Hindawi Publishing Corporation
International Journal of Stochastic Analysis
Volume 2012, Article ID 258674, 50 pages
doi:10.1155/2012/258674
Research Article
Necessary Conditions for
Optimal Control of Forward-Backward Stochastic
Systems with Random Jumps
Jingtao Shi
School of Mathematics, Shandong University, Jinan 250100, China
Correspondence should be addressed to Jingtao Shi, shijingtao@sdu.edu.cn
Received 28 September 2011; Revised 28 December 2011; Accepted 3 January 2012
Academic Editor: Jiongmin Yong
Copyright q 2012 Jingtao Shi. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
This paper deals with the general optimal control problem for fully coupled forward-backward
stochastic differential equations with random jumps FBSDEJs. The control domain is not
assumed to be convex, and the control variable appears in both diffusion and jump coefficients
of the forward equation. Necessary conditions of Pontraygin’s type for the optimal controls
are derived by means of spike variation technique and Ekeland variational principle. A linear
quadratic stochastic optimal control problem is discussed as an illustrating example.
1. Introduction
1.1. Basic Notations
Throughout this paper, we denote by Rn the space of n-dimensional Euclidean space, by Rn×d
the space of n × d matrices, and by Sn the space of n × n symmetric matrices. ·, · and | · |
denote the scalar product and norm in the Euclidean space, respectively. appearing in the
superscripts denotes the transpose of a matrix.
Let Ω, F, {Ft }t≥0 , P be a complete filtered probability space satisfying the usual
conditions, where the filtration {Ft }t≥0 is generated by the following two mutually
independent processes:
i a one-dimensional standard Brownian motion {Wt}t≥0 ;
ii a Poisson random measure N on E × R , where E ⊂ R is a nonempty open set
equipped with its Borel field BE, with compensator Ndedt
πdedt, such
that NA × 0, t
N − NA × 0, t
t≥0 is a martingale for all A ∈ BE satisfying
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πA < ∞. π is assumed to be a σ-finite measure on E, BE and is called the
characteristic measure.
Let T > 0 be fixed and U be a nonempty subset of Rk . We denote M Rn ×Rm ×Rm ×Rm .
Any generic point in M is denoted by θ ≡ x, y, z, c. Let U0, T be the set of all Ft -predictable
processes u : 0, T × Ω → U such that sup0≤t≤T E|ut|i < ∞, for all i 1, 2, . . .. Any u· ∈
U0, T is called an admissible control process. We denote by L2 E, BE, π; Rm or L2 the set
of integrable functions c : E → Rm with norm ce2L2 : E |ce|2 πde < ∞. We define
L2FT Ω; Rm : ξ : Ω −→ Rm | ξ is FT -measurable, E|ξ|2 < ∞ ,
T
2
2
m
m
ϕt dt < ∞ ,
LF 0, T ; R : ϕ : 0, T × Ω −→ R | ϕ is Ft -adapted, E
0
L2F,p 0, T ; Rm ϕ : 0, T × Ω −→ R | ϕ is Ft -predictable, E
m
:
T
2
ϕt
dt < ∞ ,
1.1
0
ϕ : 0, T × Ω × E −→ Rm |
Fp2 0, T ; Rm :
ϕ is Ft -predictable, such that, E
T 0
E
2
ϕt,
e πdedt < ∞ .
We denote
M2 0, T : L2F 0, T ; Rn × L2F 0, T ; Rm × L2F,p 0, T ; Rm × Fp2 0, T ; Rm .
1.2
Clearly, M2 0, T is a Banach space. Any process in M2 0, T is denoted by Θ· ≡
x·, y·, z·, c·, ·, whose norm is defined by
Θ·2M2 0,T 2
: E sup |xt| sup yt 2
t∈0,T t∈0,T T
2
|zt| dt E
0
T 0
E
2
|ct, e| πdedt .
1.3
1.2. Formulation of the Optimal Control Problem and Basic Assumptions
For any u· ∈ U0, T and a ∈ Rn , we consider the following fully coupled forward-backward
stochastic control system:
dxt b t, xt, yt, zt, ct, ·, ut dt σ t, xt, yt, zt, ct, ·, ut dWt
g t, xt−, yt−, zt, ct, ·, ut, e Ndedt,
E
−dyt f t, xt, yt, zt, ct, ·, ut dt − ztdWt − ct, eNdedt,
E
x0 a,
yT hxT ,
1.4
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3
with the cost functional given by
T
Ju· E
l t, xt, yt, zt, ct, ·, ut dt φxT γ y0 .
1.5
0
Here, Θ· ≡ x·, y·, z·, c·, · takes value in M Rn × Rm × Rm × Rm and
b, σ : 0, T × M × U −→ Rn ,
g : 0, T × M × U × E −→ Rn ,
φ : R −→ R,
γ : R −→ R,
n
1.6
h : Rn −→ Rm ,
f : 0, T × M × U −→ Rm ,
l : 0, T × M × U −→ R.
m
For any u· ∈ U0, T and a ∈ Rn , we refer to Θ· ≡ x·, y·, z·, c·, · as the state
process corresponding to the admissible control u· if FBSDEJ 1.4 admits a unique adapted
solution. For controlled FBSDEJ 1.4 with cost functional 1.5, we consider the following.
Problem C. Find u· ∈ U0, T , such that
Ju· inf
Ju·.
1.7
u·∈U0,T Any u· ∈ U0, T satisfying 1.7 is called an optimal control process of Problem
C, and the corresponding state process, denoted by Θ· ≡ x·, y·, z·, c·, ·, is called
optimal state process. We also refer to x·, y·, z·, c·, ·, u· as an optimal 5-tuple of
Problem C.
Our main goal in this paper is to derive some necessary conditions for the optimal
control of Problem C, which is called the stochastic maximum principle of Pontraygin’s type.
For this target, we first introduce the following basic assumption throughout this paper.
H0 For any u· ∈ U0, T and a ∈ Rn , FBSDEJ 1.4 admits a unique adapted solution
Θ· ≡ x·, y·, z·, c·, · ∈ M2 0, T . Moreover, the following estimate holds:
Θ·M2 0,T 2
≡ E sup |xt| sup yt 2
t∈0,T T
t∈0,T ⎡
≤ CE⎣|a|2 |h0|2 2
|zt| dt E
0
T
0
2
|bt, 0, ut|dt
0
T
0
|σt, 0, ut|2 dt T T
E
2
|ct, e| πdedt
0
T 0
E
2
ft, 0, utdt
⎤
2
gt, 0, ut, e πdedt⎦.
1.8
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Further, if Θ·
≡ x·,
y·,
z·, c·, · ∈ M2 0, T is the unique adapted solution of 1.4 with
: Rn → Rm and u
· ∈ U0, T , respectively, then the following stability
h· and u· replaced by h
estimate holds:
2
≡ E sup |xt − xt|
2 sup yt − yt
Θ· − Θ·
M2 0,T t∈0,T ⎡
T
t∈0,T 2
|zt − zt| dt E
0
0
≤ CE⎣|hxT − hxT
|2 T T
2
E
|ct, e − ct, e| πdedt
2
u
t dt
bt, Θt, ut − b t, Θt,
0
T
2
u
t dt
ft, Θt, ut − f t, Θt,
0
T 2
u
t dt
σt, Θt, ut − σ t, Θt,
0
⎤
T 2
u
t, e πdedt⎦.
gt, Θt, ut, e − g t, Θt,
0
E
1.9
By adopting the ideas from Wu 1
, we know that under certain G-monotonicity conditions
for the coefficients, the existence and uniqueness of FBSDEJ 1.4 guaranteed which leads to
hypothesis H0. Our main goal in this paper is to derive some necessary conditions for the
optimal control of Problem C. Hence, we impose the well-posedness of the state equation
1.4 as an assumption to avoid some technicalities not closely to our main results.
1.3. Developments of Stochastic Optimal Control and Contributions of
the Paper
It is well known that the optimal control problem is one of the central themes of modern
control sciences. Necessary conditions for the optimal control of the forward continuous
stochastic control system, that is, the so-called stochastic maximum principle of Pontraygin’s
type, were extensively studied since early 1960s. When Brownian motion is the unique noise
source, Peng 2
see also Yong and Zhou 3
obtained the maximum principle for the
general case, that is, the control variable entering the diffusion coefficient and control domain
being not convex.
Forward-backward stochastic control systems where the controlled systems are
described by forward-backward stochastic differential equations FBSDEs are widely used in
mathematical economics and mathematical finance, which includes the usual forward SDEs
as a special case. They are encountered in stochastic recursive utility optimization problems
see 4–8
and principal-agent problems see 9, 10
. Peng 11
first obtained the necessary
conditions for optimal control for the partially coupling case when the control domain is
convex. And then Xu 12
studied the nonconvex control domain case and obtained the
corresponding necessary conditions. But he needs to assume that the diffusion coefficient
in the forward control system does not contain the control variable. Ji and Zhou 8
applied
International Journal of Stochastic Analysis
5
the Ekeland variational principle to establish a maximum principle for a partially coupled
forward-backward stochastic control system, while the forward state is constrained in a
convex set at the terminal time. Wu 13
recently established a general maximum principle
for optimal control problems derived by forward-backward stochastic systems, where control
domains are nonconvex and forward diffusion coefficients explicitly depend on control
variables. Moreover, some financial optimization problems for large investors see 14–16
and some asset pricing problems with forward-backward differential utility see 7
directly
lead to fully coupled FBSDEs. Wu 17
first see also Meng 18
obtained the necessary
conditions for optimal control of fully coupled forward-backward stochastic control systems
when the control domain is convex. And then Shi and Wu 19
studied the nonconvex control
domain case and obtained the corresponding necessary conditions under some G-monotonic
assumptions. But they also similar to Xu 12
need to assume that the control variable
does not appear in the diffusion coefficient of the forward equation. Very recently, Yong 20
completely solved the problem of finding necessary conditions for optimal control of fully
coupled FBSDEs. He considered an optimal control problem for general coupled FBSDEs
with mixed initial-terminal conditions and derived the necessary conditions for the optimal
controls when the control domain is not assumed to be convex and the control variable
appears in the diffusion coefficient of the forward equation.
However, recently more and more research attentions are drawn towards the optimal
control problem for discontinuous stochastic systems or stochastic systems with random
jumps. The reason is clear for its applicable aspect. For example, there is compelling evidence
that the dynamics of prices of financial instruments exhibit jumps that cannot be adequately
captured solely by diffusion processes i.e., processes satisfying some Itô-type stochastic
differential equations SDE for short, see Merton 21
and Cont and Tankov 22
. Several
empirical studies demonstrate the existence of jumps in stock markets, the foreign exchange
market, and bond markets. Jumps constitute also a key feature in the description of credit risk
sensitive instruments. Therefore, models that incorporate jumps have become increasingly
popular in finance and several areas of science and engineering, this leads to paying more
attention to stochastic differential equations with jumps SDEJs for short. As consequences,
optimal control problems involving systems of SDEJs are widely studied. Situ 23
first
obtained the maximum principle for forward stochastic control system with jumps but
the jump coefficient does not contain the control variable. Tang and Li 24
discussed a
more general case, where the control is allowed into both diffusion and jump coefficients,
and the control domain is not necessarily convex; also some general state constraints
are imposed. Maximum principles for forward-backward stochastic control systems with
random jumps are studied in Øksendal and Sulem 25
, Shi and Wu 26
, where the FBSDEJs
are partially coupled and the control domains are convex. Recently, Shi 27
obtained the
necessary condition of optimal control as well as a sufficient condition of optimality under
the assumption that the diffusion and jump coefficients do not contain the control variable
and the control domain need not be convex. Necessary conditions for fully coupled forwardbackward stochastic control systems with random jumps were studied in Shi and Wu 28
see also Meng and Sun 29
where the control domains are convex.
In this paper, we consider the general optimal control problem for fully coupled
FBSDEJ 1.4. Here, by the word “general” we mean the allowance of the control variable
into both diffusion and jump coefficients of the forward equation and the control domain
is not assumed to be convex. It is well worth mentioning that the idea of second-order spike
variation technique, developed by Peng 2
, plays an important role in deriving the necessary
conditions of general stochastic optimal control of jump-diffusion process in Tang and Li 24
.
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International Journal of Stochastic Analysis
Following the standard idea of deriving necessary conditions for optimal control processes,
due to the fact the control domain U is not assumed to be convex, one needs to use spike
variation for the control process and then to try obtaining a Taylor-type expansion for the
state process x·, y·, z·, c·, · and the cost functional 1.5 with respect to the spike
variation of the control process, followed by some suitable duality relations to get a maximum
principle of Pontryagin-type. However, the derivation of Taylor expansion of the state process
x·, y·, z·, c·, · with respect to the spike variation of the control process is technically
difficult. The main reasons are that both z·, c·, · and u· appear in the diffusion and
jump coefficients of the forward equation of 1.4 and that the regularity/integrability of the
continuous martingale process z· and the discontinuous martingale process c·, · as part
of the state process seems to be not enough in the case when a second-order expansion is
necessary. Note that in 25–29
, due to the special structure of the problems, the second-order
expansion is not necessary when one tries to derive the necessary conditions for optimal
controls.
We overcome the above difficulty by the newly developed reduction method by Wu 13
and Yong 20
in the continuous case, independently. In fact, some ideas have been proposed
early in Kohlmann and Zhou 30
and Ma and Yong 31
. Let us make it more precise. We
first introduce the controlled initial value problem for a system of SDEJs, where x·, y· is
regarded as the state process and z·, c·, ·, u· is regarded as the control process. Next, we
regard the original terminal condition yT hxT as terminal state constraint and then
we translate Problem C into a high-dimensional reduced optimal control problem described
in Section 3. The advantage of this
by standard SDEJ with state constraint see Problem C
reduced optimal control problem is that one needs not much regularity/integrability of
processes z·, c·, · since it is treated as a control process. We apply the Ekeland variational
principle to deal with this high-dimensional reduced optimal control problem with state
constraint. Finally, necessary conditions for the optimal control of Problem C are derived
by the equivalence of Problems C and C.
The rest of this paper is organized as follows. In Section 2, under some suitable
assumptions, we give the main result of this paper, together with some discussions on special
cases. In Section 3, after we make the reduction of our optimal control problems for FBSDEJs
Problem C, a proof of our main result will be presented. Section 4 is devoted to a linear
quadratic stochastic optimal control problem as an illustrating example. Finally, in Section 5
we give the concluding remarks and compare our theorem with some existing results.
2. The Main Result and Some Special Cases
In this section, under some suitable assumptions, we will state the necessary conditions for
the optimal control of our Problem C. Also, some interesting special cases will be discussed.
Let us introduce the following further assumptions beyond H0.
H1 For any θ, u ∈ M × U, maps t → bt, θ, u, σt, θ, u, gt, θ, u, e, ft, θ, u is
Ft -progressively measurable and there exists a constant L > 0such that
2
2 b t, x, y, z, c, u − b t, x,
y,
z, c, u y,
z, c, u σ t, x, y, z, c, u − σ t, x,
2
2
y,
z, c, u, e L2 f t, x, y, z, c, u − f t, x,
y,
z, c, u g t, x, y, z, c, u, e − g t, x,
2 ≤ L |x − x|
y ∈ Rn ×Rm , a.s.,
2 y − y , ∀t, z, c, u ∈ 0, T ×Rm ×Rn ×U, x, y , x,
International Journal of Stochastic Analysis
7
b t, x, y, z, c, u 2 σ t, x, y, z, c, u 2 g t, x, y, z, c, u, e 2 2 f t, x, y, z, c, u 2
L
2 ≤ L 1 |x|2 y , ∀t, z, c, u ∈ 0, T × Rm × Rn × U, x, y ∈ Rn × Rm , a.s.
2.1
H2 Maps θ → bt, θ, u, σt, θ, u, gt, θ, u, e, ft, θ, u is twice continuously differential,
with the (partial) derivatives up to the second order being uniformly bounded, Lipschitz
continuous in θ ∈ M, and continuous in u ∈ U.
H3 The map h is twice differentiable with the derivatives up to the second order being uniformly
bounded and uniformly Lipschitz continuous; the maps φ, γ are twice differentiable with
the derivatives up to the second order being Lipschitz continuous; the map θ → lt, θ, u is
twice differentiable with the derivatives up to the second order being Lipschitz continuous
in θ ∈ M, and continuous in u ∈ U.
Next, to simplify our presentation, we now introduce some abbreviation notations.
First, we make the following convention: for any differentiable map a ≡ a1 , a2 , . . . , al :
n
l
R → R , let
⎛
a1x1 a1x2
· · · a1xn
⎞
⎟
⎜ 2
⎜ax1 a2x2 · · · a2xn ⎟
n
l×n
⎟
ax ⎜
.. . .
.. ⎟ : R −→ R .
⎜ ..
⎠
⎝ .
.
.
.
alx1 alx2 · · · alxn
2.2
In particular, for l 1, ax ax1 , ax2 , . . . , axn ∈ R1×n is an n-dimensional row vector. Also, for
any twice differentiable function a : Rn → R, the Hessian matrix is given by
⎞
ax1 x1 ax1 x2 · · · ax1 xn
⎜ax2 x1 ax2 x2 · · · ax2 xn ⎟
⎟
⎜
: Rn −→ Sn ,
≡ ax ⎜ .
..
.. ⎟
..
⎠
⎝ ..
x
.
.
.
axn x1 axn x2 · · · axn xn
⎛
axx
2.3
hereafter, Sn stands for the set of all real n × n symmetric matrices. On the other hand, for a
twice differentiable function a : Rn × Rm → R denoting x, y ∈ Rn × Rm , we denote
⎞
ax1 y1 ax1 y2 · · · ax1 ym
⎜ax2 y1 ax2 y2 · · · ax2 ym ⎟
⎟
⎜
: Rn × Rm −→ Rn×m .
≡ ax ⎜ .
..
.. ⎟
..
⎠
⎝ ..
y
.
.
.
axn y1 axn y2 · · · axn ym
⎛
axy
2.4
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International Journal of Stochastic Analysis
Now, let x·, y·, z·, c·, ·, u· be an optimal 5-tuple of Problem C. For a· b·, σ·, f·, l·, we denote
ax t, · ax t, xt, yt, zt, ct, ·, ut ,
ay t, · ay t, xt, yt, zt, ct, ·, ut ,
az t, · az t, xt, yt, zt, ct, ·, ut ,
ac t, · ac t, xt, yt, zt, ct, ·, ut ,
2.5
g x t, e gx t, xt, yt, zt, ct, e, ut, e ,
g y t, e gy t, xt, yt, zt, ct, e, ut, e ,
g z t, e gz t, xt, yt, zt, ct, e, ut, e ,
g c t, e gc t, xt, yt, zt, ct, e, ut, e .
2.6
and let
We also introduce
⎛
⎜
B X t, · ⎝
bx t, ·
by t, ·
⎞
⎟
⎠,
−f x t, · −f y t, ·
!
σ x t, · σ y t, ·
,
ΣX t, · 0
0
g x t, e g y t, e
ΥX t, e .
0
0
2.7
For a· bi ·, σ i ·, f j ·, l·, 1 ≤ i ≤ n, 1 ≤ j ≤ m, let
axx t, · axx t, xt, yt, zt, ct, ·, ut ,
ayy t, · ayy t, xt, yt, zt, ct, ·, ut ,
axy t, · axy t, xt, yt, zt, ct, ·, ut ,
ayx t, · ayx t, xt, yt, zt, ct, ·, ut ,
2.8
g xx t, e gxx t, xt, yt, zt, ct, ·, ut, e ,
g yy t, e gyy t, xt, yt, zt, ct, ·, ut, e ,
g xy t, e gxy t, xt, yt, zt, ct, ·, ut, e ,
g yx t, e gyx t, xt, yt, zt, ct, ·, ut, e .
2.9
and let
Our main result in this paper is the following theorem for Problem C.
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9
Theorem 2.1. Suppose that (H0)–(H3) hold, and let x·, y·, z·, c·, ·, u· be an optimal
5-tuple of Problem C. Then there exists a unique adapted solution p·, q·, k·, r·, · ∈
L2F 0, T ; Rn × L2F 0, T ; Rm × L2F,p 0, T ; Rn × Fp2 0, T ; Rn to the following FBSDEJ:
"
#
g x t, e rt, eπde − lx t, · dt
−dpt bx t, · pt − f x t, · qt σ x t, · kt E
− ktdWt − rt, eNdedt,
E
"
#
dqt −by t, · pt f y t, · qt − σ y t, · kt − g y t, e rt, eπde ly t, · dt
E
"
− bz t, · pt f z t, · qt − σ z t, · kt
#
− g z t, e rt, eπde lz t, · dWt
$ E
−bc t, e pt− f c t, e qt−
E
%
−σ c t, e kt − g c t, e rt, e lc t, e Ndedt,
q0 Eγy y 0 .
pT −φx xT − hx xT qT ,
2.10
Let P ·, Q·, K·, · ∈ L2F 0, T ; Snm × L2F,p 0, T ; Snm × Fp2 0, T ; Snm be the unique
adapted solution to the following matrix-valued BSDEJ:
&
−dP t B X t, · P t P tBX t, · ΣX t, · P tΣX t, · ΣX t, · Qt QtΣX t, ·
$
ΥX t, e P tΥX t, e ΥX t, e Kt, eΥX t, e
E
'
%
ΥX t, e Kt, e Kt, eΥX t, e πde H XX t, · dt
− QtdWt − Kt, eNdedt,
E
−φxx xT − hxx xT qT 0
,
P T 0
0
2.11
where H XX t, · H XX t, xt, yt, zt, ct, ·, ut, pt, qt, kt, rt, ·, and the Hamiltonian
function H : 0, T × M × U × Rn × Rm × Rn × Rn → R is defined as follows:
H t, x, y, z, c·, u, p, q, k, r·
( ) (
)
: p, b t, x, y, z, c·, u q, −f t, x, y, z, c·, u
(
)
(
)
k, σ t, x, y, z, c·, u re, g t, x, y, z, ce, u, e πde − l t, x, y, z, c·, u .
E
2.12
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International Journal of Stochastic Analysis
Then,
H t, xt, yt, zt, ct, ·, ut, pt, qt, kt, rt, ·
− H t, xt, yt, zt, ct, ·, u, pt, qt, kt, rt, ·
1 σ t, xt, yt, zt, ct, ·, ut − σ t, xt, yt, zt, ct, ·, u P1 t
2
× σ t, xt, yt, zt, ct, ·, ut − σ t, xt, yt, zt, ct, ·, u
1
−
g t, xt, yt, zt, ct, e, ut, e − g t, xt, yt, zt, c t, g , u, e
2 E
−
2.13
× P1 t K1 t, e
× g t, xt, yt, zt, ct, e, ut, e − g t, xt, yt, zt, ct, e, u, e πde ≥ 0,
∀u ∈ U, a.e., a.s.,
with P · ≡
P1 · P2 ·
P2 · P3 ·
, and K·, · ≡
K1 ·,· K2 ·,·
K2 ·,· K3 ·,·
.
Remark 2.2. In fact, the second-order
equation
2.11 can be split into the following
Qadjoint
1 · Q2 ·
three BSDEJs if we add that Q· ≡ Q · Q · :
2
3
&
−dP1 t bx t, · P1 t P1 tbx t, · σ x t, · P1 tσ x t, ·
σ x t, · Q1 t Q1 tσ x t, · − f x t, · P2 t − P2 tf x t, ·
$
g x t, e P1 tg x t, e g x t, e K1 t, eg x t, e
E
'
%
g x t, e K1 t, e K1 t, eg x t, e πde H xx t, · dt
− Q1 tdWt − K1 t, eNdedt,
2.14
E
P1 T −φxx xT − hxx xT qT ,
&
−dP2 t P1 tby t, · bx t, ·P2 t σ x t, · P1 tσ y t, ·
σ x t, ·Q2 t Q1 t σ y t, · − f x t, ·P3 t − P2 t f y t, ·
$
g x t, e P1 tg y t, e g x t, e K1 t, eg y t, e
E
'
g x t, eK2 t, e K1 t, eg y t, e πde H xy t, · dt
− Q2 tdWt − K2 t, eNdedt,
E
P2 T 0,
%
2.15
International Journal of Stochastic Analysis
&
−dP3 t by t, · P2 t P2 tby t, · σ y t, ·P1 tσ y t, ·
σ y t, · Q2 t Q2 tσ y t, · − f y t, · P3 t − P3 tf y t, ·
$
g y t, eP1 tg y t, e g y t, eK1 t, eg y t, e
E
'
%
g y t, e K2 t, e K2 t, eg y t, e πde H yy t, · dt
11
2.16
− Q3 tdWt −
E
K3 t, eNdedt,
P3 T 0.
Note that this kind of three second-order adjoint equations appears in Wu 13
whereas
not in Yong 20
.
Let us now look at some special cases. It can be seen that our theorem recovers the
known results.
1 A Classical Stochastic Optimal Control Problem with Random Jumps
Consider a controlled SDEJ:
dxt bt, xt, utdt σt, xt, utdWt E
gt, xt−, ut, eNdedt,
2.17
x0 a,
with the cost functional
T
Ju· E
lt, xt, utdt φxT .
2.18
0
In this case,
b t, x, y, z, c·, u bt, x, u,
σ t, x, y, z, c·, u σt, x, u,
g t, x, y, z, c·, u, · gt, x, u, ·,
f t, x, y, z, c·, u 0,
l t, x, y, z, c·, u, · lt, x, u,
γ y 0.
2.19
Hence, we have, by some direct computation/observation,
y· z· c·, · 0,
P2 · Q2 · K2 ·, · 0,
q· 0,
P3 · Q3 · K3 ·, · 0.
2.20
12
International Journal of Stochastic Analysis
Consequently, 2.10 and 2.11 or equivalently, 2.14 become
"
#
−dpt bx t, · pt σ x t, · kt g x t, e rt, eπde − lx t, · dt
E
− ktdWt − rt, eNdedt,
2.21
E
pT φx xT ,
−dP1 t &
bx t, · P1 t P1 tbx t, · σ x t, · P1 tσ x t, ·
σ x t, · Q1 t Q1 tσ x t, ·
$
g x t, e P1 tg x t, e g x t, e K1 t, eg x t, e
E
'
%
g x t, e K1 t, e K1 t, eg x t, e πde H xx t, · dt
− Q1 tdWt − K1 t, eNdedt,
2.22
E
P1 T −φxx xT ,
where the Hamiltonian function H : 0, T × Rn × U × Rn × Rn × Rn → R is defined as
(
)
H t, x, u, p, k, r· : p, bt, x, u k, σt, x, u
(
)
re, gt, x, u, e πde − lt, x, u, a.s.
2.23
E
The maximum condition 2.13 is reduced to the following:
H t, xt, ut, pt, kt, rt, · − H t, xt, u, pt, kt, rt, ·
1
− σt, xt, ut − σt, xt, u P1 tσt, xt, ut − σt, xt, u
2
1
−
gt, xt, ut, e − gt, xt, u, e P1 t K1 t, e
2 E
× gt, xt, ut, e − gt, xt, u, e πde ≥ 0, ∀u ∈ U, a.e., a.s.
2.24
These are the necessary conditions for the stochastic optimal control problem with random
jumps see 2.37 of Tang and Li 24
. When the jump coefficient g is independent of u·,
our result is reduced to Situ 23
. Where there are no random jumps, our result recovers the
well-known maximum principle for the classical stochastic optimal control problem of Peng
2
and Yong and Zhou 3
. And when U is convex, the classical result of Bensoussan 32
is
recovered.
International Journal of Stochastic Analysis
13
2 A Stochastic Optimal Control Problem for BSDEJs with Random Jumps
Consider a controlled BSDEJ:
−dyt f t, yt, zt, ct, ·, ut dt − ztdWt −
E
ct, eNdedt,
2.25
yT hT ,
with the cost functional
T
Ju· E
l t, yt, zt, ct, ·, ut dt γ y0 .
2.26
0
In this case,
b t, x, y, z, c·, u σ t, x, y, z, c·, u g t, x, y, z, c·, u, · 0,
f t, x, y, z, c·, u f t, y, z, c·, u , l t, x, y, z, c·, u g t, y, z, c·, u, · ,
φx 0.
2.27
Hence, we have
x· 0,
p· k· r·, · 0,
P1 · Q1 · K1 ·, · 0,
P2 · Q2 · K2 ·, · 0.
2.28
Consequently, 2.10 and 2.11 or equivalently, 2.16 become
%
$
%
$
dqt f y t, · qt ly t, · dt − f z t, · qt lz t, · dWt
$
%
−
f c t, e qt lc t, e Ndedt,
E
q0 Eγy y0 ;
&
−dP3 t − f y t, · P3 t − P3 tf y t, ·
$
g y t, eP1 tg y t, e g y t, eK1 t, eg y t, e g y t, e K2 t, e
E
−
E
2.29
'
%
K2 t, eg y t, e πde − qt f yy t, · − lyy t, · dt − Q3 tdWt
K3 t, eNdedt,
P3 T 0.
The maximum condition 2.13 is reduced to the following:
(
)
qt, f t, yt, zt, ct, ·, ut − f t, yt, zt, ct, ·, u
l t, yt, zt, ct, ·, ut − l t, yt, zt, ct, ·, u ≤ 0,
∀u ∈ U, a.e., a.s.
2.30
14
International Journal of Stochastic Analysis
As we know, this new result has not been published elsewhere. When there are no random
jumps, our result partially recovers that of Dokuchaev and Zhou 33
.
3 A Stochastic Optimal Control Problem for FBSDEJs with Random Jumps
Consider a controlled FBSDEJ:
dxt bt, xt, utdt σt, xt, utdWt E
gt, xt−, ut, eNdedt,
−dyt f t, xt, yt, zt, ct, ·, ut dt − ztdWt −
x0 a,
E
ct, eNdedt,
2.31
yT hxT ,
with the cost functional
T
l t, xt, yt, zt, ct, ·, ut dt φxT γ y0 .
Ju· E
2.32
0
In this case, we have
b t, x, y, z, c·, u bt, x, u,
σ t, x, y, z, c·, u σt, x, u,
g t, x, y, z, c·, u, · gt, x, u, ·.
2.33
Consequently, 2.10 becomes
"
#
−dpt bx t, · pt − f x t, · qt σ x t, · kt g x t, e rt, eπde − lx t, · dt
E
− ktdWt −
E
rt, eNdedt,
%
$
%
$
dqt f y t, · qt ly t, · dt − f z t, · qt lz t, · dWt
$
%
−
f c t, e qt lc t, e Ndedt,
E
pT φx xT hx xT qT ,
q0 Eγy y0 ,
2.34
International Journal of Stochastic Analysis
15
and 2.14, 2.15, 2.16 become
−dP1 t bx t, · P1 t P1 tbx t, · σ x t, · P1 tσ x t, ·
σ x t, · Q1 t Q1 tσ x t, · − f x t, · P2 t − P2 tf x t, ·
$
g x t, e P1 tg x t, e g x t, e K1 t, eg x t, e
E
%
g x t, e K1 t, e K1 t, eg x t, e πde H xx t, · dt
− Q1 tdWt − K1 t, eNdedt,
E
P1 T −φxx xT − hxx xT qT ,
−dP2 t bx t, ·P2 t σ x t, ·Q2 t − f x t, ·P3 t − P2 t f y t, ·
$
g x t, e P1 tg y t, e g x t, e K1 t, eg y t, e
E
%
g x t, eK2 t, e K1 t, eg y t, e πde H xy t, · dt
− Q2 tdWt − K2 t, eNdedt,
2.35
E
P2 T 0,
−dP3 t −f y t, · P3 t − P3 tf y t, · − qt f yy t, · − lyy t, · dt
− Q3 tdWt − K3 t, eNdedt,
E
P3 T 0,
where the Hamiltonian function H : 0, T × M × U × Rn × Rm × Rn × Rn → R is defined as
follows:
) (
)
(
H t, x, y, z, c·, u, p, q, k, r· : p, bt, x, u q, −f t, x, y, z, c·, u k, σt, x, u
(
)
re, gt, x, u, e πde − l t, x, y, z, c·, u .
E
2.36
The maximum condition 2.13 remains the same. When U is convex, we essentially recover
the results in Shi and Wu 26
and Øksendal and Sulem 25
partial information case, and
when σ, g are independent of u· and U is not assumed to be convex, our result is reduced
to that of Shi 27
. Note that in both of these cases, our second adjoint equations are new.
When there are no random jumps, our result partially recovers that of Wu 13
. Note
that in 3.27 of 13
, some additional parameters have to be involved and to be determined.
And when U is convex, our result is reduced to that of Peng 11
and Wang and Wu 34
partial observation case. Result of Xu 12
is recovered when σ is independent of u· and
U is not assumed to be convex.
16
International Journal of Stochastic Analysis
4 A Stochastic Optimal Control Problem for Fully Coupled FBSDEJs with Random Jumps
Consider the controlled fully coupled FBSDEJ 1.4 with the cost functional 1.5. When U
is convex, we essentially recover the results in Shi and Wu 28
and Meng and Sun 29
partial information case. When there are no random jumps, our result becomes a special
case of Yong 20
, because in 20
the author considered the mixed initial-terminal conditions,
and some additional necessary conditions for the optimal control are derived. When σ is
independent of u·, our result recovers those of Shi and Wu 19, 35
partial observation
case. And when U is convex, our result is reduced to that of Wu 17
.
3. Problem Reduction and the Proof of the Main Theorem
This section is devoted to the proof of our main theorem. The proof is lengthy and technical.
Therefore, we divide it into several steps to make the idea clear.
Step 1 problem reduction. Consider the following initial value problem for a control system
of SDEJs:
dxt b t, xt, yt, zt, ct, ·, ut dt σ t, xt, yt, zt, ct, ·, ut dWt
g t, xt−, yt−, zt, ct, ·, ut, e Ndedt,
E
dyt −f t, xt, yt, zt, ct, ·, ut dt ztdWt ct, eNdedt,
3.1
E
x0 a,
y0 y0 ,
where x·, y· is regarded as the state process and z·, c·, ·, u· as the control process.
It is standard that see 36
, for any z·, c·, ·, u· ∈ L2F,p 0, T ; Rm × Fp2 0, T ;
Rm × U0, T and y0 ∈ Rm , there exists a unique strong solution
x·, y· ≡ x ·, y0 , z·, c·, ·, u· , y ·, y0 , z·, c·, ·, u· ∈ L2F 0, T ; Rn × L2F 0, T ; Rm 3.2
to 3.1, depending on the 4-tuple y0 , z·, c·, ·, u·. Next, we regard the original terminal
condition as the terminal state constraint:
yT hxT .
3.3
Let A be the set of all 4-tuples y0 , z·, c·, ·, u· ∈ Rm × L2F,p 0, T ; Rm × Fp2
0, T ; Rm × U0, T such that the unique corresponding state process x·, y· satisfies the
constraint 3.3. Note that hypothesis H0 implies that, for any u· ∈ U0, T , there exists a
unique 4-tuple y0 , z·, c·, · ∈ Rm × L2F,p 0, T ; Rm × Fp2 0, T ; Rm such that state equation
3.1 admits a unique state process x·, y· ∈ L2F 0, T ; Rn × L2F 0, T ; Rm satisfying the
International Journal of Stochastic Analysis
17
state constraint 3.3. Hence, H0 and H1 imply A /
∅. We rewrite the cost functional 1.5
as follows:
T
J y0 , z·, c·, ·, u· E
l t, xt, yt, zt, ct, ·, ut dt φxT γ y0 .
3.4
0
Now, we can pose the following.
Find y , z·, c·, ·, u· ∈ A, such that
Problem C.
0
J y0 , z·, c·, ·, u· inf
y0 ,z·,c·,·,u·∈A
J y0 , z·, c·, ·, u· .
3.5
We, respectively, refer to z·, c·, ·, u· as an optimal control process, to x·, y·
as the corresponding optimal state process, and to y 0 , z·, c·, ·, u· as an optimal 4-tuple
of Problem C.
are equivalent. The advantage of Problem C
is that
It is clear that Problems C and C
one does not need much regularity/integrability on z·, c·, · since it is treated as part of a
control process; the disadvantage is that one has to treat terminal constraint 3.3.
Step 2 applying Ekeland’s variational principle. For convenience, let us cite Ekeland’s variational principle, whose proof can be found in Ekeland 37
or Yong and Zhou 3
.
Lemma 3.1. Let V, d·, · be a complete metric space, and let f· : V → R be lower-semicontinuous, bounded below. If for all ρ > 0 there exists u ∈ V satisfying
fu ≤ inf fv ρ,
3.6
v∈V
then there exists uρ ∈ V , satisfying the following:
i fuρ ≤ fu,
√
ii du, uρ ≤ ρ,
iii for all v ∈ V, fv √
ρdv, uρ ≥ fuρ .
with the corresponding
Let y0 , z·, c·, ·, u· be an optimal 4-tuple of Problem C,
optimal state process x·, y·. For any δ > 0, we define
J δ y0 , z·, c·, ·, u·
2 1/2
+2
. * J y0 , z·, c·, ·, u· − J y0 , z·, c·, ·, u· δ EyT − hxT 3.7
∀ y0 , z·, c·, ·, u· ∈ Rm × L2F,p 0, T ; Rm × Fp2 0, T ; Rm × U0, T .
If we define the Ekeland’s distance on U0, T by
.
dE u·, u
· Ω
u
t}|dP,
|{t ∈ 0, T , ut /
∀u·, u
· ∈ U0, T ,
3.8
18
International Journal of Stochastic Analysis
with |A| being the Lebesgue measure of set A ⊆ 0, T , then L2F,p 0, T ; Rm × Fp2 0, T ; Rm ×
U0, T is a complete metric space under the following metric:
$
%1/2
,
·2
z· − z·2 c·, · − c·, ·2π dE u·, u
∀z·, c·, ·, u·, z·, c·, ·, u
· ∈ L2F,p 0, T ; Rm × Fp2 0, T ; Rm × U0, T ,
3.9
. T
. T where z·2 E 0 |zt|2 dt, c·, ·2π E 0 E |ct, e|2 πdedt, and y0 , z·, c·, ·, u· →
J δ y0 , z·, c·, ·, u· is continuous. Also it is clear that
J δ y0 , z·, c·, ·, u·
> 0, ∀ y0 , z·, c·, ·, u· ∈ Rm × L2F,p 0, T ; Rm × Fp2 0, T ; Rm × U0, T ,
J δ y 0 , z·, c·, ·, u·
δ≤
inf
J δ y0 , z·, c·, ·, u· δ.
3.10
y0 ,z·,c·,·,u·∈Rm ×L2F,p 0,T ;Rm ×Fp2 0,T ;Rm ×U0,T Hence, by Lemma 3.1, there exists a 4-tuple y0δ , zδ ·, cδ ·, ·, uδ · ∈ Rm × L2F,p 0, T ; Rm ×
Fp2 0, T ; Rm × U0, T such that
i J δ y0δ , zδ ·, cδ ·, ·, uδ · ≤ J δ y 0 , z·, c·, ·, u· δ,
2 2
2
ii zδ · − z· cδ ·, · − c·, · dE uδ ·, u· ≤ δ,
π
2 2
2 #1/2
, "
iii − δ zδ · − z· cδ ·, · − c·, · dE uδ ·, u·
π
δ
δ
δ δ
≤ J y0 , z·, c·, ·, u· − J y0 , z ·, cδ ·, ·, uδ · ,
∀ y0 , z·, c·, ·, u· ∈ Rm × L2F,p 0, T ; Rm × Fp2 0, T ; Rm × U0, T .
3.11
Thus, y0δ , zδ ·, cδ ·, ·, uδ · is a global minimum point of the following penalized cost
functional
"
2 2
2 #1/2
, .
J δ y0 , z·, c·, ·, u· δ zδ · − z· cδ ·, · − c·, · dE uδ ·, u·
π
3.12
In other words, if we pose a penalized optimal control problem with the state constraint 3.3
and the cost functional 3.12, then y0δ , zδ ·, cδ ·, ·, uδ · is an optimal 4-tuple of the
problem. Note that this problem does not have state constraints, and the optimal 4-tuple
y0δ , zδ ·, cδ ·, ·, uδ · approaches y0 , z·, c·, ·, u· as δ → 0.
International Journal of Stochastic Analysis
19
Step 3 nontriviality of the multiplier. Note that the state process xδ ·, yδ · corresponding
to the 4-tuple y0δ , zδ ·, cδ ·, ·, uδ · satisfies the following SDEJ:
dxδ t b t, xδ t, yδ t, zδ t, cδ t, ·, uδ t dt
σ t, xδ t, yδ t, zδ t, cδ t, ·, uδ t dWt
g t, xδ t−, yδ t−, zδ t, cδ t, ·, uδ t, e Ndedt,
E
dyδ t −f t, xδ t, yδ t, zδ t, cδ t, ·, uδ t dt zδ tdWt xδ 0 a,
3.13
E
cδ t, eNdedt,
yδ 0 y0δ ,
with
2
Eyδ T − h xδ T 2
2
≤ J δ y0δ , zδ ·, cδ ·, ·, uδ · ≤ J δ y 0 , z·, c·, ·, u· δ2 −→ 0,
3.14
as δ −→ 0.
We now regard Θδ · ≡ xδ ·, yδ ·, zδ ·, cδ ·, · as the unique solution to the
following FBSDEJ:
dxδ t b t, xδ t, yδ t, zδ t, cδ t, ·, uδ t dt
σ t, xδ t, yδ t, zδ t, cδ t, ·, uδ t dWt
g t, xδ t−, yδ t−, zδ t, cδ t, ·, uδ t, e Ndedt,
E
−dyδ t f t, xδ t, yδ t, zδ t, cδ t, ·, uδ t dt − zδ tdWt −
x 0 a,
δ
y T h yδ T hδT ,
E
3.15
cδ t, eNdedt,
δ
with
hδT ≡ yδ T − h xδ T .
3.16
Next, for any ηT ∈ L2FT Ω; Rm , with
2
EηT ≤ 1,
3.17
20
International Journal of Stochastic Analysis
by H0, the following FBSDEJ
dxδ,ε t b t, xδ,ε t, yδ,ε t, zδ,ε t, cδ,ε t, ·, uδ t dt
σ t, xδ,ε t, yδ,ε t, zδ,ε t, cδ,ε t, ·, uδ t dWt
g t, xδ,ε t−, yδ,ε t−, zδ,ε t, cδ,ε t, ·, uδ t, e Ndedt,
E
−dyδ,ε t f t, xδ,ε t, yδ,ε t, zδ,ε t, cδ,ε t, ·, uδ t dt
δ,ε
− z tdWt − cδ,ε t, eNdedt,
E
xδ,ε 0 a,
yδ,ε T h yδ,ε T hδT εηT ,
3.18
admits a unique adapted solution Θδ,ε · ≡ xδ,ε ·, yδ,ε ·, zδ,ε ·, cδ,ε ·, ·. Note that FBSDEJ
3.18 is nothing but FBSDEJ 3.15 with only hδT replaced by hδT εηT , and uδ · remains
unchanged. Thus, by 1.9, we have
δ,ε
Θ · − Θδ ·
M2 0,T ≤ Kε2 ,
as ε −→ 0,
3.19
which implies
T T 2
2
2
δ,ε
δ,ε
lim yδ,ε 0 − yδ 0 E
z t − zδ t dt E
c t, e − cδ t, e πdedt
ε→0
0
0
E
0.
3.20
Hence, with y0δ,ε yδ,ε 0, by taking y0 , z·, c·, ·, u· y0δ,ε , zδ,ε ·, cδ,ε ·, ·, uδ · in the
last relation in 3.11, we have
,
−K δε ≤ J δ y0δ,ε , zδ,ε ·, cδ,ε ·, ·, uδ · − J δ y0δ , zδ ·, cδ ·, ·, uδ ·
2
2
J δ y0δ,ε , zδ,ε ·, cδ,ε ·, ·, uδ · − J δ y0δ , zδ ·, cδ ·, ·, uδ ·
J δ y0δ,ε , zδ,ε ·, cδ,ε ·, ·, uδ · J δ y0δ , zδ ·, cδ ·, ·, uδ ·
%2
$ J y0δ,ε , zδ,ε ·, cδ,ε ·, ·, uδ · − J y0 , z·, c·, ·, u· δ
J δ y0δ,ε , zδ,ε ·, cδ,ε ·, ·, uδ · J δ y0δ , zδ ·, cδ ·, ·, uδ ·
International Journal of Stochastic Analysis
21
$ %2
J y0δ , zδ ·, cδ ·, ·, uδ · − J y0 , z·, c·, ·, u· δ
− J δ y0δ,ε , zδ,ε ·, cδ,ε ·, ·, uδ · J δ y0δ , zδ ·, cδ ·, ·, uδ ·
$
2 2 %
E yδ,ε T − h xδ,ε T − yδ T − h xδ T J δ y0δ,ε , zδ,ε ·, cδ,ε ·, ·, uδ · J δ y0δ , zδ ·, cδ ·, ·, uδ ·
%
$ δ,ε δ,ε
δ,ε
δ
δ δ
δ
δ
≡ Φδ,ε
0 J y0 , z ·, c ·, ·, u · − J y0 , z ·, c ·, ·, u ·
.
δ,ε
δ,ε
δ
−
h
E Φδ,ε
,
y
−
h
x
T
T
T
T
$ %
≡ Φδ0 o1 J y0δ,ε , zδ,ε ·, cδ,ε ·, ·, uδ · − J y0δ , zδ ·, cδ ·, ·, uδ ·
$.%
εE ΦδT o1, ηT ,
3.21
where o1 stands for certain scalars or vectors that go to 0 as ε → 0,
.
Φδ,ε
0 J δ y0δ,ε , zδ,ε ·, cδ,ε ·, ·, uδ ·
2
J δ y0δ , zδ ·, cδ ·, ·, uδ ·
1
$ β J y0δ,ε , zδ,ε ·, cδ,ε ·, ·, uδ · − J y0 , z·, c·, ·, u·
×
0
Φδ,ε
T
.
3.22
δ δ
%
δ
δ
1 − β J y0 , z ·, c ·, ·, u · − J y0 , z·, c·, ·, u· dβ δ ,
yδ,ε T − h xδ,ε T yδ T − h xδ T ,
J δ y0δ,ε , zδ,ε ·, cδ,ε ·, ·, uδ · J δ y0δ , zδ ·, cδ ·, ·, uδ ·
J y0δ , zδ ·, cδ ·, ·, uδ · − J y 0 , z·, c·, ·, u· δ
.
Φδ0 ∈ 0, 1
,
J δ y0δ , zδ ·, cδ ·, ·, uδ ·
yδ T − h xδ T δ .
ΦT ∈ L2FT Ω; Rm .
δ δ
δ
δ
δ
J y0 , z ·, c ·, ·, u ·
3.23
We point out that Φδ0 , ΦδT is independent of ηT , and
Φδ0 ≥ 0,
2
2
δ
Φ0 EΦδT 1.
3.24
22
International Journal of Stochastic Analysis
Then there is a subsequence of Φδ0 , ΦδT , still denoted by Φδ0 , ΦδT , such that
lim Φδ0 , ΦδT Φ0 , ΦT ,
δ→0
with |Φ0 |2 E|ΦT |2 1.
3.25
We claim that
Φ0 /
0.
3.26
To show this, let us observe that by 3.19, one has
$ %
Φδ0 o1 J y0δ,ε , zδ,ε ·, cδ,ε ·, ·, uδ · − J δ y0δ , zδ ·, cδ ·, ·, uδ · ≤ εK Φδ0 o1 .
3.27
Hence, dividing by ε in 3.21 and then letting ε → 0 yields
.
,
−K δ ≤ K Φδ0 E ΦδT , ηT ,
3.28
with the constant K independent of ηT . Now, if Φ0 0, then the above leads to
.
−r δ ≤ E ΦδT , ηT ,
∀ηT ∈ L2FT Ω; Rm ,
2
EηT ≤ 1,
3.29
with r δ → 0, as δ → 0, uniformly in E|ηT |2 ≤ 1. Then,
2 !1/2
. sup E ΦδT , ηT ≤ r δ −→ 0,
EΦδT E|ηT |2 ≤1
as δ −→ 0.
3.30
Thus, for δ > 0 small enough, we must have
2
2
1 Φδ0 EΦδT < 1.
3.31
It is a contradiction which proves 3.26.
We refer to Φ0 , ΦT as the Lagrange multiplier of the corresponding optimal 4-tuple
y0 , z·, c·, ·, u·. Equation 3.26 shows the nontriviality of the Lagrange multiplier
Φ0 , ΦT .
International Journal of Stochastic Analysis
23
Step 4 spike variations. For notational simplicity, we now denote
⎞
⎛
!
!
!
z
a
xT x
X0 ,
,
XT XT X
,
v· ⎝c·⎠,
yT y0
y
u
!
b t, x, y, z, c·, u ,
Bt, X, v· −f t, x, y, z, c·, u
!
σ t, x, y, z, c·, u
Σt, X, v· ,
z
!
g t, x, y, z, c·, u, ·
,
Υt, X, v· c·
!
0
ΓX0 , XT φxT γ y0 , ΠX0 , XT ,
yT − hxT 3.32
Rl Rn × Rm , V0, T L2F,p 0, T ; Rm × Fp2 0, T ; Rm × U0, T ,
H Rl × L2FT Ω; Rl ≡ Rl × X2l , H0 Rn × L2FT Ω; Rm ≡ Rn × X2m .
Consequently,
J y0 , z·, c·, ·, u· J y0 , v·, · ,
J δ y0 , z·, c·, ·, u· J δ y0 , v·, · .
3.33
Note that H and H0 are Hilbert spaces. We identify H∗ H and H∗0 H0 . Also, from
the above,
Γ : H −→ R,
Π : H −→ H0 .
3.34
We denote the gradient DΓ and the Hessian D2 Γ of Γ as follows:
DΓX0 , XT ≡ ΓX0 X0 , XT , ΓXT X0 , XT ∈ LH; R ≡ H∗ H,
!
ΓX0 X0 X0 , XT ΓX0 XT X0 , XT ∈ Ls H; H,
D2 ΓX0 , XT ≡
ΓXT X0 X0 , XT ΓXT XT X0 , XT 3.35
where LH1 ; H2 is the set of all linear bounded operators from H1 to H2 , and Ls H; H is
the set of all linear bounded self-adjoint operators from H to itself. Clearly,
ΓX0 X0 , XT 0, γy y0
∈ Rl ,
ΓXT X0 , XT φx xT , 0 ∈ X2l ,
24
International Journal of Stochastic Analysis
ΓX0 X0 X0 , XT ΓX0 XT X0 , XT ΓXT X0 X0 , XT ΓXT XT X0 , XT !
0
0 ∈ Smn ,
0 γyy y0
!
0 0
∈ L X2l ; Rl ,
0 0
!
0 0
∈ L Rl ; X2l ,
0 0
!
φxx xT 0
∈ L X2l ; X2l .
0
0
3.36
Also, by the Fréchet differentiability, for Π : H → H0 , we have
DΠX0 , XT ≡ ΠX0 X0 , XT , ΠXT X0 , XT ∈ LH; H0 ,
!
ΠX0 X0 X0 , XT ΠX0 XT X0 , XT ∈ LH; LH; H0 ,
D2 ΠX0 , XT ≡
ΠXT X0 X0 , XT ΠXT XT X0 , XT 3.37
T ∈ H0 . Then,
≡ Φ
0, Φ
To make the above more precise, let us take any Φ
/
0 .
yT − hxT , Φ
T .
ΠX0 , XT , Φ
3.38
Thus,
"
#
"/
0#
≡ D ΠX0 , XT , Φ
DΠX0 , XT Φ
/
0 /
0 ,
, ΠX0 , XT , Φ
ΠX0 , XT , Φ
X0
XT
ΠXT X0 , XT Φ
∈ LH; R,
≡ ΠX0 X0 , XT Φ,
with
≡
ΠX0 X0 , XT Φ
≡
ΠXT X0 , XT Φ
"
/
0
ΠX0 , XT , Φ
/
0
ΠX0 , XT , Φ
0, 0,
X0
XT
T ;
T, Φ
−hx xT Φ
"/
0#
2
2
D ΠX0 , XT Φ ≡ D
ΠX0 , XT , Φ
#
3.39
International Journal of Stochastic Analysis
/
0
0
⎞
⎛/
,
X
,
X
ΠX
Φ
Φ
ΠX
,
,
0
T
0
T
⎟
⎜
/
0 X0 X0 /
0 X0 XT ⎟
⎜
⎠
⎝
ΠX0 , XT , Φ
ΠX0 , XT , Φ
XT X0
XT XT
ΠX0 XT X0 , XT Φ
ΠX0 X0 X0 , XT Φ
≡
∈ Ls H; H,
ΠXT X0 X0 , XT Φ ΠXT XT X0 , XT Φ
25
3.40
with
≡
ΠX0 X0 X0 , XT Φ
≡
ΠXT X0 X0 , XT Φ
≡
ΠX0 XT X0 , XT Φ
/
/
/
ΠX0 , XT , Φ
0
X0 X0
ΠX0 , XT , Φ
0
XT X0
ΠX0 , XT , Φ
/
0
≡ ΠX0 , XT , Φ
ΠXT XT X0 , XT Φ
XT XT
!
0 0
,
0 0
!
0 0
,
0 0
!
0 0
,
0 0
0
X0 XT
T 0
−hxx xT Φ
.
0
0
3.41
Next, for fixed X0 ∈ Rl , v· ∈ V0, T and any ε ∈ 0, 1, let
!
0
.
X0δ,ε X0δ √
,
εy0
vδ t, ·, t ∈ 0, T \ Sε ,
δ,ε
v t, · vt, ·, t ∈ Sε ,
3.42
3.43
for some measurable set Sε ⊆ 0, T with |Sε | εT . It is clear that vδ,ε ·, · ∈ V0, T .
Let X δ,ε · be the state process of 3.1 corresponding to X0δ,ε , vδ,ε ·, ·. Further, let
X1δ,ε · and X2δ,ε · be, respectively, the solutions to the following SDEJs:
%
$
δ
dX1δ,ε t BX
t, ·X1δ,ε tdt ΣδX ΔΣδ t, ·ISε t dWt
$
%
δ
ΥX
t, eX1δ,ε t− ΔΥδ t, eISε t Ndedt,
E
!
0
;
X1δ,ε 0 √
εy0
#
"
1 δ
δ
dX2δ,ε t BX
t, ·X2δ,ε t ΔBδ t, ·ISε t BXX
t, ·X1δ,ε t2 dt
2
"
#
1 δ
δ,ε
δ,ε
δ,ε
2
δ
δ
ΣX t, ·X2 t ΔΣX t, ·X1 tISε t ΣXX t, ·X1 t dWt
2
3.44
26
International Journal of Stochastic Analysis
"
#
1 δ
δ
δ
ΥX
t, eX2δ,ε t− ΔΥX
t, eX1δ,ε t−ISε t ΥXX
t, eX1δ,ε t−2 Ndedt,
2
E
X2δ,ε 0 0,
3.45
where ISε · denotes the indicator function of the set Sε , and for any X ∈ Rl ,
δ
BX
t, · BX t, X δ t, vδ t, · ,
ΣδX t, · ΣX t, X δ t, vδ t, · ,
δ
ΥX
t, · ΥX t, X δ t, vδ t, · ,
ΔBδ t, · B t, X δ t, vt, · − B t, X δ t, vδ t, · ,
ΔΣδ t, · Σ t, X δ t, vt, · − Σ t, X δ t, vδ t, · ,
ΔΥδ t, · Υ t, X δ t, vt, · − Υ t, X δ t, vδ t, · ,
ΔΣδX t, · ΣX t, X δ t, vt, · − ΣX t, X δ t, vδ t, · ,
δ
ΔΥX
t, · ΥX t, X δ t, vt, · − ΥX t, X δ t, vδ t, · ,
⎛-
1,δ
BXX
t, ·X, X
.⎞
⎜
⎟
⎜
⎟
⎜.⎟
⎜
⎟
2,δ
δ
B t, ·X, X ⎟,
BXX
t, ·X 2 ⎜
⎜ XX
⎟
⎜
⎟
..
⎜
⎟
.
⎝.⎠
l,δ
BXX t, ·X, X
⎛-
Σ1,δ
XX t, ·X, X
1,δ
ΥXX
t, ·X, X
1 ≤ i ≤ l,
i
δ
δ
Σi,δ
,
t,
X
·
Σ
v
·
t,
t,
t,
XX
XX
1 ≤ i ≤ l,
i,δ
i
ΥXX
t, X δ t, vδ t, · ,
t, · ΥXX
1 ≤ i ≤ l.
.⎞
⎜
⎟
⎜
⎟
⎜.⎟
⎜
⎟
2,δ
Σ t, ·X, X ⎟,
ΣδXX t, ·X 2 ⎜
⎜ XX
⎟
⎜
⎟
..
⎜
⎟
.
⎝.⎠
Σl,δ
XX t, ·X, X
⎛-
i,δ
i
BXX
t, X δ t, vδ t, · ,
t, · BXX
.⎞
⎜
⎟
⎜
⎟
⎜.⎟
⎜
⎟
2,δ
δ
Υ t, ·X, X ⎟,
ΥXX
t, ·X 2 ⎜
⎜ XX
⎟
⎜
⎟
..
⎜
⎟
.
⎝.⎠
l,δ
ΥXX
t, ·X, X
3.46
International Journal of Stochastic Analysis
27
We point out that everything is reduced to the classical optimal control problems for
SDEJs. Then by Lemma 2.1 of Tang and Li 24
, the following estimates hold for any integers
k ≥ 1:
2k
2k
sup EX1δ,ε t sup EX δ,ε t − X δ t ≤ Cεk ,
0≤t≤T
0≤t≤T
0≤t≤T
0≤t≤T
2k
2k
sup EX2δ,ε t sup EX δ,ε t − X δ t − X1δ,ε t ≤ Cε2k ,
3.47
2k
sup EX δ,ε t − X δ t − X1δ,ε t − X2δ,ε t o ε2k .
0≤t≤T
Now, from the last relation in 3.11, noting 3.19, we obtain
−
, *√
+
δ εy0 εT K
≤ J δ y0δ,ε , vδ,ε ·, · − J δ y0δ , vδ ·, ·
2
2
J δ y0δ,ε , vδ,ε ·, · − J δ y0δ , vδ ·, ·
J δ y0δ,ε , vδ,ε ·, · J δ y0δ , vδ ·, ·
$ %2 $ %2
J y0δ,ε , vδ,ε ·, · − J y 0 , v·, · δ − J y0δ , vδ ·, · − J y 0 , v·, · δ
J δ y0δ,ε , vδ,ε ·, · J δ y0δ , vδ ·, ·
" 2
2 #
E Π X0δ,ε , X δ,ε T − Π X0δ , X δ T J δ y0δ,ε , vδ,ε ·, · J δ y0δ , vδ ·, ·
%
$ δ,ε
δ,ε
δ
δ
−
J
y
≡ Φδ,ε
J
y
,
v
,
v
·
·
·,
·,
0
0
0
2
1
0
δ,ε
δ,ε
δ
δ
E
, Π X0 , X T − Π X0 , X T Φδ,ε
T
$ %
≡ Φδ0 o1 J y0δ,ε , vδ,ε ·, · − J y0δ , vδ ·, ·
/
! 0
0
δ,ε
δ,ε
δ
δ
−
Π
X
,
E
,
Π
X
,
X
,
X
T
T
0
0
ΦδT o1
3.48
δ,ε
δ,ε
δ,ε
δ
δ
where
Φ0 , ΦT is defined in 3.25 and Φ0 , ΦT is defined similarly to 3.22, with X0 a
replaced by 3.42. We have shown that along a sequence,
y0δ,ε
lim Φδ0 , ΦδT Φ0 , ΦT ,
δ→0
0.
with Φ0 /
3.49
28
International Journal of Stochastic Analysis
Note that
J
y0δ,ε , vδ,ε ·, ·
−J
y0δ , vδ ·, ·
T$ %
E
l t, X δ,ε t, vδ,ε t, · − l t, X δ t, vδ t, · dt
0
3.50
%
$ δ,ε
δ,ε
δ
δ
E Γ X0 , X T − Γ X0 , X T ≡ I1 I2 .
We first deal with term I1 . We have
T$ %
I1 E
l t, X δ,ε t, vδ,ε t, · − l t, X δ t X1δ,ε t X2δ,ε t, vδ,ε t, · dt
0
T$ l t, X δ t X1δ,ε t X2δ,ε t, vδ,ε t, ·
0
%
−l t, X δ t X1δ,ε t X2δ,ε t, vδ t, · dt
T$ %
E
l t, X δ t X1δ,ε t X2δ,ε t, vδ t, · − l t, X δ t, vδ t, · dt
E
0
T $- .%
E
lX t, X δ t X1δ,ε t X2δ,ε t, vδ,ε t, · , X δ,ε t − X δ t − X1δ,ε t − X2δ,ε t dt
0
1
E
2
E
T0
T0
lXX t, X δ t X1δ,ε t X2δ,ε t, vδ,ε t, ·
.
× X δ,ε t − X δ t − X1δ,ε t − X2δ,ε t , X δ,ε t − X δ t − X1δ,ε t − X2δ,ε t dt
D2 lδ,ε,1,2 X δ,ε t − X δ t − X1δ,ε t − X2δ,ε t ,
.
X δ,ε t − X δ t − X1δ,ε t − X2δ,ε t dt
T$ %
E
l t, X δ t, vδ,ε t, · − l t, X δ t, vδ t, · dt
0
T- .
E
lX t, X δ t, vδ,ε t, · − lX t, X δ t, vδ t, · , X1δ,ε t X2δ,ε t dt
0
1
E
2
E
T -$ %
lXX t, X δ t, vδ,ε t, · − lXX t, X δ t, vδ t, ·
0
.
× X1δ,ε t X2δ,ε t , X1δ,ε t X2δ,ε t dt
T0
.
D2 lδ,ε X1δ,ε t X2δ,ε t , X1δ,ε t X2δ,ε t dt
T- .
E
lX t, X δ t, vδ t, · , X1δ,ε t X2δ,ε t dt
0
T- .
1
E
lXX t, X δ t, vδ t, · X1δ,ε t X2δ,ε t , X1δ,ε t X2δ,ε t dt
2
0
T
.
E
D2 lδ X1δ,ε t X2δ,ε t , X1δ,ε t X2δ,ε t dt,
0
3.51
International Journal of Stochastic Analysis
29
with
.
D2 lδ,ε,1,2 1 $ β lXX t, β X δ t X1δ,ε t X2δ,ε t 1 − β X1δ,ε t, vδ t, ·
0
.
D2 lδ,ε %
−lXX t, X δ t X1δ,ε t X2δ,ε t, vδ t, · dβ,
1 $ β lXX t, βX δ t 1 − β X δ t X1δ,ε t X2δ,ε t , vδ,ε t, ·
0
− lXX t, βX δ t 1 − β X δ t X1δ,ε t X2δ,ε t , vδ t, ·
%
−lXX t, X δ t, vδ,ε t, · lXX t, X δ t, vδ t, · dβ,
.
D l 2 δ
1 $ β lXX t, βX δ t 1 − β X δ t X1δ,ε t X2δ,ε t , vδ t, ·
0
%
−lXX t, X δ t, vδ t, · dβ.
3.52
Next we deal with term I2 . Similarly, we have
X0δ,ε − X0δ
I2 E DΓ
X δ,ε T − X δ T 1
2
1
X0δ,ε − X0δ
X0δ,ε − X0δ
2 δ
D Γ
,
2
X δ,ε T − X δ T X δ,ε T − X δ T 2
1
X0δ,ε − X0δ
X0δ,ε − X0δ
2 δ,ε
,
D Γ
X δ,ε T − X δ T X δ,ε T − X δ T ≡ E ΓδX0 X0δ,ε − X0δ ΓδXT X δ,ε T − X δ T δ
3.53
.
1 - δ δ,ε
ΓX0 X0 X0 − X0δ , X0δ,ε − X0δ
2
.
1 - δ δ,ε
ΓXT XT X T − X δ T , X δ,ε T − X δ T 2
12
X0δ,ε − X0δ
X0δ,ε − X0δ
2 δ,ε
,
,
D Γ
X δ,ε T − X δ T X δ,ε T − X δ T with
. DΓδ ΓδX0 , ΓδXT ≡ ΓX0 X0δ , X δ T , ΓXT X0δ , X δ T ,
⎞
⎛
δ
δ
δ
Γ
0
X
,
X
T
X
X
0
Γ
0
0
.
0
X0 X0
⎠,
D2 Γδ ≡⎝
0 ΓδXT XT
0
ΓXT XT X0δ , X δ T 1 $
%
2 δ,ε .
D Γ β D2 Γ βX0δ 1 − β X0δ,ε , βXTδ 1 − β XTδ,ε − D2 Γ X0δ , X δ T dβ.
0
3.54
30
International Journal of Stochastic Analysis
Also, we have
2
0
δ,ε
δ,ε
δ
δ
, Π X0 , X T − Π X0 , X T E
Φδ,ε
T
2
1
0
δ,ε
δ
, Π 0, X T − Π 0, X T ≡E
Φδ,ε
T
1
2
0
δ
δ,ε
δ
E
ΠXT 0, X T , X T − X T Φδ,ε
T
2
1
0
1
δ
δ,ε
δ
δ,ε
δ
X T − X T , X T − X T ΠXT XT 0, X T Φδ,ε
2
T
2
1
0
2 δ,ε
δ,ε
δ
δ,ε
δ
,
X T − X T , X T − X T D Π
Φδ,ε
T
1
3.55
with
D Π
2
δ,ε
.
1 $
%
β ΠXT XT 0, βX δ T 1 − β X δ,ε T − ΠXT XT 0, X δ T dβ.
3.56
0
Under assumptions H2, H3, we have
δ,ε 2 δ,ε,1,2 δ,ε
δ,ε
Φ0 D l
≤ K X δ,ε t − X δ t − X1 t − X2 t,
δ,ε 2 δ,ε δ,ε 2 δ δ,ε
δ,ε
Φ0 D l Φ0 D l ≤ K X1 t X2 t,
0
δ,ε
δ,ε 2 δ,ε 2 δ,ε
δ,ε
δ
δ
Φ0 D Γ D Π
δ,ε ≤ K X0 − X0 X T − X T .
ΦT Hence, by 3.47, we have
δ,ε
δ,ε
2 δ,ε,1,2
δ,ε
δ
E Φδ,ε
,
X
D
l
−
X
−
X
−
X
t
t
t
t
0
2
1
.
X δ,ε t − X δ t − X1δ,ε t − X2δ,ε t .
2 δ,ε
Φδ,ε
X1δ,ε t X2δ,ε t , X1δ,ε t X2δ,ε t 0 D l
.
δ,ε
δ,ε
δ,ε
δ,ε
2 δ
,
X
Φδ,ε
X
D
l
X
X
t
t
t
t
0
2
2
1
1
1
2
X0δ,ε − X0δ
X0δ,ε − X0δ
δ,ε
2 δ,ε
Φ0 D Γ
,
X δ,ε T − X δ T X δ,ε T − X δ T 3.57
International Journal of Stochastic Analysis
1
2
0
X δ,ε T − X δ T , X δ,ε T − X δ T D2 Πδ,ε
δ,ε
ΦT
31
3
≤ KE X δ,ε t − X δ t − X1δ,ε t − X2δ,ε t
3 3 !
3 X1δ,ε t X2δ,ε t X0δ,ε − X0δ X δ,ε T − X δ T ≤ Kε3/2 .
3.58
Consequently, combining 3.50 with 3.55, from 3.48, we can derive that
−
, *√
+
δ εy0 εT K
%
$ δ,ε
δ,ε
δ
δ
≤ Φδ,ε
0 J y0 , v ·, · − J y0 , v ·, ·
1
E
Φδ,ε
0
2
0
δ,ε
δ
, Π 0, X T − Π 0, X T Φδ,ε
T
T"
E
l t, X δ t, vδ,ε t, · − l t, X δ t, vδ t, ·
0
.
- lX t, X δ t, vδ t, · , X1δ,ε t X2δ,ε t
.#
1- lXX t, X δ t, vδ t, · X1δ,ε t, X1δ,ε t dt
2
" E ΓX0 X0δ , X δ T X0δ,ε − X0δ
.
1ΓX0 X0 X0δ , X δ T X0δ,ε − X0δ , X0δ,ε − X0δ
2
ΓXT X0δ , X δ T X δ,ε T − X δ T .#
1δ
δ
δ,ε
δ
δ,ε
δ
ΓXT XT X0 , X T X T − X T , X T − X T 2
1
E
ΠXT
1
2
2
0
δ,ε
δ
, X T − X T Φδ,ε
T
0, X T δ
1
ΠXT XT
0, X T δ
2
0
δ,ε
δ
δ,ε
δ
O ε3/2
X T − X T , X T − X T δ,ε
ΦT
32
International Journal of Stochastic Analysis
T &
E
0
%
$
δ
δ,ε
δ
δ
Φδ,ε
0 l t, X t, v t, · − l t, X t, v t, ·
.
- Φδ,ε
lX t, X δ t, vδ t, · , X1δ,ε t X2δ,ε t
0
.'
1 δ,ε - δ,ε
δ,ε
δ
δ
Φ0 lXX t, X t, v t, · X1 t, X1 t dt
2
/
0 !0
√
δ
δ
ε Φδ,ε
,
X
Γ
,
X
T
X
0
0
0
y0
/
0 ! 0 !0
ε
δ
δ
X
Γ
,
X
,
Φδ,ε
T
X0 X0
0
0
y0
y0
2
2
1
0
δ,ε
δ,ε
δ,ε
δ
δ
δ
Φ0 ΓXT X0 , X T ΠXT 0, X T , X1 T X2 T Φδ,ε
T
2
1
0
1
δ,ε
δ,ε
δ,ε
δ
δ
δ
X1 T , X1 T Φ0 ΓXT XT X0 , X T ΠXT XT 0, X T Φδ,ε
2
T
O ε3/2 .
3.59
δ,ε ·, Ξ
δ,ε ·, Ψ
δ,ε ·, · be the adapted solution to the following BSDEJ:
Step 5 duality. Let Φ
"
#
δ,ε t Υδ t, e Ξ
δ,ε t Bδ t, · Φ
δ,ε t Σδ t, · Ψ
δ,ε t, eπde − Φδ,ε lδ t, · dt
−dΦ
X
X
X
0 X
E
δ,ε tdWt − Ξ
δ,ε t, eNdedt,
−Ψ
E
0
δ,ε
δ,ε
δ
δ
δ
T − Φ ΓXT X , X T ΠXT 0, X T ,
dΦ
0
0
Φδ,ε
T
3.60
.
δ
δ,ε ·, Ξ
δ,ε ·, Ψ
δ,ε ·, · goes to
where lX
t, · lX t, X δ t, vδ t, ·. Then, as ε, δ → 0, 0, Φ
·, which is the adapted solution to the following BSDEJ:
Φ·,
Ψ·,
Ξ·,
"
#
−dΦt BX t, · Φt ΣX t, · Ψt ΥX t, e Ξt, eπde − Φ0 lX t, · dt
E
eNdedt,
− ΨtdWt
− Ξt,
E
"
− Φ0 ΓXT X 0 , XT ΠXT 0, XT dΦT
!#
0
,
ΦT
3.61
International Journal of Stochastic Analysis
33
where
.
BX t, · BX t, Xt, vt, · ,
.
ΥX t, · BX t, Xt, vt, · ,
.
ΣX t, · ΣX t, Xt, vt, · ,
. lX t, · lX t, Xt, vt, · ,
3.62
Φ0 , ΦT is determined by 3.25.
δ,ε ·, X δ,ε · X δ,ε ·, we can get
Applying Itô’s formula to Φ
2
1
1
−E
Φδ,ε
0 ΓXT
X0δ , X δ T ΠXT 0, X T δ
2
0
δ,ε
δ,ε
, X1 T X2 T Φδ,ε
T
.
δ,ε T , X δ,ε T X δ,ε T E Φ
2
1
/
!0
δ,ε 0, √ 0
E Φ
εy0
E
T &/
0
δ
δ,ε t
δ,ε t − Σδ t, · Ψ
− BX
t, · Φ
X
−
E
δ
δ,ε t, eπde Φδ,ε lδ t, ·, X δ,ε t X δ,ε t
ΥX
t, e Ξ
0 X
2
1
0
0
/
1 δ
δ,ε
δ,ε
δ,ε
2
δ,ε
δ
δ
Φ t, BX t, · X1 t X2 t ΔB t, ·ISε t BXX t, ·X1 t
2
/
δ,ε t, Σδ t, · X δ,ε t X δ,ε t
Ψ
X
2
1
0
1
ΔΣδX t, ·X1δ,ε t ΔΣδ t, · ISε t ΣδXX t, ·X1δ,ε t2
2
/
δ,ε t, e, Υδ t, e X δ,ε t X δ,ε t
Ξ
X
2
1
E
0
'
1 δ
δ,ε
δ,ε
2
δ
δ
ΔΥX t, eX1 t ΔΥ t, e ISε t ΥXX t, eX1 t πde dt
2
/
!0
δ,ε 0, √ 0
E Φ
εy0
T &/
0
1 δ
δ,ε
2
δ,ε
δ
Φ t, ΔB t, ·ISε t BXX t, ·X1 t
E
2
0
.
δ,ε
δ,ε
δ
Φδ,ε
0 lX t, ·, X1 t X2 t
34
International Journal of Stochastic Analysis
/
0
δ,ε t, ΔΣδ t, ·ISε t 1 Σδ t, ·X δ,ε t2
Ψ
1
2 XX
/
0
'
δ,ε t, e, ΔΥδ t, eISε t 1 Υδ t, eX δ,ε t2 πde dt
Ξ
1
2 XX
E
O ε3/2 .
3.63
Then by 3.59, we have
−
, *√
+
δ εy0 εT K
≤E
T&
%
$
δ
δ,ε
δ
δ
−
l
t,
X
l
t,
X
Φδ,ε
v
·
v
·
t,
t,
t,
t,
0
0
Φδ,ε
0
-
δ
lX
t, ·, X1δ,ε t
X2δ,ε t
.
.
1 δ,ε - δ
δ,ε
δ,ε
Φ0 lXX t, ·X1 t, X1 t dt
2
/
!0
0
δ,ε
− E Φ 0, √
εy0
−E
T &/
0
δ,ε t, ΔBδ t, ·ISε t 1 Bδ t, ·X δ,ε t2
Φ
1
2 XX
0
.
δ,ε
δ,ε
δ
l
− Φδ,ε
·,
X
X
t,
t
t
0 X
2
1
/
0
1 δ
δ,ε
2
δ,ε
δ
Ψ t, ΔΣ t, ·ISε t ΣXX t, ·X1 t
2
/
0
'
δ,ε t, e, ΔΥδ t, eISε t 1 Υδ t, eX δ,ε t2 πde dt
Ξ
1
2 XX
E
/
√
δ
δ
E
ε Φδ,ε
0 ΓX0 X0 , X T ,
!0
0
y0
/
0 ! 0 !0
ε
δ,ε
δ
δ
,
Φ0 ΓX0 X0 X0 , X T y0
y0
2
1
1
δ
δ
X
Φδ,ε
Γ
,
X
T
X
X
T T
0
0
2
ΠXT XT
0, X T δ
0
Φδ,ε
T
2
X1δ,ε T , X1δ,ε T T&
%
$
δ
δ,ε
δ
δ
Φδ,ε
E
0 l t, X t, v t, · − l t, X t, v t, ·
0
O ε3/2
International Journal of Stochastic Analysis
35
.
1 δ,ε - δ
δ,ε
δ,ε
Φ0 lXX t, ·X1 t, X1 t dt
2
T &/
0
δ,ε t, ΔBδ t, ·ISε t 1 Bδ t, ·X δ,ε t2
Φ
1
2 XX
0
/
0
δ,ε t, ΔΣδ t, ·ISε t 1 Σδ t, ·X δ,ε t2
Ψ
1
2 XX
/
0
'
1 δ
δ,ε
2
δ,ε
δ
Ξ t, e, ΔΥ t, eISε t ΥXX t, eX1 t πde dt
2
E
/
!0
√
0
δ,ε
δ
δ
δ,ε
E
ε Φ0 ΓX0 X0 , X T − Φ 0,
y0
−E
/
0 ! 0 !0
ε
δ
δ
X
Γ
,
X
,
Φδ,ε
T
X0 X0
0
0
y0
y0
2
1
1
δ
δ
Φδ,ε
0 ΓXT XT X0 , X T 2
2
0
δ,ε
δ,ε
δ
ΠXT XT 0, X T O ε3/2 .
X1 T , X1 T δ,ε
ΦT
3.64
.
Note that Y δ,ε · X1δ,ε ·X1δ,ε · satisfies
dY δ,ε t
&
δ
δ
BX
t, ·Y δ,ε t Y δ,ε tBX
t, · ΣδX t, ·Y δ,ε tΣδX t, ·
%
$
ΔΣδ t, ·ΔΣδ t, · ΣδX t, ·X1δ,ε tΔΣδ t, · ΔΣδ t, ·X1δ,ε tΣδX t, · ISε t
&
δ
δ
ΥX
t, eY δ,ε tΥX
t, e
E
$
δ
ΥX
t, eX1δ,ε tΔΥδ t, e
δ
t, eX1δ,ε tΥX
t, e
'
'
%
ΔΥ t, eΔΥ t, · ISε t πde dt
ΔΥ
ΣδX t, ·Y δ,ε t Y δ,ε tΣδX t, ·
%
$
X1δ,ε tΔΣδ t, · ΔΣδ t, ·X1δ,ε t ISε t dWt
δ
δ
δ
δ
ΥX
t, eY δ,ε t− Y δ,ε t−ΥX
t, e ΥX
t, eY δ,ε t−ΥX
t, e
δ
E
δ
δ
$
δ
X1δ,ε t−ΔΥδ t, e ΔΥδ t, eX1δ,ε t− ΥX
t, eX1δ,ε t−ΔΥδ t, e
36
Y δ,ε 0 International Journal of Stochastic Analysis
%
δ
ΔΥδ t, eX1δ,ε t−ΥX
t, e ΔΥδ t, eΔΥδ t, e ISε t Ndedt,
!
0
0
.
0 εy0 y0
3.65
Now, let P δ,ε ·, Qδ,ε ·, K δ,ε ·, · be the adapted solution to the following BSDEJ:
−dP
δ,ε
&
δ
δ
t BX
t, · P δ,ε t P δ,ε tBX
t, · ΣδX t, · P δ,ε tΣδX t, ·
ΣδX t, · Qδ,ε t Qδ,ε tΣδX t, ·
$
δ
δ
δ
δ
ΥX
t, e P δ,ε tΥX
t, e ΥX
t, e K δ,ε t, eΥX
t, e
E
'
%
δ,ε,Φδ,ε
δ,ε
δ
δ,ε
δ
0
πde
H
ΥX
K
e
e
K
eΥ
e
·
dt
t,
t,
t,
t,
X t,
XX
− Qδ,ε tdWt −
dP
δ,ε
T −
Φδ,ε
0 ΓXT XT
E
K δ,ε t, eNdedt,
X0δ , X δ T ΠXT XT 0, X T δ
0
Φδ,ε
T
,
3.66
where
δ,ε,Φδ,ε
0
HXX
.
δ
δ
δ,ε t, Ξ
δ,ε t, Ψ
δ,ε t, · ,
,
X
v
·,
Φ
t, · HXX t, Φδ,ε
t,
t,
0
3.67
Ξ·
Ψ,
with HXX t, λ, X, v·, Φ,
being the Hessian of the following
0
. . /
H t, λ, X, v·, Φ, Ψ, Ξ· Φ, Bt, X, v· Ψ, Σt, X, v·
E
.
Ξe,
Υt, X, ve πde − λlt, X, v·.
3.68
International Journal of Stochastic Analysis
37
Applying Itô’s formula to P δ,ε ·Y δ,ε ·
, we have
1
Φδ,ε
0 ΓXT XT
−E
1
P
δ,ε
0 √
X0δ , X δ T 0
,
√
δ
2
0
Φδ,ε
T
2
X1δ,ε T , X1δ,ε T εy0
%
$
%
$
E tr P δ,ε T Y δ,ε T − tr P δ,ε 0Y δ,ε 0
E
T
0
εy0
0
ΠXT XT
0, X T $
tr P δ,ε t ΔΣδ t, ·ΔΣδ t, · ΣδX t, ·X1δ,ε tΔΣδ t, ·
%
ΔΣδ t, ·X1δ,ε tΣδX t, · ISε t
$
%
δ,ε,Φδ,ε
Qδ,ε t X1δ,ε tΔΣδ t, · ΔΣδ t, ·X1δ,ε t ISε t − HXX 0 t, ·Y δ,ε t
$
δ
P δ,ε t ΥX
t, eX1δ,ε tΔΥδ t, e
E
%
δ
ΔΥδ t, eX1δ,ε tΥX
t, e ΔΥδ t, eΔΥδ t, e ISε t
$
K δ,ε t, e X1δ,ε t−ΔΥδ t, e ΔΥδ t, eX1δ,ε t−
δ
δ
ΥX
t, eX1δ,ε t−ΔΥδ t, e ΔΥδ t, eX1δ,ε t−ΥX
t, e
%
δ
δ
ΔΥ t, eΔΥ t, e ISε t πde dt
E
T
&
#
"
tr P δ,ε t ΔΣδ t, ·ΔΣδ t, · ΔΥδ t, eΔΥδ t, e πde ISε t
0
E
E
K δ,ε t, eΔΥδ t, eΔΥδ t, e ISε tπde
δ,ε,Φδ,ε
0
−HXX
E
T
'
t, ·Y δ,ε t dt O ε3/2
&
tr ΔΣδ t, · P δ,ε tΔΣδ t, ·ISε t
0
E
%
$
ΔΥδ t, e P δ,ε t K δ,ε t, e ΔΥδ t, eISε tπde
/
0'
δ,ε,Φδ,ε
− HXX 0 t, ·X1δ,ε t, X1δ,ε t dt O ε3/2 .
3.69
38
International Journal of Stochastic Analysis
Consequently, from 3.64, we have
−
, *√
+
δ εy0 εT K
T
%
$
δ
δ,ε
δ
δ
≤E
Φδ,ε
0 l t, X t, v t, · − l t, X t, v t, ·
0
. .
δ,ε t, ΔΣδ t, ·
δ,ε t, ΔBδ t, · − Ψ
− Φ
−
E
.
δ,ε t, e, ΔΥδ t, e πde
Ξ
1
− ΔΣδ t, · P δ,ε tΔΣδ t, ·
2
'
%
$
1
δ
δ,ε
δ,ε
δ
−
ΔΥ t, e P t K t, e ΔΥ t, eπde ISε tdt
2 E
2
1
0
√
δ,ε
δ
δ
δ,ε
0,
ε Φ0 ΓX0 X0 , X T − Φ
E
y0
2
1
0
0
ε δ,ε
δ
δ
δ,ε
,
O ε3/2 .
Φ0 ΓX0 X0 X0 , X T − P 0
2
y0
y0
3.70
Step 6 variational inequality. In 3.70, dividing
sending δ → 0, we get
√
ε and then sending ε → 0 followed by
!0
0
y0
!
!0
0
0 − Φ0,
, ∀y0 ∈ Rm ,
y0
γy y0
3.71
!
0 .
Eγy y0
3.72
/
0 ≤ E Φ0 ΓX0 X0 , XT − Φ0,
/
E Φ0
which implies
Φ0
Φ0
Using a standard argument of Tang and Li 24
in 3.70, we have the following
variational inequality:
% .
$
Φ0 l t, Xt, vt, · − l t, Xt, vt, · − Φt,
B t, Xt, vt, · − B t, Xt, vt, ·
/
0
− Ψt, Σ t, Xt, vt, · − Σ t, Xt, vt, ·
International Journal of Stochastic Analysis
.
e, Υ t, Xt, vt, e − Υ t, Xt, vt, e πde
−
Ξt,
39
E
" "
#
#
1 − Σ t, Xt, vt, · − Σ t, Xt, vt, ·
P t Σ t, Xt, vt, · − Σ t, Xt, vt, ·
2
$ %
1
−
Υ t, Xt, vt, e − Υ t, Xt, vt, e P t Kt, e
2 E
%
$ × Υ t, Xt, vt, e − Υ t, Xt, vt, e πde ≥ 0, ∀v·, a.e., a.s.
3.73
Step 7 finalizing the proof . Since Φ0 /
0, by rescaling, we let Φ0 1. Then the first-order
adjoint equation 3.61 reduces to
#
"
−dΦt BX t, · Φt ΣX t, · Ψt ΥX t, e Ξt, eπde − lX t, · dt
E
eNdedt,
− ΨtdWt
− Ξt,
E
"
− ΓXT X 0 , XT ΠXT 0, XT ΦT
0
ΦT
3.74
!#
.
Let δ, ε → 0 in 3.66, then P δ,ε ·, Qδ,ε ·, K δ,ε ·, · → P ·, Q·, K·, · which is the
adapted solution to the following second-order adjoint equation:
&
−dP t BX t, · P t P tBX t, · ΣX t, · P tΣX t, · ΣX t, · Qt QtΣX t, ·
$
ΥX t, e P tΥX t, e ΥX t, e Kt, eΥX t, e
E
'
%
ΥX t, e Kt, e Kt, eΥX t, e πde H XX t, · dt
− QtdWt −
E
Kt, eNdedt,
"
P T − ΓXT XT X 0 , XT ΠXT XT 0, XT 0
ΦT
!#
,
3.75
where
.
· ,
H XX t, · H XX t, Xt, vt, ·, Φt,
Ψt,
Ξt,
3.76
40
International Journal of Stochastic Analysis
Ξ·
Ψ,
being the Hessian of the following
with H XX t, X, v·, Φ,
0
. . /
H t, X, v·, Φ, Ψ, Ξ· Φ, Bt, X, v· Ψ, Σt, X, v·
.
Ξe,
Υt, X, ve πde − lt, X, v·.
3.77
E
Also, from 3.72, we have
!
0 .
Eγy y0
Φ0
3.78
Note that
ΓXT X 0 , XT !
φx xT ,
0
ΓXT XT X 0 , XT !
φxx xT 0
.
0
0
3.79
On the other hand, since
/ Π 0, XT ,
0
ΦT
!0
yT − h XT ΦT ,
3.80
we have
ΠXT
0, XT ΠXT XT 0, XT 0
ΦT
0
ΦT
!
−hx xT ΦT
ΦT
,
−hxx xT ΦT 0
.
0
0
!
3.81
Let
≡
Φ·
!
p·
,
q·
k·
≡
Ψ·
,
k·
· ≡
Ξ·,
!
r·, ·
.
r·, ·
3.82
Then it follows from 3.74, 3.78, and 3.82 that
p0
q0
!
!
0 ,
Eγy y0
! pT −φx xT hx xT ΦT
.
qT −ΦT
By 3.82, we can rewrite the equations in 3.74 as
3.83
International Journal of Stochastic Analysis
! ! bx t, · −f x t, ·
kt
σ x t, · 0
pt
pt
−d
qt
qt
by t, · −f y t, ·
kt
σ y t, · 0
!
g x t, e 0
l t, ·
rt, e
dt
πde − x
r
e
g
0
e
t,
t,
l
y t, ·
E
y
!
kt
rt, e − dWt −
Ndedt,
t, e
kt
E r
41
3.84
and the variational inequality 3.73 now takes the form:
l t, xt, yt, z, c·, u − l t, xt, yt, zt, ct, ·, ut
(
)
− pt, b t, xt, yt, z, c·, u − b t, xt, yt, zt, ct, ·, ut
(
)
qt, f t, xt, yt, z, c·, u − f t, xt, yt, zt, ct, ·, ut
(
)
− kt, σ t, xt, yt, z, c·, u − σ t, xt, yt, zt, ct, ·, ut
)
(
−
rt, e, g t, xt, yt, z, ce, u − g t, xt, yt, zt, ct, e, ut πde
E
. − kt,
z − zt − r t, e, ce − ct, eπde
-
E
!
1 σ t, xt, yt, z, c·, u − σ t, xt, yt, zt, ct, ·, ut
P t
z − zt
2
!
σ t, xt, yt, z, c·, u − σ t, xt, yt, zt, ct, ·, ut
×
z − zt
−
1
−
2
E
!
g t, xt, yt, z, ce, u, e − g t, xt, yt, zt, ct, e, ut, e
ce − ct, e
× P t Kt, e
!
g t, xt, yt, z, ce, u, e − g t, xt, yt, zt, ct, e, ut, e
×
πde
ce − ct, e
≥ 0,
∀z, c·, u, a.e., a.s.
3.85
Thus, taking u u·, z zt εz0 and c· ct, · εc0 ·, then dividing by ε and sending
ε → 0, we get
( ) (
)
0 ≤ lz t, xt, yt, zt, ct, ·, ut , z0 − pt, bz t, xt, yt, zt, ct, ·, ut z0
(
) (
)
qt, fz t, xt, yt, zt, ct, ·, ut z0 − kt, σz t, xt, yt, zt, ct, ·, ut z0
42
International Journal of Stochastic Analysis
.
) (
rt, e, gz t, xt, yt, zt, ct, e, ut z0 − kt,
z0
E
( ) (
)
lc t, xt, yt, zt, ct, ·, ut , c0 · − pt, bc t, xt, yt, zt, ct, ·, ut c0 ·
(
) (
)
qt, fc t, xt, yt, zt, ct, ·, ut c0 · − kt, σc t, xt, yt, zt, ct, ·, ut c0 ·
)
(
−
rt, e, gc t, xt, yt, zt, ct, e, ut c0 e
E
− r t, e, c0 eπde, ∀z0 ∈ Rm , c0 · ∈ L2 E, BE, π; Rm .
−
E
3.86
Hence,
lz t, xt, yt, zt, ct, ·, ut − bz t, xt, yt, zt, ct, ·, ut pt
kt
fz t, xt, yt, zt, ct, ·, ut qt − σz t, xt, yt, zt, ct, ·, ut kt
− gz t, xt, yt, zt, ct, e, ut rt, eπde,
E
rt, · lc t, xt, yt, zt, ct, ·, ut − bc t, xt, yt, zt, ct, ·, ut pt
fc t, xt, yt, zt, ct, ·, ut qt − σc t, xt, yt, zt, ct, ·, ut kt
− gc t, xt, yt, zt, ct, e, ut rt, eπde.
E
3.87
Combining 3.83, 3.84 with 3.87, we arrive at 2.10.
Next, we have
"
0 !#
P T − ΓXT XT X 0 , XT ΠXT XT 0, XT ΦT
−φxx xT hxx xT ΦT 0
0
0
−φxx xT − hxx xT qT 0
.
0
0
3.88
Then 2.11 follows from 3.75.
Further, by taking z z·, c· c·, · in the variational inequality 3.85, we obtain
l t, xt, yt, zt, ct, ·, u − l t, xt, yt, zt, ct, ·, ut
(
)
− pt, b t, xt, yt, zt, ct, ·, u − b t, xt, yt, zt, ct, ·, ut
(
)
qt, f t, xt, yt, zt, ct, ·, u − f t, xt, yt, zt, ct, ·, ut
International Journal of Stochastic Analysis
43
)
(
− kt, σ t, xt, yt, zt, ct, ·, u − σ t, xt, yt, zt, ct, ·, ut
)
(
−
rt, e, g t, xt, yt, zt, ct, e, u, e − g t, xt, yt, zt, ct, e, ut, e πde
E
!
1 σ t, xt, yt, zt, ct, ·, u − σ t, xt, yt, zt, ct, ·, ut
P t
0
2
!
σ t, xt, yt, zt, ct, ·, u − σ t, xt, yt, zt, ct, ·, ut
×
0
−
1
−
2
E
!
g t, xt, yt, zt, ct, ·, u, e − g t, xt, yt, zt, ct, e, ut, e
0
× P t Kt, e
!
g t, xt, yt, zt, ct, ·, u, e − g t, xt, yt, zt, ct, e, ut, e
×
πde
0
≥ 0,
∀u ∈ U, a.e., a.s.,
3.89
which gives 2.13. This completes the proof of our main result—Theorem 2.1.
4. A Linear Quadratic Example
In this section, we briefly discuss a linear quadratic case, which serves as an illustrating
example of our main result. Consider the following linear controlled FBSDEJ:
*
+
dxt b1 xt b2 yt b3 zt b4 ct, · b5 ut dt
+
*
σ1 xt σ2 yt σ3 zt σ4 ct, · σ5 ut dWt
+
*
g1 xt− g2 yt− g3 zt g4 ct, e g5 ut Ndedt,
*
E
+
−dyt f1 xt f2 yt f3 zt f4 ct, · f5 ut dt
− ztdWt − ct, eNdedt,
4.1
E
x0 a,
yT h1 xT ,
with the quadratic cost functional given by
T
%
$
1
2
2
2
2
2
2
2
l1 x t l2 y t l3 z t l4 c t, · l5 u t dt φ1 x T γ1 y 0 .
Ju· E
2
0
4.2
In the above, bi , σi , gi , fi , li , i 1, . . . , 5 and h1 , φ1 , γ1 are all real constants of R. The control
domain U ⊆ R could be very arbitrary; in particular, U does not have to be a convex set. We let
44
International Journal of Stochastic Analysis
U0, T be the set of all Ft -predictable processes u : 0, T ×Ω → U such that sup0≤t≤T E|ut|i <
∞, ∀i 1, 2, . . ..
For any u· ∈ U0, T , we denote
⎛ ⎞
x
Ξ ⎝ y ⎠,
z
⎞
−f1 x − f2 y − f3 z − f4 c· − f5 u
At, Ξ, c·, u ⎝ b1 x b2 y b3 z b4 c· b5 u ⎠,
σ1 x σ2 y σ3 z σ4 c· σ5 u
⎛
4.3
and assume that H4.1
(
)
At, Ξ, c·, u − A t, Ξ , c ·, u , Ξ − Ξ
)
( g1 x − x g2 y − y g3 z − z g4 ce − c e , ce − c e πde
E
!
2
2
2
2
≤ −β1 x − x − β2 y − y z − z ce − c e πde ,
4.4
E
h1 ≥ μ1 > 0,
where β1 , β2 , and μ1 are given constants with β1 > 0, μ1 > 0, β2 ≥ 0. Then by Wu 1
, there
exists a unique adapted solution x·, y·, z·, c·, · to 4.1.
Now, suppose that the corresponding stochastic control problem admits an optimal
5-tuple x·, y·, z·, c·, ·, u·. Now let us look at Theorem 2.1. The first-order adjoint
equation 2.10 reads
"
#
g1 rt, eπde − l1 xt dt
−dpt b1 pt − f1 qt σ1 kt E
− ktdWt − rt, eNdedt,
E
#
dqt −b2 pt f2 qt − σ2 kt − g2 rt, eπde l2 yt dt
E
#
"
−b3 pt f3 qt − σ3 kt − g3 rt, eπde l3 zt dWt
E
+
*
−b4 pt− f4 qt− − σ4 kt − g4 rt, e l4 ct, e Ndedt,
"
4.5
E
pT −φ1 xT − h1 qT ,
q0 γ1 y 0 .
Similarly, we denote
⎛ ⎞
q
ϑ ⎝p⎠,
k
⎛
⎞
−b1 p f1 q − σ1 k − g1 r· l1 x
Bt, ϑ, r· ⎝−b2 p f2 q − σ2 k − g2 r· l2 y⎠.
−b2 p f3 q − σ3 k − g3 r· l3 z
4.6
International Journal of Stochastic Analysis
45
Since controlled FBSDEJ satisfies H4.1, we can easily check that the above FBSDEJ 4.4
satisfies the following condition H4.2:
)
Bt, ϑ, r·, u − B t, ϑ , r ·, u , ϑ − ϑ
(
−b4 p − p f4 q − q − σ4 k − k
E
)
−g4 re − r e l4 ce, re − r e πde
!
2
2
2 2
≥ β1 q − q β2 p − q k − k re − r e πde ,
(
4.7
E
− h1 ≤ −μ1 ,
where β1 , β2 , and μ1 are the same as in H4.1. Then by Wu 38
, FBSDEJ 4.4 admits a unique
adapted solution q·, p·, k·, r·, ·.
The 2 × 2 matrix-valued second-order adjoint equation 2.11 becomes
−dP t &
B P t P tB Σ P tΣ Σ Qt QtΣ Υ P tΥ
$
'
%
Υ Kt, eΥ Υ Kt, e Kt, eΥ πde − L dt
E
− QtdWt − Kt, eNdedt,
!
−φ1 0
,
0 0
P T 4.8
E
where
B
!
b1 b2
,
−f1 −f2
Σ
!
σ1 σ2
,
0 0
Υ
!
g 1 g2
,
0 0
L
As before, 4.5 can be split into the following three scalar BSDEJs:
&
−dP1 t 2b1 P1 t σ12 P1 t 2σ1 Q1 t − 2f1 P2 t g12 P1 t
$
'
%
2
g1 K1 t, e 2g1 K1 t, e πde − l1 dt
E
!
l1 0
.
0 l2
4.9
46
International Journal of Stochastic Analysis
− Q1 tdWt −
E
K1 t, eNdedt,
P1 T −φ1 ,
4.10
&
−dP2 t b2 P1 t b1 P2 t σ1 σ2 P1 t σ1 Q2 t σ2 Q1 t − f1 P3 t − f2 P2 t
g1 g2 P1 t E
'
+
*
g1 g2 K1 t, e g1 K2 t, e g2 K1 t, e πde dt
− Q2 tdWt −
E
K2 t, eNdedt,
P2 T 0,
4.11
&
−dP3 t 2b2 P2 t σ22 P1 t 2σ2 Q2 t − 2f2 P3 t g22 P1 t
$
'
%
g22 K1 t, e 2g2 K2 t, e πde − l2 dt
E
− Q3 tdWt − K3 t, eNdedt,
4.12
E
P3 T 0.
Note that since all the coefficients in 4.5 are constants and the terminal condition is
deterministic, we must have
Q· 0,
K·, · 0,
4.13
and P · is deterministic. Consequently,
!
−φ1 0
.
0 0
4.14
Ṗ1 t − 2b1 σ12 g12 P1 t 2f2 P2 t l1 ,
P1 T −φ1 ,
Ṗ2 t − b2 σ1 σ2 − g1 g2 P1 t b1 − f2 P2 t − f1 P3 t,
P2 T 0,
2
2
2
Ṗ3 t − σ2 g2 P1 t − σ2 P2 t 2f2 P3 t − l2 ,
P3 T 0,
4.15
*
+
Ṗ t − B P t P tB Σ P tΣ Υ P tΥ − L ,
P T And at the same time 4.7–4.10 reduce to
and Qi · 0, Ki ·, · 0, i 1, 2, 3. Clearly, linear ODE 4.11 admits a unique explicit
solution and then ODEs 4.12 do.
International Journal of Stochastic Analysis
47
Since the Hamiltonian function 2.12 reads
H x, y, z, c·, u, p, q, k, r·
b1 x b2 y b3 z b4 c· b5 u p − f1 x f2 y f3 z f4 c· f5 u q
σ1 x σ2 y σ3 z σ4 c· σ5 u k
g1 x g2 y g3 z g4 ce g5 u reπde
4.16
E
1 2
l1 x l2 y2 l3 z2 l4 c2 · l5 u2 ,
−
2
the maximum condition 2.13 now takes the form noting that K1 ·, · 0
"
#
b5 pt − f5 qt σ5 kt g5 rt, eπde ut − u
E
% 1$
%
1 $
− l5 u2 t − u2 − σ52 g52 ut − u
2 P1 t ≥ 0,
2
2
4.17
∀u ∈ U, t ∈ 0, T .
This implies that
b5 pt − f5 qt σ5 kt E
g5 rt, eπde
%
1
1$
l5 ut u
σ52 g52 ut − u
P1 t,
2
2 b5 pt − f5 qt σ5 kt g5 rt, eπde
≥
∀u ∈ U, u > ut, t ∈ 0, T ;
4.18
E
%
1
1$
≤ l5 ut u
σ52 g52 ut − u
P1 t,
2
2
∀u ∈ U, u ≤ ut, t ∈ 0, T .
Now, let us look at an interesting special case. Noting that some related problems
without random jumps were discussed by Shi and Wu 39
by a purely completion-of-squares
technique.
Suppose that
f1 > 0,
b1 f2 ,
σ1 f3 ,
b2 ≤ 0,
g1 f4 ,
σ3 ≤ 0,
σ2 −b3 ,
g4 ≤ 0,
g2 −b4 ,
g3 −σ4 .
4.19
It is very easy to check that the assumptions H4.1 and H4.2 hold. Now the first-order
adjoint equation 4.4 reads
"
#
g1 rt, eπde − l1 xt dt
−dpt b1 pt − f1 qt σ1 kt E
− kt dWt − rt, eNdedt,
E
48
International Journal of Stochastic Analysis
"
#
dqt −b2 pt b1 qt − σ2 kt − g2 rt, eπde l2 yt dt
E
#
"
σ2 pt σ1 qt − σ3 kt σ4 rt, eπde l3 zt dWt
E
+
*
g2 pt− g1 qt− − σ4 kt − g4 rt, e l4 ct, e Ndedt,
E
pT −φ1 xT − h1 qT ,
q0 γ1 y0 ,
4.20
and the second-order adjoint ODEs 4.12 reduce to
Ṗ1 t − 2b1 σ12 g12 P1 t 2b1 P2 t l1 ,
P1 T −φ1 ,
Ṗ2 t − b2 σ1 σ2 − g1 g2 P1 t − f1 P3 t,
P2 T 0,
2
2
2
P3 T 0.
Ṗ3 t − σ2 g2 P1 t − σ2 P2 t 2b1 P3 t − l2 ,
4.21
If, in addition, b1 σ2 0 noting that this does not mean that the state equation 4.1
is decoupled, then 4.21 further reduce to
Ṗ1 t − σ12 g12 P1 t l1 ,
P1 T −φ1 ,
Ṗ2 t − b2 − g1 g2 P1 t − f1 P3 t,
P2 T 0,
Ṗ3 t −g22 P1 t − l2 ,
4.22
4.23
4.24
P3 T 0.
We can first solve P1 · explicitly from 4.22:
P1 t −e
σ12 g12 T −t
T
l1 e
−σ12 g12 T −s
ds φ1 ,
4.25
t
and then solve P3 · explicitly from 4.24. Finally, explicit P2 · can be obtained from 4.23.
The maximum condition 4.18 remains the same.
However, we cannot obtain the explicit optimal control from the maximum condition
4.18 in general, since this depends heavily on the adjoint processes p·, q·, k·, r·, ·.
The explicit, adapted solution to fully coupled FBSDEJ such as 4.20 is an interesting and
challenging open problem. And the explicit optimal control in its state feedback form is rather
difficult to obtain even in some very simple cases when there are no random jumps. We will
consider some relevant issues in the future work.
5. Concluding Remarks
In this paper, we have discussed a general optimal control problem for fully coupled forwardbackward stochastic differential equations with random jumps FBSDEJs. The control
domain is not assumed to be convex, and the control variable appears in both diffusion and
jump coefficients of the forward equation. Enlightened by Tang and Li 24
, Wu 13
, Yong
International Journal of Stochastic Analysis
49
20
, necessary conditions of Pontraygin’s type for the optimal controls are derived by means
of spike variation technique and Ekeland variational principle. And the general maximum
principle for forward stochastic control systems with random jumps 24
and the maximum
principles for forward-backward stochastic control systems with random jumps 25, 26
are
recovered in this paper. A linear quadratic stochastic optimal control problem is discussed as
an illustrating example.
Acknowledgment
This work was supported by China Postdoctoral Science Foundation Funded Project no.
20100481278, Postdoctoral Innovation Foundation Funded Project of Shandong Province
no. 201002026, National Natural Sciences Foundations of China no. 11126209 and
Shandong Province no. ZR2011AQ012, and Independent Innovation Foundation of
Shandong University IIFSDU, no. 2010TS060.
References
1
Z. Wu, “Forward-backward stochastic differential equations with Brownian motion and Poisson process,” Acta Mathematicae Applicatae Sinica, vol. 15, no. 4, pp. 433–443, 1999.
2
S. G. Peng, “A general stochastic maximum principle for optimal control problems,” SIAM Journal on
Control and Optimization, vol. 28, no. 4, pp. 966–979, 1990.
3
J. M. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, vol. 43, Springer,
New York, NY, USA, 1999.
4
N. El Karoui, S. G. Peng, and M. C. Quenez, “Backward stochastic differential equations in finance,”
Mathematical Finance, vol. 7, no. 1, pp. 1–71, 1997.
5
N. El Karoui, S. G. Peng, and M. C. Quenez, “A dynamic maximum principle for the optimization of
recursive utilities under constraints,” The Annals of Applied Probability, vol. 11, no. 3, pp. 664–693, 2001.
6
M. Schroder and C. Skiadas, “Optimal consumption and portfolio selection with stochastic
differential utility,” Journal of Economic Theory, vol. 89, no. 1, pp. 68–126, 1999.
7
F. Antonelli, E. Barucci, and M. E. Mancino, “Asset pricing with a forward-backward stochastic
differential utility,” Economics Letters, vol. 72, no. 2, pp. 151–157, 2001.
8
S. L. Ji and X. Y. Zhou, “A maximum principle for stochastic optimal control with terminal state
constraints, and its applications,” Communications in Information and Systems, vol. 6, no. 4, pp. 321–
337, 2006.
9
N. Williams, “On dynamic principal-agent problems in continuous time,” Working paper, 2008,
http://www.ssc.wisc.edu/∼nwilliam/dynamic-pa1.pdf.
10
J. Cvitanić, X. H Wan, and J. F. Zhang, “Optimal contracts in continuous-time models,” Journal of
Applied Mathematics and Stochastic Analysis, vol. 2006, Article ID 95203, 27 pages, 2006.
11
S. G. Peng, “Backward stochastic differential equations and applications to optimal control,” Applied
Mathematics and Optimization, vol. 27, no. 2, pp. 125–144, 1993.
12
W. S. Xu, “Stochastic maximum principle for optimal control problem of forward and backward
system,” Journal of the Australian Mathematical Society Series B, vol. 37, no. 2, pp. 172–185, 1995.
13
Z. Wu, “A general maximum principle for optimal control problems of forward-backward stochastic
control systems,” Working paper, Shandong University, China, 2009.
14
J. Cvitanić and J. Ma, “Hedging options for a large investor and forward-backward SDE’s,” The Annals
of Applied Probability, vol. 6, no. 2, pp. 370–398, 1996.
15
D. Cuoco and J. Cvitanić, “Optimal consumption choices for a “large” investor,” Journal of Economic
Dynamics & Control, vol. 22, no. 3, pp. 401–436, 1998.
16
R. Buckdahn and Y. Hu, “Hedging contingent claims for a large investor in an incomplete market,”
Advances in Applied Probability, vol. 30, no. 1, pp. 239–255, 1998.
17
Z. Wu, “Maximum principle for optimal control problem of fully coupled forward-backward
stochastic systems,” Systems Science and Mathematical Sciences, vol. 11, no. 3, pp. 249–259, 1998.
50
International Journal of Stochastic Analysis
18
Q. X. Meng, “A maximum principle for optimal control problem of fully coupled forward-backward
stochastic systems with partial information,” Science in China. Series A, vol. 52, no. 7, pp. 1579–1588,
2009.
19
J. T. Shi and Z. Wu, “The maximum principle for fully coupled forward-backward stochastic control
system,” Acta Automatica Sinica, vol. 32, no. 2, pp. 161–169, 2006.
20
J. M. Yong, “Optimality variational principle for controlled forward-backward stochastic differential
equations with mixed initial-terminal conditions,” SIAM Journal on Control and Optimization, vol. 48,
no. 6, pp. 4119–4156, 2010.
21
R. C. Merton, “Option pricing when underlying stock returns are discontinuous,” Journal of Financial
Economics, vol. 3, no. 1-2, pp. 125–144, 1976.
22
R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC, Boca Raton,
Fla, USA, 2004.
23
R. Situ, “A maximum principle for optimal controls of stochastic systems with random jumps,” in
Proceedings of the National Conference on Control Theory and Applications, Qingdao, China, 1991.
24
S. J. Tang and X. J. Li, “Necessary conditions for optimal control of stochastic systems with random
jumps,” SIAM Journal on Control and Optimization, vol. 32, no. 5, pp. 1447–1475, 1994.
25
B. Øksendal and A. Sulem, “Maximum principles for optimal control of forward-backward stochastic
differential equations with jumps,” SIAM Journal on Control and Optimization, vol. 48, no. 5, pp. 2945–
2976, 2009.
26
J. T. Shi and Z. Wu, “Maximum principle for forward-backward stochastic control system with
random jumps and applications to finance,” Journal of Systems Science & Complexity, vol. 23, no. 2,
pp. 219–231, 2010.
27
J. T. Shi, “Global maximum principle for the forward-backward stochastic optimal control problem
with Poisson jumps,” Early View in Asian Journal of Control, vol. 16, no. 1, pp. 1–11, 2014.
28
J. T. Shi and Z. Wu, “Maximum principle for fully coupled forward-backward stochastic control
system with random jumps,” in Proceedings of the 26th Chinese Control Conference, pp. 375–380,
Zhangjiajie, China, 2007.
29
Q. X. Meng and Y. Z. Sun, “Optimal control problem of fully coupled forward-backward stochastic
systems with Poisson jumps under partial information,” in Proceedings of the 30th Chinese Control
Conference (CCC ’11), pp. 1317–1322, Yantai, China, 2011.
30
M. Kohlmann and X. Y. Zhou, “Relationship between backward stochastic differential equations and
stochastic controls: a linear-quadratic approach,” SIAM Journal on Control and Optimization, vol. 38,
no. 5, pp. 1392–1407, 2000.
31
J. Ma and J. M. Yong, Forward-Backward Stochastic Differential Equations and Their Applications, vol. 1702
of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1999.
32
A. Bensoussan, “Lectures on stochastic control,” in Nonlinear Filtering and Stochastic Control, vol. 972
of Lecture Notes in Mathematics, pp. 1–62, Springer, Berlin, Germany, 1982.
33
N. Dokuchaev and X. Y. Zhou, “Stochastic controls with terminal contingent conditions,” Journal of
Mathematical Analysis and Applications, vol. 238, no. 1, pp. 143–165, 1999.
34
G. C. Wang and Z. Wu, “The maximum principles for stochastic recursive optimal control problems
under partial information,” IEEE Transactions on Automatic Control, vol. 54, no. 6, pp. 1230–1242, 2009.
35
J. T. Shi and Z. Wu, “Maximum principle for partially-observed optimal control of fully-coupled
forward-backward stochastic systems,” Journal of Optimization Theory and Applications, vol. 145, no.
3, pp. 543–578, 2010.
36
N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, vol. 24, NorthHolland, Amsterdam, The Netherlands, 2nd edition, 1989.
37
I. Ekeland, “Nonconvex minimization problems,” Bulletin of The American Mathematical Society, vol. 1,
no. 3, pp. 443–474, 1979.
38
Z. Wu, “Fully coupled FBSDE with Brownian motion and Poisson process in stopping time duration,”
Journal of the Australian Mathematical Society, vol. 74, no. 2, pp. 249–266, 2003.
39
J. T. Shi and Z. Wu, “The LQ problem for fully coupled forward-backward stochastic control system,”
Journal of Shandong University, vol. 40, no. 1, pp. 11–17, 2005.
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