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Discrete & Continuous Dynamical Systems - A

November 2015 , Volume 35 , Issue 11

Special issue on analysis and control of stochastic partial differential equations

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Preface
Baojun Bian, Shanjian Tang and Qi Zhang
2015, 35(11): i-iv doi: 10.3934/dcds.2015.35.11i +[Abstract](641) +[PDF](132.3KB)
Abstract:
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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Global existence for the stochastic Degasperis-Procesi equation
Yong Chen and Hongjun Gao
2015, 35(11): 5171-5184 doi: 10.3934/dcds.2015.35.5171 +[Abstract](859) +[PDF](395.6KB)
Abstract:
This paper is concerned with the Cauchy problem of stochastic Degasperis-Procesi equation. Firstly, the local well-posedness for this system is established. Then the precise blow-up scenario for solutions to the system is derived. Finally, the gloabl well-posedness to the system is presented.
The obstacle problem for quasilinear stochastic PDEs with non-homogeneous operator
Laurent Denis, Anis Matoussi and Jing Zhang
2015, 35(11): 5185-5202 doi: 10.3934/dcds.2015.35.5185 +[Abstract](742) +[PDF](428.6KB)
Abstract:
We prove the existence and uniqueness of solution of the obstacle problem for quasilinear Stochastic PDEs with non-homogeneous second order operator. Our method is based on analytical technics coming from the parabolic potential theory. The solution is expressed as a pair $(u,\nu)$ where $u$ is a predictable continuous process which takes values in a proper Sobolev space and $\nu$ is a random regular measure satisfying minimal Skohorod condition. Moreover, we establish a maximum principle for local solutions of such class of stochastic PDEs. The proofs are based on a version of Itô's formula and estimates for the positive part of a local solution which is non-positive on the lateral boundary.
Invariant foliations for stochastic partial differential equations with dynamic boundary conditions
Zhongkai Guo
2015, 35(11): 5203-5219 doi: 10.3934/dcds.2015.35.5203 +[Abstract](574) +[PDF](435.9KB)
Abstract:
Invariant foliations are geometric structures useful for describing and understanding qualitative behaviors of nonlinear dynamical systems. They decompose the state space into regions of different dynamical regimes, and thus help depict dynamics. We investigate invariant foliations for a class of stochastic partial differential equations with random dynamical boundary conditions, and then provide an approximation for these foliations when the noise intensity is sufficiently small.
Large deviation principle for stochastic heat equation with memory
Yueling Li, Yingchao Xie and Xicheng Zhang
2015, 35(11): 5221-5237 doi: 10.3934/dcds.2015.35.5221 +[Abstract](811) +[PDF](401.5KB)
Abstract:
In this work, using the weak convergence argument, we prove a Freidlin-Wentzell's large deviation principle for a class of stochastic heat equations with memory and Dirichlet boundary conditions, where the nonlinear term is allowed to be of polynomial growth.
Exponential convergence of non-linear monotone SPDEs
Feng-Yu Wang
2015, 35(11): 5239-5253 doi: 10.3934/dcds.2015.35.5239 +[Abstract](702) +[PDF](451.2KB)
Abstract:
For a Markov semigroup $P_t$ with invariant probability measure $\mu$, a constant $\lambda>0$ is called a lower bound of the ultra-exponential convergence rate of $P_t$ to $\mu$, if there exists a constant $C\in (0,\infty)$ such that $$ \sup_{\mu(f^2)\le 1}||P_tf-\mu(f)||_\infty \le C e^{-\lambda t},\ \ t\ge 1.$$ By using the coupling by change of measure in the line of [F.-Y. Wang, Ann. Probab. 35(2007), 1333--1350], explicit lower bounds of the ultra-exponential convergence rate are derived for a class of non-linear monotone stochastic partial differential equations. The main result is illustrated by the stochastic porous medium equation and the stochastic $p$-Laplace equation respectively. Finally, the $V$-uniformly exponential convergence is investigated for stochastic fast-diffusion equations.
Stochastic Korteweg-de Vries equation driven by fractional Brownian motion
Guolian Wang and Boling Guo
2015, 35(11): 5255-5272 doi: 10.3934/dcds.2015.35.5255 +[Abstract](714) +[PDF](443.9KB)
Abstract:
We consider the Cauchy problem for the Korteweg-de Vries equation driven by a cylindrical fractional Brownian motion (fBm) in this paper. With Hurst parameter $H\geq\frac{7}{16}$ of the fBm, we obtain the local existence results with initial value in classical Sobolev spaces $H^s$ with $s\geq -\frac{9}{16}$. Furthermore, we give the relation between the Hurst parameter $H$ and the index $s$ to the Sobolev spaces $H^s$, which finds out the regularity between the driven term fBm and the initial value for the stochastic Korteweg-de Vries equation.
On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions: The critical case
Freddy Delbaen, Ying Hu and Adrien Richou
2015, 35(11): 5273-5283 doi: 10.3934/dcds.2015.35.5273 +[Abstract](658) +[PDF](371.7KB)
Abstract:
In F. Delbaen, Y. Hu and A. Richou (Ann. Inst. Henri Poincaré Probab. Stat. 47(2):559--574, 2011), the authors proved that uniqueness of solution to quadratic BSDE with convex generator and unbounded terminal condition holds among solutions whose exponentials are $L^p$ with $p$ bigger than a constant $\gamma$ ($p>\gamma$). In this paper, we consider the critical case: $p=\gamma$. We prove that the uniqueness holds among solutions whose exponentials are $L^\gamma$ under the additional assumption that the generator is strongly convex. These exponential moments are natural as they are given by the existence theorem.
Backward doubly stochastic differential equations with polynomial growth coefficients
Qi Zhang and Huaizhong Zhao
2015, 35(11): 5285-5315 doi: 10.3934/dcds.2015.35.5285 +[Abstract](720) +[PDF](504.4KB)
Abstract:
In this paper we study the solvability of backward doubly stochastic differential equations (BDSDEs for short) with polynomial growth coefficients and their connections with SPDEs. The corresponding SPDE is in a very general form, which may depend on the derivative of the solution. We use Wiener-Sobolev compactness arguments to derive a strongly convergent subsequence of approximating SPDEs. For this, we prove some new estimates to the solution and its Malliavin derivative of the corresponding approximating BDSDEs. These estimates lead to the verifications of the conditions in the Wiener-Sobolev compactness theorem and the solvability of the BDSDEs and the SPDEs with polynomial growth coefficients.
Degenerate backward SPDEs in bounded domains and applications to barrier options
Nikolai Dokuchaev
2015, 35(11): 5317-5334 doi: 10.3934/dcds.2015.35.5317 +[Abstract](599) +[PDF](442.9KB)
Abstract:
Backward stochastic partial differential equations of parabolic type in bounded domains are studied in the setting where the coercivity condition is not necessary satisfied. Generalized solutions based on the representation theorem are suggested. Some regularity is derived from the regularity of the first exit times of non-Markov characteristic processes. Uniqueness, solvability and regularity results are obtained. Applications to pricing and hedging of European barrier options are considered.
On forward and backward SPDEs with non-local boundary conditions
Nikolai Dokuchaev
2015, 35(11): 5335-5351 doi: 10.3934/dcds.2015.35.5335 +[Abstract](605) +[PDF](427.7KB)
Abstract:
We study linear stochastic partial differential equations of parabolic type with non-local in time or mixed in time boundary conditions. The standard Cauchy condition at the terminal time is replaced by a condition that mixes the random values of the solution at different times, including the terminal time, initial time and continuously distributed times. For the case of backward equations, this setting covers almost surely periodicity. Uniqueness, solvability and regularity results for the solutions are obtained. Some possible applications to portfolio selection are discussed.
On the Cauchy-Dirichlet problem in a half space for backward SPDEs in weighted Hölder spaces
Wenning Wei
2015, 35(11): 5353-5378 doi: 10.3934/dcds.2015.35.5353 +[Abstract](610) +[PDF](565.6KB)
Abstract:
This paper is concerned with solution in weighted Hölder spaces for backward stochastic partial differential equations (BSPDEs) in a half space. Considering the solution as functional with value in Banach spaces of stochastic processes, and using the methods of partial differential equations (PDEs), we establish the existence and uniqueness of classical solution for BSPDE in functional weighted Hölder spaces.
Backward stochastic Schrödinger and infinite-dimensional Hamiltonian equations
Qing Xu
2015, 35(11): 5379-5412 doi: 10.3934/dcds.2015.35.5379 +[Abstract](730) +[PDF](520.6KB)
Abstract:
The paper is concerned with a semi-linear backward stochastic Schrödinger equation in $\mathbb{R}^d$ or in its bounded domain of a $C^2$ boundary. Galerkin's finite-dimensional approximation method is used and the harmonic role of the Laplacian is shown. The existence, uniqueness and regularity are given for the weak solution of the equation. A more general backward stochastic Hamiltonian partial differential equation is also discussed.
Constrained viscosity solution to the HJB equation arising in perpetual American employee stock options pricing
Baojun Bian, Shuntai Hu, Quan Yuan and Harry Zheng
2015, 35(11): 5413-5433 doi: 10.3934/dcds.2015.35.5413 +[Abstract](630) +[PDF](508.1KB)
Abstract:
We consider the valuation of a block of perpetual ESOs and the optimal exercise decision for an employee endowed with them and with trading restrictions. A fluid model is proposed to characterize the exercise process. The objective is to maximize the overall discount returns for the employee through exercising the options over time. The optimal value function is defined as the grant-date fair value of the block of options, and is then shown by the dynamic programming principle to be a continuous constrained viscosity solution to the associated Hamilton-Jacobi-Bellman (HJB) equation, which is a fully nonlinear second order elliptic partial differential equation (PDE) in the plane. We prove the comparison principle and the uniqueness. The numerical simulation is discussed and the corresponding optimal decision turns out to be a threshold-style strategy. These results provide an appropriate method to estimate the cost of the ESOs for the company and also offer favorable suggestions on selecting right moments to exercise the options over time for the employee.
Some linear-quadratic stochastic differential games for equations in Hilbert spaces with fractional Brownian motions
Tyrone E. Duncan
2015, 35(11): 5435-5445 doi: 10.3934/dcds.2015.35.5435 +[Abstract](844) +[PDF](342.6KB)
Abstract:
A noncooperative, two person, zero sum, stochastic differential game is formulated and solved that is described by a linear stochastic equation in a Hilbert space with a fractional Brownian motion and a quadratic payoff functional for the two players. The stochastic equation can model stochastic partial differential equations not only with distributed strategies and noise but also with control strategies and noise restricted to the boundary of the domain. The optimal strategies for the two players are given explicitly. The verification method is a generalization of completion of squares and provides the optimal strategies directly without solving partial differential equations or backward stochastic differential equations. Some examples of games described by stochastic partial differential equations are given.
Switching game of backward stochastic differential equations and associated system of obliquely reflected backward stochastic differential equations
Ying Hu and Shanjian Tang
2015, 35(11): 5447-5465 doi: 10.3934/dcds.2015.35.5447 +[Abstract](708) +[PDF](451.9KB)
Abstract:
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.
A Dynkin game under Knightian uncertainty
Hyeng Keun Koo, Shanjian Tang and Zhou Yang
2015, 35(11): 5467-5498 doi: 10.3934/dcds.2015.35.5467 +[Abstract](730) +[PDF](586.3KB)
Abstract:
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).
A stochastic maximum principle with dissipativity conditions
Carlo Orrieri
2015, 35(11): 5499-5519 doi: 10.3934/dcds.2015.35.5499 +[Abstract](704) +[PDF](440.1KB)
Abstract:
In this paper we prove a version of the maximum principle, in the sense of Pontryagin, for the optimal control of a finite dimensional stochastic differential equation, driven by a multidimensional Wiener process. We drop the usual Lipschitz assumption on the drift term and substitute it with dissipativity conditions, allowing polynomial growth. The control enters both the drift and the diffusion term and takes values in a general metric space.
Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations
Shanjian Tang and Fu Zhang
2015, 35(11): 5521-5553 doi: 10.3934/dcds.2015.35.5521 +[Abstract](835) +[PDF](569.1KB)
Abstract:
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.

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