Appl Math Optim 51:279–332 (2005)
2005 Springer Science+Business Media, Inc.
Generalized Directional Gradients, Backward Stochastic Differential
Equations and Mild Solutions of Semilinear Parabolic Equations
and Gianmario Tessitore
Dipartimento di Matematica, Politecnico di Milano,
piazza Leonardo da Vinci 32, 20133 Milano, Italy
Dipartimento di Matematica, Universit`a di Parma,
via d’Azeglio 85, 43100 Parma, Italy
Abstract. We study a forward–backward system of stochastic differential equa-
tions in an inﬁnite-dimensional framework and its relationships with a semilinear
parabolic differential equation on a Hilbert space, in the spirit of the approach of
Pardoux–Peng. We prove that the stochastic system allows us to construct a unique
solution of the parabolic equation in a suitable class of locally Lipschitz real func-
tions. The parabolic equation is understood in a mild sense which requires the notion
of a generalized directional gradient, that we introduce by a probabilistic approach
and prove to exist for locally Lipschitz functions. The use of the generalized direc-
tional gradient allows us to cover various applications to option pricing problems
and to optimal stochastic control problems (including control of delay equations
and reaction–diffusion equations), where the lack of differentiability of the coefﬁ-
cients precludes differentiability of solutions to the associated parabolic equations
of Black–Scholes or Hamilton–Jacobi–Bellman type.
Key Words. Backward stochastic differential equations, Partial differential equa-
tions in Hilbert spaces, Quadratic variation, Hamilton–Jacobi–Bellman equations.
AMS Classiﬁcation. Primary 60H30, 35R15, secondary 83E20, 49E20.
The ﬁrst author was partially supported by the European Community’s Human Potential Programme un-
der Contract HPRN-CT-2002-00279, QP-Applications. Both authors were partially supported by the European
Community’s Human Potential Programme under Contract HPRN-CT-2002-00281, Evolution Equations.