs , us , vs )ds −
δZs dBs ,
t
t
where
F (s, Xs0,x0 , δy, δz, u, v)
= 1s≤τn [H1 (s, x, δy + Ys1n , δz + Zs1n , u, v) − H1 (s, x, Ys1n , Zs1n , u, v)]
+ 1τn <s≤τm H1 (s, x, Ys1n , Zs1n , u, v).
The function F satisfies the hypothesis (12) and actually F (s, Xs0,x0 , 0, 0, u, v) =
−1τn <s≤τm H1 (s, Xs0,x0 , Ys1n , Zs1n , u, v). Besides, for 1 < q̄ < q, by Hölder inequality,
we have,
h R
q̄ i
T
0,x0
E
F
(s,
X
,
0,
0,
u
,
v
)ds
s
s
s
T −δ
h R
q̄2 R
q̄2
T
T
1n 2
2 2
0,x0 2
≤ Cq̄ E
|Z
|
ds
C
C
(1
+
|X
|
)ds
s
s
τn
τn σ f
R
q̄2 i
T
+ τn Ch (1 + |Xs0,x0 |γ + |Ys1n |)ds
q̄q i q−q̄
h R
q2 i qq̄ h R
q−
q̄
q
T
T
1n 2
2 2
0,x0 2
≤ Cq̄ { E
|Z
|
ds
E
C
C
(1
+
|X
|
)ds
s
s
T −δ
τn σ f
h R
q̄2 i
h R
q̄2 i
T
T
0,x0 γ
1n
+E
C
(1
+
|X
|
)ds
+
E
|Y
|ds
}
h
s
s
τn
τn
which converges to 0 as n converges to infinity, considering (Y 1n , Z 1n ) ∈ STq −δ,T ×
HTq −δ,T for q > 1 and X 0,x0 has moments of any order, and τn → T , P-a.s.
Since, g 1m − g 1n → 0 in Lq̄ for any q̄ > 1 as n, m → ∞, through Lemma 3.8, we
obtain that, (δY, δZ) → 0 in STq −δ,T × HTq −δ,T for q < q̄ as n, m → ∞. It is easy to
check that the limit of this sequence is the solution to BSDE (11) for t ∈ [T − δ, T ].
Applying the same method, we will find the solution for BSDE (11) for t ∈
[T − 2δ, T − δ]. Repeating the same method backwardly finitely many times, we
finally find the solution for BSDE (11) on the global interval [0, T ].
4. Existence of Nash equilibria.
Theorem 4.1. Let us assume that, Assumptions 1, 2 and 3 are fulfilled, Then there
exist two deterministic functions ς i (t, x), i = 1, 2, with polynomial growth and two
pairs of P-measurable processes (Y i , Z i ), i = 1, 2, with values in R1+m such that:
For i = 1, 2,
(a) P-a.s., ∀s ≤ T , Ysi = ς i (s, Xs0,x0 ) and Z i (ω) := (Zti (ω))t≤T is dt-square
integrable ;
(b) For any s ≤ T ,
RECURSIVE SDG AND RELATED BSDE
(
301
−dYsi = Hi s, Xs0,x0 , Ysi , Zsi , (u∗ , v ∗ )(s, Xs0,x0 , Ys1 , Ys2 , Zs1 , Zs2 ) ds − Zsi dBs ,
YTi = g i (XT0,x0 ).
(22)
Besides, the control ((u∗ , v ∗ )(s, Xs0,x0 , Ys1 , Ys2 , Zs1 , Zs2 ))s≤T is admissible and a Nash
equilibrium point for the recursive NZSDG.
Proof. The existence of solutions for BSDE (22) has been shown in [16]. We focus
on the second conclusion below.
For s ≤ T , let us set u∗s = u∗ (s, Xs0,x0 , Ys1 , Ys2 , Zs1 , Zs2 ) and vs∗ = v ∗ (s, Xs0,x0 , Ys1 ,
2
Ys , Zs1 , Zs2 ), then (u∗ , v ∗ ) ∈ M. From the definition of costs (10), we obviously
have, Y01 = J 1 (u∗ , v ∗ ).
∗
Next let u be an arbitrary element of M1 and let us show that Y 1 ≤ Y 1,(u,v ) ,
∗
1,(u,v )
which yields Y01 = J 1 (u∗ , v ∗ ) ≤ Y0
= J 1 (u, v ∗ ).
∗
The control (u, v ) is admissible and thanks to Theorem 3.1, there exists a pair
∗
∗
of P-measurable processes Y 1,(u,v ) , Z 1,(u,v ) ∈ STq̄ (dPu,v∗ ) × HTq̄ (dPu,v∗ ), for any
q̄ > 1, such that
1,(u,v ∗ )
Yt
= g 1 (XT0,x0 ) +
Z
T
∗
∗
H1 (s, Xs0,x0 , Ys1,(u,v ) , Zs1,(u,v ) , us , vs∗ )ds
t
Z
−
T
∗
Zs1,(u,v ) dBs , ∀t ≤ T.
(23)
t
∗
Afterwards, we aim to compare Y 1 and Y 1,(u,v ) . So let us denote by
4Y = Y 1 − Y 1,(u,v
∗
)
∗
and 4Z = Z 1 − Z 1,(u,v ) .
For k ≥ 0, we define the stopping time τk as follows:
Rs
τk := inf{s ≥ 0, |4Ys | + 0 |4Zr |2 dr ≥ k} ∧ T.
The sequence of stopping times (τk )k≥0 is of stationary type and converges to T .
Rt
Next setting at = qCh 0 1 + |Xr0,x0 |α dr, α ∈ (0, 2) and applying Itô-Meyer formula
to eat |(4Y )+ |q (1 < q < q̄) between t ∧ τk and τk , we obtain: ∀t ≤ T ,
R τk a
eat∧τk |(4Yt∧τk )+ |q + c(q) t∧τ
e s |(4Ys )+ |q−2 14Ys >0 |4Zs |2 ds
k
R τk
+ t∧τk qCh (1 + |Xs0,x0 |α )eas |(4Ys )+ |q ds
R τk a
aτk
= e |(4Yτk )+ |q + q t∧τ
e s |(4Ys )+ |q−1 14Ys >0 ×
k
1,(u,v ∗ )
1,(u,v ∗ )
×R H1 (s, Xs0,x0 , Ys1 , Zs1 , u∗s , vs∗ ) − H1 (s, Xs0,x0 , Ys
, Zs
, us , vs∗ ) ds
τk
−q t∧τ
eas |(4Ys )+ |q−1 14Ys >0 4Zs dBs ,
k
where c(q) =
q(q−1)
.
2
Since
∗
∗
14Ys >0 H1 (s, Xs0,x0 , Ys1 , Zs1 , u∗s , vs∗ ) − H1 (s, Xs0,x0 , Ys1,(u,v ) , Zs1,(u,v ) , us , vs∗ )
= 14Ys >0 H1 (s, Xs0,x0 , Ys1 , Zs1 , u∗s , vs∗ ) − H1 (s, Xs0,x0 , Ys1 , Zs1 , us , vs∗ )
∗
∗
+ H1 (s, Xs0,x0 , Ys1 , Zs1 , us , vs∗ ) − H1 (s, Xs0,x0 , Ys1,(u,v ) , Zs1,(u,v ) , us , vs∗ )
≤ 14Ys >0 4Zs σ −1 (s, Xs0,x0 )f (s, Xs0,x0 , us , vs∗ ) + Ch (1 + |Xs0,x0 |α )|(4Ys )+ |
302
RUI MU AND ZHEN WU
which is obtained by the generalized Isaacs’ assumption 4 and the monotonic assumption (3)-(i) on h1 . Then, we have that, for any t ≤ T ,
Z τk
∗
|eat∧τk (4Yt∧τk )+ |q ≤ eaτk |(4Yτk )+ |q − q
eas |(4Ys )+ |q−1 14Ys >0 4Zs dBsu,v .
t∧τk
(24)
as
By definition of the stopping time τk , and the fact that e has moment of any order
since α ∈ (0, 2),we have
h Z τk
i
∗
u,v ∗
E
|(4Ys )+ |q−1 14Ys >0 4Zs dBsu,v = 0.
t∧τk
∗
Then taking expectation on both sides of (24) under the probability Pu,v , we
obtain,
∗ ∗ Eu,v eat∧τk |(4Yt∧τk )+ |q ≤ Eu,v eaτk |(4Yτk )+ |q
(25)
∗
∗
Next taking into account of Y 1,(u,v ) ∈ STq̄ (dPu,v ) and Y 1 has a representation
through ς 1 which is deterministic and of polynomial growth, we deduce that,
q u,v ∗
as
1,(u,v ∗ )
1
E
sup e
|Ys
| + |Ys |
s≤T
≤ Eu,v
∗
q̄ h q̄ i
∗
∗
< ∞.
e q̄−q as + Eu,v sup |Ys1,(u,v ) | + |Ys1 |
s≤T
∗
The sequence (eaτk |(4Yτk )+ |q )k converges to eaT |4YT |q = 0, Pu,v -a.s., as k → ∞,
∗
then, it also converges to 0 in L1 (dPu,v ) since it is uniformly integral by the above
inequality.
Therefore, by passing n to the limit on both sides of (25) and using Fatou’s
∗
lemma, we can show that Eu,v [eat |(4Yt )+ |q ] = 0, ∀t ≤ T , which implies that
∗
1,(u,v ∗ )
, P-a.s., since the probability Pu,v and P are equivalent. Thus,
Yt1 ≤ Yt
∗
1,(u,v )
Y01 = J 1 (u∗ , v ∗ ) ≤ Y0
= J 1 (u, v ∗ ). In the same way, we can show that for
2,(u∗ ,v)
any arbitrary element v ∈ M2 , Y02 = J 2 (u∗ , v ∗ ) ≤ Y0
= J 1 (u∗ , v), which tell
∗ ∗
us that, the pair (u , v ) is a Nash equilibrium point for this game problem.
5. An example. In this section, we will give a simple example which satisfies our
assumptions. We can clearly see that the Nash equilibrium point exists for this
recursive game.
Let us consider a one-dimensional game problem and we assume the admissible
control (u, v) takes value from U1 ×U2 = [−1, 1]×[−1, 1]. In addition, in BSDE (11),
let us take σ(t, x) = 1, f (t, x, u, v) = 1, h1 (t, x, y, u, v) = yu + v 2 , h2 (t, x, y, u, v) =
yv + u2 and g i (x) = x, i = 1, 2. We can verify that Assumptions 1-3 are satisfied.
Besides, the recursive type costs J i (u, v) of the players are defined by the initial
i,(u,v)
values Y0
of the following BSDEs: for i = 1, 2,

Z Th
Z T
i

1,(u,v)
0,x0
1,(u,v)
1,(u,v)
2


Y
=
X
+
Z
+
Y
u
+
v
ds
−
Zs1,(u,v) dBs ;
s
 t
s
s
s
T
t
t
Z Th
Z T
i


2,(u,v)
0,x

= XT 0 +
Zs2,(u,v) + Ys2,(u,v) vs + u2s ds −
Zs2,(u,v) dBs .
 Yt
t
t
From Theorem 3.1, the cost is well-posed. We then obtain Hamiltonians as
follows: H1 (t, x, y1 , p, u, v) = p + y1 u + v 2 and H2 (t, x, y2 , q, u, v) = q + y2 v + u2 .
RECURSIVE SDG AND RELATED BSDE
303
Then, it is not difficult to deduce that
u∗ (y1 ) = (−1) × 1{y1 ≥0} + 1 × 1{y1 <0}
and
v ∗ (y2 ) = (−1) × 1{y2 ≥0} + 1 × 1{y2 <0}
are admissible controls which satisfy the generalized Isaacs assumption 4-(i). As
we can see, the control (u∗ , v ∗ ) is discontinuous on (y1 , y2 ). However, considering
y1 u∗ (y 1 ) = −|y 1 |, y2 v ∗ (y 2 ) = −|y 2 | and |v ∗ (y 2 )|2 = |u∗ (y 1 )|2 =1 are continuous
functions on (y 1 , y 2 ), therefore, the mapping (y 1 , y 2 , p, q) 7−→ (H1∗ , H2∗ )(y 1 , y 2 , p, q)
is continuous with
H1∗ (y 1 , y 2 , p, q) = p + y1 u∗ (y 1 ) + |v ∗ (y 2 )|2
and
H2∗ (y 1 , y 2 , p, q) = q + y2 v ∗ (y 2 ) + |u∗ (y 1 )|2 ,
which tell us Assumption 4-(ii) is satisfied.
Finally, from Theorem 4.1, we know (u∗ (Y 1 ), v ∗ (Y 2 )) is one pair of Nash equilibrium point of this game, where (Y 1 , Y 2 ) are solutions of the following BSDEs:

Z T
Z T
1

0,x0
1
1 ∗
1
∗
2 2


Zs + Ys u (Ys ) + |v (Ys )| ds −
Zs1 dBs ;
 Yt = XT +
t
Z


0,x0
2

 Yt = XT +
t
T
2
Zs + Ys2 v ∗ (Ys2 ) + |u∗ (Ys1 )|2 ds −
t
Z
T
Zs2 dBs .
t
Acknowledgments. We are grateful to the anonymous referees for their useful
comments and suggestions.
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Received February 2016; revised January 2017.
E-mail address: [email protected]
E-mail address: [email protected]