Systems & Control Letters 60 (2011) 344–349
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Systems & Control Letters
journal homepage: www.elsevier.com/locate/sysconle
Existence of optimal controls for systems driven by FBSDEs
✩
Khaled Bahlali
a
, Boulekhrass Gherbal
b
, Brahim Mezerdi
b,∗
a
IMATH, UFR Sciences, USTV, B.P. 20132, 83957 La Garde, Cedex, France
b
Laboratory of Applied Mathematics, University Med Khider, BP 145, Biskra 07000, Algeria
article info
Article history:
Received 19 April 2010
Received in revised form
29 January 2011
Accepted 28 February 2011
Available online 29 March 2011
Keywords:
Forward backward stochastic differential
equation
Stochastic control
Weak convergence
Existence
abstract
We prove the existence of optimal relaxed controls as well as strict optimal controls for systems governed
by non linear forward–backward stochastic differential equations (FBSDEs). Our approach is based on
weak convergence techniques for the associated FBSDEs in the Jakubowski S-topology and a suitable
Skorokhod representation theorem.
© 2011 Elsevier B.V. All rights reserved.
1. Introduction
In this paper, we study the existence of optimal controls for
systems driven by FBSDEs of the form
X
t
= x +
∫
t
0
b(s, X
s
, U
s
)ds +
∫
t
0
σ(s, X
s
)dW
s
,
Y
t
= g (X
T
) +
∫
T
t
f (s, X
s
, Y
s
, U
s
)ds
−
∫
T
t
Z
s
dW
s
− (M
T
− M
t
)
(1.1)
where b, σ , f and g are given functions, (W
t
, t ≥ 0) is a standard
Brownian motion, defined on some filtered probability space
(Ω, F , F
t
, P ), satisfying the usual conditions. X , Y , Z are square
integrable adapted processes and M is a square integrable
martingale which is orthogonal to W . The control variable U
t
,
called strict control, is a measurable, F
t
-adapted process with
values in a compact metric space A. The expected cost on the time
interval [0, T ] is of the form
J (U ) = E
[
l(Y
0
) +
∫
T
0
h(t , X
t
, Y
t
, U
t
)dt
]
. (1.2)
✩
Partially supported by CMEP, PHC Tassili no. 07 MDU 705 and Marie Curie ITN
no. 213841-2.
∗
Corresponding author. Tel.: +213 33 74 77 88; fax: +213 33 74 77 88.
E-mail addresses: bahlali@univ-tln.fr (K. Bahlali), bgherbal@yahoo.fr
(B. Gherbal), bmezerdi@yahoo.fr (B. Mezerdi).
The objective of the controller is to minimize this cost function,
over the class U of admissible controls, that is, adapted processes
with values in some set A, called the action space. A control u is
called optimal if it satisfies J ( u) = inf{J (u), u ∈ U}.
In the case of forward Itô’s SDEs, the existence of such a strict
optimal control follows from the convexity of the image of the
action space A by the mapping (b(t , x, .), σ
2
(t , x, .), h(t , x, .)),
which is known as the Roxin-type convexity condition, see for
instance [1–3]. Without this convexity condition, an optimal
control may fail to exist in U. It should be noted that the set U is not
equipped with a compact topology. The idea is then to introduce a
new class R of admissible controls, in which the controller chooses
at time t , a probability measure q
t
(da) on the control set A, rather
than an element u
t
∈ A. These are called relaxed controls.
Using compactification techniques, Fleming [4] derived the
first existence result of an optimal relaxed control for SDEs
with uncontrolled diffusion coefficient. The case of SDEs with a
controlled diffusion coefficient has been solved by El-Karoui et
al. [1], where the optimal relaxed control is shown to be Markovian.
Linear backward stochastic differential equations (BSDEs) have
been studied in the early seventies by Bismut [5], in connection
with the stochastic version of the Pontriagin maximum principle.
More precisely, the adjoint process in the maximum principle
satisfies a linear BSDE. The first existence and uniqueness result
for non linear BSDEs has been proved by Pardoux and Peng [6].
This important paper has given rise to a huge literature on BSDEs
and has become a powerful tool in many fields such as financial
mathematics, optimal control, stochastic games, semi linear and
quasi linear partial differential equations, differential geometry
0167-6911/$ – see front matter © 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.sysconle.2011.02.011