Interior gradient and Hessian estimates for the Dirichlet problem of semi-linear degenerate elliptic systems: A probabilistic approach

In this paper, we give interior gradient and Hessian estimates for systems of semi-linear degenerate elliptic partial differential equations on bounded domains, using both tools of backward stochastic differential equations and quasi-derivatives.     

A local strict comparison theorem and converse comparison theorems for reflected backward stochastic differential equations

A local strict comparison theorem and some converse comparison theorems are proved for reflected backward stochastic differential equations under suitable conditions.     

Regioselective addition of Ti enolates to conjugated enones: New insights into the mechanism based on experimental and DFT investigation

The regioselective addition mechanism of the Ti(IV) enolates derived from α-diazo-β-keto carbonyl compounds and α-diazo-β-keto phosphonates to conjugated enones has been studied on the basis of a hypothetical bridging chloride-controlled theory, by density functional theory (DFT), and experimentally. The DFT results indicate that, for the Ti(IV) enolate 3 derived from α-diazo-β-keto carbonyl compounds, the free energy of the bridging chloride-controlled 1,2-addition transition state is 2.4 kcal/mol higher than that of 1,4-addition, and the calculated enthalpies of 1,2-addition is 4.36 kcal/mol more than that of 1,4-addition. For the Ti(IV) enolate 4 derived from α-diazo-β-keto phosphonates, in contrary, the free energy of the bridging chloride-controlled 1,2-addition transition state is 1.1 kcal/mol lower than that of 1,4-addition, and the calculated enthalpy of 1,2-addition is 3.46 kcal/mol less than that of 1,4-addition. Our findings demonstrate that the nucleophilic addition of these Ti(IV) enolates to conjugated enones was carried out not only kinetically but also irreversibly for the first time.     

Strong solution of backward stochastic partial differential equations in C 2 domains

This paper is concerned with the strong solution to the Cauchy–Dirichlet problem for backward stochastic partial differential equations of parabolic type. Existence and uniqueness theorems are obtained, due to an application of the continuation method under fairly weak conditions on variable coefficients and C ² domains. The problem is also considered in weighted Sobolev spaces which allow the derivatives of the solutions to blow up near the boundary. As applications, a comparison theorem is obtained and the semi-linear equation is discussed in the C ² domain.     

Maximum principle for quasi-linear backward stochastic partial differential equations

In this paper we are concerned with the maximum principle for quasi-linear backward stochastic partial differential equations (BSPDEs for short) of parabolic type. We first prove the existence and uniqueness of the weak solution to quasi-linear BSPDEs with the null Dirichlet condition on the lateral boundary. Then using the De Giorgi iteration scheme, we establish the maximum estimates and the global maximum principle for quasi-linear BSPDEs. To study the local regularity of weak solutions, we also prove a local maximum principle for the backward stochastic parabolic De Giorgi class.     

L p Theory for Super-Parabolic Backward Stochastic Partial Differential Equations in the Whole Space

This paper is concerned with semi-linear backward stochastic partial differential equations (BSPDEs for short) of super-parabolic type. An L ^p -theory is given for the Cauchy problem of BSPDEs, separately for the case of p∈(1,2] and for the case of p∈(2,∞). A comparison theorem is also addressed.     

线性脉冲控制系统的能控性与最优控制

首先讨论了一类线性随机脉冲控制系统的精确能控性质,给出了该类控制系统的脉冲精确能控的等价的代数判据.然后提出了一个确定性的二维线性脉冲控制系统的时间-脉冲强度最优控制问题;利用动态规划原理,给出了脉冲最优控制的反馈形式和值函数的显式表达式;说明了值函数在整个平面上是连续的,在左右两个半平面的内部还是连续可微的.     

SEMI-LINEAR SYSTEMS OF BACKWARD STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS IN $R^n$

This paper explores the diffeomorphism of a backward stochastic ordinary differential equation (BSDE) to a system of semi-linear backward stochastic partial differential equations (BSPDEs), under the inverse of a stochastic flow generated by an ordinary stochastic differential equation (SDE). The author develops a new approach to BSPDEs and also provides some new results. The adapted solution of BSPDEs in terms of those of SDEs and BSDEs is constructed. This brings a new insight on BSPDEs, and leads to a probabilistic approach. As a consequence, the existence, uniqueness, and regularity results are obtained for the (classical, Sobolev, and distributional) solution of BSPDEs.The dimension of the space variable x is allowed to be arbitrary n, and BSPDEs are allowed to be nonlinear in both unknown variables, which implies that the BSPDEs may be nonlinear in the gradient. Due to the limitation of space, however, this paper concerns only classical solution of BSPDEs under some more restricted assumptions.     

Mean Field Games with Common Noises and Conditional Distribution Dependent FBSDEs

In this paper, the authors consider the mean field game with a common noise and allow the state coefficients to vary with the conditional distribution in a nonlinear way. They assume that the cost function satisfies a convexity and a weak monotonicity property. They use the sufficient Pontryagin principle for optimality to transform the mean field control problem into existence and uniqueness of solution of conditional distribution dependent forward-backward stochastic differential equation (FBSDE for short). They prove the existence and uniqueness of solution of the conditional distribution dependent FBSDE when the dependence of the state on the conditional distribution is sufficiently small, or when the convexity parameter of the running cost on the control is sufficiently large. Two different methods are developed. The first method is based on a continuation of the coefficients, which is developed for FBSDE by [Hu, Y. and Peng, S., Solution of forward-backward stochastic differential equations, Probab. Theory Rel., 103(2), 1995,273–283]. They apply the method to conditional distribution dependent FBSDE. The second method is to show the existence result on a small time interval by Banach fixed point theorem and then extend the local solution to the whole time interval.     

Gradient Convergence of Deep Learning-Based Numerical Methods for BSDEs

Chinese Annals of Mathematics, Series B - The authors prove the gradient convergence of the deep learning-based numerical method for high dimensional parabolic partial differential equations and...