A second-order maximum principle for singular optimal stochastic controls
Shanjian Tang
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.
keywords: first and second adjoint processes vector-valued measure theory. second-order maximum principle Singular optimal stochastic control spike variation
Baojun Bian Shanjian Tang Qi Zhang
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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Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process
Shaokuan Chen Shanjian Tang
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.
keywords: stochastic differential equation Poisson point process Backward stochastic integral partial differential equation backward stochastic differential equation stochastic flow Itô-Wentzell formula.
A Dynkin game under Knightian uncertainty
Hyeng Keun Koo Shanjian Tang Zhou Yang
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).
keywords: Backward stochastic partial differential equation variational inequality Dynkin game super-parabolicity.
Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations
Shanjian Tang Fu Zhang
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.
keywords: backward SDE dynamic programming. Path-dependent optimal stochastic control viscosity solution path-dependent Bellman equation
Switching game of backward stochastic differential equations and associated system of obliquely reflected backward stochastic differential equations
Ying Hu Shanjian Tang
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.
keywords: backward stochastic differential equations oblique reflection. Switching game

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