Abstracts - Facultatea de Matematică

space D � [0, T ]; R d+p� in the Meyer–Zheng topology. Using weak convergence in the
Meyer–Zheng topology, we shall give a general result on the existence of a weak solution
Y , with driving process X admitting a given distribution, defined on some filtered
probability space (Ω, F, P ; F). By examples, we can show that there are, indeed, weak
solutions which are not strong, i.e., are not solutions in the usual sense adapted to the
filtration F X generated by X. We will also discuss pathwise uniqueness and uniqueness
in law of the solution and conclude, similar to the Yamada–Watanabe theorem, that
pathwise uniqueness and weak existence ensure the existence of a (uniquely determined)
strong solution. Applying these concepts, finding a unique strong solution is divided into
two subtasks: To prove pathwise uniqueness and to prove weak existence for BSDE (1).
It turns out that pathwise uniqueness holds whenever every weak solution of BSDE (1)
has a.s. continuous paths, and this condition is even necessary if the driving process X
is a continuous local martingale satisfying the previsible representation property.
References:
[1] Buckdahn, R.; Engelbert, H.-J.; Ră¸scanu, A., On weak solutions of backward
stochastic differential equations, Theory Probab. and its Appl. Vol. 49, No.1, 70–108
(2004).
[2] Buckdahn, R.; Engelbert, H.-J., A backward stochastic differential equations
without strong solution, Theory Probab. and its Appl. Vol. 50, No.2 (2005).
[3] Buckdahn, R.; Engelbert, H.-J., On the notion of weak solutions of backward
stochastic differential equations, pp. 21, to appear in: Proceedings of the Fourth Colloquium
on Backward Stochastic Differential Equations and Their Applications, Shanghai,
P.R. China, May 29 – June 1, 2005.
[4] Buckdahn, R.; Engelbert, H.-J., On the continuity of weak solutions of backward
stochastic differential equations, manuscript, pp. 13, submitted.
Regular and singular optimal controls
H. O. FATTORINI
University of California, Department of Mathematics
Los Angeles, California 90095-1555
hof@math.ucla.edu
Let E be a Banach space, S(t) a strongly continuous semigroup in E. We deal
mostly with the time optimal control problem for
y ′ (t) = Ay(t) + u(t), y(0) = ζ (1)
with point target condition y(T ) = y and control constraint �u(t)� = 1 a. e. A
multiplier space Z is a space Z ⊃ E ∗ such that S(t) ∗ Z ⊆ E ∗ (t > 0). Pontryagin’s
maximum principle for a control u(t) in 0 ≤ t ≤ T is
〈S(T − t) ∗ z, u(t)) = max
�u�≤1 〈S(T − t)∗ z, u〉 a.e. in 0 = t = T (2)
3