We study stochastic parabolic and elliptic PDEs driven by purely spatial white noise. Even the simplest equations driven by this noise often do not have a square-integrable solution and must be solved in special weighted spaces. We demonstrate that the Cameron-Martin version of the Wiener chaos decomposition is an effective tool to study both stationary and evolution equations driven by space-only noise. The paper presents results about solvability of such equations in weighted Wiener chaos spaces and studies the long-time behavior of the solutions of evolution equations with space-only noise.
The purpose of this paper is to give a survey of a class of maximal inequalities for purely discontinuous martingales, as well as for stochastic integral and convolutions with respect to Poisson measures, in infinite dimensional spaces. Such maximal inequalities are important in the study of stochastic partial differential equations with noise of jump type.
The purpose of this paper is to establish the theory of stochastic pseudo-differential operators and give its applications in stochastic partial differential equations. First, we introduce some concepts on stochastic pseudo-differential operators and prove their fundamental properties. Also, we present the boundedness theory, invertibility of stochastic elliptic operators and the Garding inequality. Moreover, as an application of the theory of stochastic pseudo-differential operators, we give a Calderon-type uniqueness theorem on the Cauchy problem of stochastic partial differential equations.…
Under general conditions we show that the solution of a stochastic parabolic partial differential equation of the form \[ \partial_t u = \mathrm{div} (A \nabla u) + f(t,x, u) + g_i (t,x,u) \dot{w}^i_t \] is almost surely Hölder continuous in both space and time variables.
In this Note, we present a Calderón-type uniqueness theorem on the Cauchy problem of stochastic partial differential equations. To this aim, we introduce the concept of stochastic pseudo-differential operators, and establish their boundedness and other fundamental properties. The proof of our uniqueness theorem is based on a new Carleman-type estimate.
In this paper we introduce a class of forward-backward stochastic differential equations on tensor fields of Riemannian manifolds, which are related to semi-linear parabolic partial differential equations on tensor fields. Moreover, we will use these forward-backward stochastic differential equations to give a stochastic characterization of incompressible Navier-Stokes equations on Riemannian manifolds, where some extra conditions used in [22] are not required.
We consider the problem of constructing weak solutions to the Itô and to the Stratonovich stochastic differential equations having critical-order singularities in the drift and critical-order discontinuities in the dispersion matrix.
A class of (possibly) degenerate stochastic integro-differential equations of parabolic type is considered, which includes the Zakai equation in nonlinear filtering for jump diffusions. Existence and uniqueness of the solutions are established in Bessel potential spaces.
We introduce a class of second order backward stochastic differential equations and show relations to fully non-linear parabolic PDEs. In particular, we provide a stochastic representation result for solutions of such PDEs and discuss Monte Carlo methods for their numerical treatment.
We prove maximal inequalities for $L_q$-valued martingales obtained by stochastic integration with respect to compensated random measures. A version of these estimates for integrals with respect to compensated Poisson random measures were first obtained by Dirksen (arXiv:1208:3885) using arguments based on inequalities for sums of independent Banach-space-valued random variables, geometric properties of Banach spaces, and decoupling inequalities. Our proofs are completely different and rely almost exclusively on classical stochastic analysis for real semimartingales.
Random invariant manifolds are geometric objects useful for understanding complex dynamics under stochastic influences. Under a nonuniform hyperbolicity or a nonuniform exponential dichotomy condition, the existence of random pseudo-stable and pseudo-unstable manifolds for a class of \emph{random} partial differential equations and \emph{stochastic} partial differential equations is shown. Unlike the invariant manifold theory for stochastic \emph{ordinary} differential equations, random norms are not used. The result is then applied to a nonlinear stochastic partial differential equation with …
In this paper, we deal with a class of backward doubly stochastic differential equations (BDSDEs, in short) involving subdifferential operator of a convex function and driven by Teugels martingales associated with a Lévy process. We show the existence and uniqueness result by means of Yosida approximation. As an application, we give the existence of stochastic viscosity solution for a class of multivalued stochastic partial differential-integral equations (MSPIDEs, in short).
In the G-framework, we establish existence of an optimal stochastic relaxed control for stochastic differential equations driven by a G-Brownian motion.
We prove the existence of random dynamical systems and random attractors for a large class of locally monotone stochastic partial differential equations perturbed by additive Lévy noise. The main result is applicable to various types of SPDE such as stochastic Burgers type equations, stochastic 2D Navier-Stokes equations, the stochastic 3D Leray-$α$ model, stochastic power law fluids, the stochastic Ladyzhenskaya model, stochastic Cahn-Hilliard type equations, stochastic Kuramoto-Sivashinsky type equations, stochastic porous media equations and stochastic $p$-Laplace equations.
In this paper, stability theorems for stochastic differential equations and backward stochastic differential equations driven by G-Brownian motion are obtained. We show the existence and uniqueness of solutions to forward-backward stochastic differential equations driven by G-Brownian motion. Stability theorem for forward-backward stochastic differential equations driven by G-Brownian motion is also presented.
In this paper, we consider the Cauchy problem of semi-linear degenerate backward stochastic partial differential equations (BSPDEs in short) under general settings without technical assumptions on the coefficients. For the solution of semi-linear degenerate BSPDE, we first give a proof for its existence and uniqueness, as well as regularity. Then the connection between semi-linear degenerate BSPDEs and forward backward stochastic differential equations (FBSDEs in short) is established, which can be regarded as an extension of Feynman-Kac formula to non-Markov frame.
We study the "periodic homogenization" for a class of nonlocal partial differential equations of parabolic-type with rapidly oscillating coefficients, related to stochastic differential equations driven by multiplicative isotropic $α$-stable Lévy noise ($1<α<2$) which is nonlinear in the noise component. Our homogenization method is probabilistic. It turns out that, under suitable regularity assumptions, the limit of the solutions satisfies a nonlocal partial differential equation with constant coefficients, which are associated to a symmetric $α$-stable Lévy process.
In this paper, we study forward-backward doubly stochastic differential equations driven by Brownian motions and Poisson process (FBDSDEP in short). Both the probabilistic interpretation for the solutions to a class of quasilinear stochastic partial differential-integral equations (SPDIEs in short) and stochastic Hamiltonian systems arising in stochastic optimal control problems with random jumps are treated with FBDSDEP. Under some monotonicity assumptions, the existence and uniqueness results for measurable solutions of FBDSDEP are established via a method of continuation. Furthermore, the c…
In this paper we show that solutions of stochastic partial differential equations driven by Brownian motion can be approximated by stochastic partial differential equations forced by pure jump noise/random kicks. Applications to stochastic Burgers equations are discussed.
In this paper, we study backward doubly stochastic differential equations driven by Brownian motions and Poisson process (BDSDEP in short) with non-Lipschitz coefficients on random time interval. The probabilistic interpretation for the solutions to a class of quasilinear stochastic partial differential-integral equations (SPDIEs in short) is tthe solutionBDSDEP. Under non-Lipschitz conditions, the existence and uniqueness results for measurable solutions of BDSDEP are established via the smoothing technique. Then, the continuous dependence for solutions of BDSDEP is derived. Finally, the prob…
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.
We study a general class of fully coupled backward-forward stochastic differential equations of mean-field type (MF-BFSDE). We derive existence and uniqueness results for such a system under weak monotonicity assumptions and without the non-degeneracy condition on the forward equation. This is achieved by suggesting an implicit approximation scheme that is shown to converge to the solution of the system of MF-BFSDE. We apply these results to derive an explicit form of open-loop Nash equilibrium strategies for nonzero sum mean-field linear-quadratic stochastic differential games with random coe…
We first state a special type of Itô formula involving stochastic integrals of both standard and fractional Brownian motions. Then we use Doss-Sussman transformation to establish the link between backward doubly stochastic differential equations, driven by both standard and fractional Brownian motions, and backward stochastic differential equations, driven only by standard Brownian motions. Following the same technique, we further study associated nonlinear stochastic partial differential equations driven by fractional Brownian motions and partial differential equations with stochastic coeffic…
In this paper, by virtue of Malliavin calculus, we establish a relationship between backward doubly stochastic differential equations with random coefficients and quasilinear stochastic PDEs, and thus extend the well-known nonlinear stochastic Feynman-Kac formula of Pardoux and Peng [14] to non-Markovian case.
This essay explores the meaning of stochastic differential equations and stochastic integrals. It sets these subjects in a context of Riemann-Stieltjes integration. It is intended as a comment or supplement to \cite{MTRV}.
For a mixed stochastic differential equation containing both Wiener process and a Hölder continuous process with exponent $γ>1/2$, we prove a stochastic viability theorem. As a consequence, we get a result about positivity of solution and a pathwise comparison theorem. An application to option price estimation is given.
In this Note, assuming that the generator is uniform Lipschitz in the unknown variables, we relate the solution of a one dimensional backward stochastic differential equation with the value process of a stochastic differential game. Under a domination condition, a filtration-consistent evaluations is also related to a stochastic differential game. This relation comes out of a min-max representation for uniform Lipschitz functions as affine functions. The extension to reflected backward stochastic differential equations is also included.
We derive a stochastic Gronwall lemma with suprema over the paths in the upper bound of the assumed affine-linear growth assumption. This allows applications to Itô processes with coefficients which depend on earlier time points such as stochastic delay equations or Euler-type approximations of stochastic differential equations. We apply our stochastic Gronwall lemma with path-suprema to stochastic functional differential equations and prove a strong convergence rate for coefficient functions which depend on path-suprema.
Existence and uniqueness results of fully coupled forward stochastic differential equations without drifts and backward stochastic differential equations in a degenerate case are obtained for an arbitrarily large time duration.
We define fully coupled forward-backward stochastic differential equations on spaces related to continuous time, finite state Markov Chains. Existence and uniqueness results of the fully coupled forward-backward stochastic differential equations on Markov Chains are obtained.
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…
We review recent progress in the study of infinite-dimensional stochastic differential equations with symmetry. This paper contains examples arising from random matrix theory.
This paper extends the idea of E.Gobet, J.P.Lemor and X.Warin from the setting of Backward Stochastic Differential Equations to that of Backward Doubly Stochastic Differential equations. We propose some numerical approximation scheme of these equations introduced by E.Pardoux and S.Peng.
By the methods of probability and duality technique, we give some comparison theorems for the solutions of infinite horizon forward-backwad stochastic differential equations.
We prove constructible sufficient conditions of lack of exit by solutions of stochastic differential Ito's equations from domains with smooth boundaries
This paper is devoted to proving the strong averaging principle for slow-fast stochastic partial differential equations with locally monotone coefficients, where the slow component is a stochastic partial differential equations with locally monotone coefficients and the fast component is a stochastic partial differential equations (SPDEs) with strongly monotone coefficients. The result is applicable to a large class of examples, such as the stochastic porous medium equation, the stochastic $p$-Laplace equation, the stochastic Burgers type equation and the stochastic 2D Navier-Stokes equation, …
In this paper, we consider the fully coupled forward-backward stochastic functional differential equations (FBSFDEs) with stochastic functional differential equations as the forward equations and the generalized anticipated backward stochastic differential equations as the backward equations. We will prove the existence and uniqueness theorem for FBSFDEs. As an application, we deal with a quadratic optimal control problem for functional stochastic systems, and get the explicit form of the optimal control by virtue of FBSFDEs.
In this article, we introduce the notion of stochastic symmetry of a differential equation. It consists in a stochastic flow that acts over a solution of a differential equation and produces another solution of the same equation. In the ordinary case, we give necessary conditions in order to obtain such symmetries. These conditions involve the infinitesimal generator of the flow and the coefficients of the equation. Moreover, we show how to obtain necessary conditions in order to find an application that transforms a stochastic differential equation that one would like to solve into a target e…
In this paper, we consider the stochastic optimal control problems under G-expectation. Based on the theory of backward stochastic differential equations driven by G-Brownian motion, which was introduced in [10.11], we can investigate the more general stochastic optimal control problems under G-expectation than that were constructed in [28]. Then we obtain a generalized dynamic programming principle and the value function is proved to be a viscosity solution of a fully nonlinear second-order partial differential equation.
We study the existence of a unique solution to semilinear fractional backward doubly stochastic differential equation driven by a Brownian motion and a fractional Brownian motion with Hurst parameter less than 1/2. Here the stochastic integral with respect to the fractional Brownian motion is the extended divergence operator and the one with respect to Brownian motion is Itô's backward integral. For this we use the technique developed by R.Buckdahn to analyze stochastic differential equations on the Wiener space, which is based on the Girsanov theorem and the Malliavin calculus, and we reduce …
In this paper, we generalize to Gaussian Volterra processes the existence and uniqueness of solutions for a class of non linear backward stochastic differential equations (BSDE) and we establish the relation between the non linear BSDE and the partial differential equation (PDE). A comparison theorem for the solution of the BSDE is proved and the continuity of its law is studied.
We prove an existence and uniqueness result for generalized backward doubly stochastic differential equations driven by Lévy processes with non-Lipschitz assumptions.
We trace the evolution of the theory of stochastic partial differential equations from the foundation to its development, until the recent solution of long-standing problems on well-posedness of the KPZ equation and the stochastic quantization in dimension three.
We prove the existence of the unique solution of a general Backward Stochastic Differential Equation with quadratic growth driven by martingales. Some kind of comparison theorem is also proved.
Stability equivalence for stochastic differential equations, stochastic differential delay equations and their corresponding Euler-Maruyama methods in $G$-framework
该研究证明在 $G$-框架下，当延迟或数值步长足够小时，$G$-SDDE、辅助 $G$-SDE 及其 EM 方法在 $p$ 阶矩实际指数稳定性上彼此等价，从而支持通过模拟检验稳定性。
In this paper, a probabilistic interpretation for the viscosity solution of a parabolic partial differential equation is obtained by virtue of the solution of a class of quadratic backward stochastic differential equations (BSDEs, for short). Furthermore, we prove the convergence and the large deviation principle for the solution of this class of quadratic BSDEs, which is associated with a family of Markov processes with the diffusion coefficients that tend to be zero.
In this paper, we prove that there exists a unique strong solution to reflecting stochastic differential equations with merely measurable drift giving an affirmative answer to the longstanding problem. This is done through Zvonkin transformation and a careful analysis of the transformed reflecting stochastic differential equations on non-smooth time-dependent domains.
We extend the Yamada-Watanabe condition for pathwise uniqueness to stochastic differential equations with jumps, in the special case where small jumps are summable.
We consider stochastic differential equations driven by some Volterra processes. Under time reversal, these equations are transformed into past dependent stochastic differential equations driven by a standard Brownian motion. We are then in position to derive existence and uniqueness of solutions of the Volterra driven SDE considered at the beginning.
In this paper, we show that the integration of a stochastic differential equations driven by G-Brownian motion in R can be reduced to the integration of an ordinary differential equations parametrized by a variable in (Ω,F). We study the sample solutions of G-SDEs by an extention of G-Itô formula. And then we also get a comparison theorem for G-SDEs and its applications.
In this paper we prove a general approximation result for reflected stochastic differential equations in bounded domains satisfying conditions reorganized by Ren and Wu. Then we show that it includes Wong-Zakai approximation, mollifier approximation, etc.
