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DCDS-B

A singular optimal stochastic control problem is studied. A
second-order maximum principle is presented. The second-order
adjoint processes are involved, though the diffusion of the control
system is control independent. The range theorem of vector-valued
measures is used to prove the maximum principle. Examples are given
to illustrate the applications.

DCDS

The workshop on ``Analysis and Control of Stochastic Partial Differential Equations" was held in Fudan University on December 3--6, 2012, which was jointly organized and financially supported by Fudan University and Tongji University. Many of the contributions in the special issue were reported in the workshop, and there are also some few others which are solicited from renowned researchers in the fields of stochastic partial differential equations (SPDEs). The contents of the special issue are divided into the following three parts.

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keywords:

MCRF

In this paper we
investigate classical solution of a semi-linear system of backward
stochastic integral partial differential equations driven by a
Brownian motion and a Poisson point process. By proving an
Itô-Wentzell formula for jump diffusions as well as an abstract
result of stochastic evolution equations, we obtain the stochastic
integral partial differential equation for the inverse of the
stochastic flow generated by a stochastic differential equation
driven by a Brownian motion and a Poisson point process. By
composing the random field generated by the solution of a backward
stochastic differential equation with the inverse of the stochastic
flow, we construct the classical solution of the system of backward
stochastic integral partial differential equations. As a result, we
establish a stochastic Feynman-Kac
formula.

DCDS

We study a zero-sum Dynkin game under Knghtian uncertainty. The
associated Hamiton-Jacobi-Bellman-Isaacs equation takes the form of
a semi-linear backward stochastic partial differential variational
inequality (SBSPDVI). We establish existence and uniqueness of a
strong solution by using the Banach fixed point theorem and a
comparison theorem. A solution to the SBSPDVI is used to construct a
saddle point of the Dynkin game. In order to establish this
verification we use the generalized Itó-Kunita-Wentzell formula
developed by Yang and Tang (2013).

DCDS

In this paper we study the optimal stochastic control problem for
a path-dependent stochastic system under a recursive path-dependent
cost functional, whose associated Bellman equation from dynamic programming
principle is a path-dependent fully nonlinear partial differential
equation of second order. A novel notion of viscosity solutions is
introduced by restricting the semi-jets on an $\alpha$-Hölder space
$\mathbf{C}^{\alpha}$ for $\alpha\in(0,\frac{1}{2})$. Using Dupire's
functional Itô calculus, we prove that the value functional of the
optimal stochastic control problem is a viscosity solution to the
associated path-dependent Bellman equation. A state-dependent approximation
of the path-dependent value functional is given.

DCDS

This paper is concerned with the switching game of a one-dimensional backward
stochastic differential equation (BSDE). The associated Bellman-Isaacs equation is a system of matrix-valued BSDEs living in a special unbounded convex domain with reflection on
the boundary along an oblique direction. In this paper, we show the existence of an adapted solution to this system of BSDEs with oblique reflection by the
penalization method, the monotone convergence, and the a priori estimates.

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