Monte-Carlo Algorithms for Forward Feynman-Kac type representation for semilinear nonconservative Partial Differential Equations

路径依赖偏微分方程和非马尔可夫正倒向随机方程的黏性解

Viscosity Solutions of Path-Dependent PDEs and Non-Markovian Forward-Backward Stochastic Equations

马尔可夫正倒向随机微分方程为半线性抛物型偏微分方程提供了非线性Feynman-Kac表示公式。证明了非马尔可夫正倒向随机微分方程为半线性路径依赖偏微分方程提供了非线性Feynman-Kac公式。这将Ekren，Keller，Touzi和Zhang [4]中的结果推广到了前向动力学中扩散系数可能退化的情况。

半线性抛物型偏积分微分方程的障碍问题

The obstacle problem for semilinear parabolic partial integro-differential equations

给出了半线性抛物型偏积分微分方程障碍问题弱Sobolev解的概率解释。利用Leandre（1985）关于带跳偏微分方程解的同胚性质的结果，构造了带跳偏微分方程变分方程的随机检验函数。这导致了与相关的带跳跃的反射倒向随机微分方程（RBSDEs）的自然联系，即费曼·卡茨（Feynman Kac）公式用于PIDE的解。此外，它给出了一个应用程序的定价和套期保值的或有债权的约束条件下的财富或投资组合过程中的金融市场，包括跳跃。

Continuity of the Feynman-Kac formula for a generalized parabolic equation

It is well-known since the work of Pardoux and Peng [12] that Backward Stochastic Differential Equations provide probabilistic formulae for the solution of (systems of) second order elliptic and parabolic equations, thus providing an extension of the Feynman-Kac formula to semilinear PDEs, see also Pardoux and Rascanu [14]. This method was applied to the class of PDEs with a nonlinear Neumann boundary condition first by Pardoux and Zhang [15]. However, the proof of continuity of the extended Feynman-Kac formula with respect to x (resp. to (t,x)) is not correct in that paper. Here we consider a…

半线性偏积分微分方程的多层 Picard 逼近算法及其复杂度分析

Multilevel Picard approximation algorithm for semilinear partial integro-differential equations and its complexity analysis

本文提出一种用于半线性抛物型偏积分微分方程的多层 Picard 逼近算法，证明其收敛到唯一粘性解，并给出多项式复杂度上界以避免维度灾难，最高在万维数值算例中验证了有效性。

Backward stochastic differential equations with Young drift

We prove via a direct fixpoint argument the well-posedness of backward stochastic differential equations containing an additional drift driven by a path of finite $p$-variation with $p \in [1,2)$. An application to the Feynman-Kac representation of semilinear rough partial differential equations is given.

On existence and uniqueness properties for solutions of stochastic fixed point equations

The Feynman-Kac formula implies that every suitable classical solution of a semilinear Kolmogorov partial differential equation (PDE) is also a solution of a certain stochastic fixed point equation (SFPE). In this article we study such and related SFPEs. In particular, the main result of this work proves existence of unique solutions of certain SFPEs in a general setting. As an application of this main result we establish the existence of unique solutions of SFPEs associated with semilinear Kolmogorov PDEs with Lipschitz continuous nonlinearities even in the case where the associated semilinea…

Backward doubly stochastic differential equations and SPDEs with quadratic growth

In this paper, we initiate the study of backward doubly stochastic differential equations (BDSDEs, for short) with quadratic growth. The existence, comparison, and stability results for one-dimensional BDSDEs are proved when the generator $f(t,Y,Z)$ grows in $Z$ quadratically and the terminal value is bounded, by introducing some new ideas. Moreover, in this framework, we use BDSDEs to give a probabilistic representation for the solutions of semilinear stochastic partial differential equations (SPDEs, for short) in Sobolev spaces, and use it to prove the existence and uniqueness of such SPDEs,…

求解具有梯度依赖非线性的一般半线性抛物型偏微分方程的多层 Picard 算法

Multilevel Picard algorithm for general semilinear parabolic PDEs with gradient-dependent nonlinearities

针对系数非常数且非线性依赖梯度的一般半线性抛物型 PDE，提出多层 Picard 逼近算法，并借助 SFPE 证明解与梯度的表示、收敛性及复杂度，辅以 300 维数值验证。

多级Picard逼近在数值逼近含梯度依赖非线性项的一般半线性偏微分方程时可克服维数灾难

Multilevel Picard approximations overcome the curse of dimensionality in the numerical approximation of general semilinear PDEs with gradient-dependent nonlinearities

本文证明了Neufeld与Wu提出的用于求解含梯度依赖非线性项的一般半线性抛物型PDE的多级Picard（MLP）算法可克服维数灾难，其计算复杂度仅随维度和精度倒数呈多项式增长。

具有奇异生成元和无界终端条件的二次 BSDE：理论与应用

Quadratic BSDEs with Singular Generators and Unbounded Terminal Conditions: Theory and Applications

本文研究生成元关于 $y$ 奇异的二次 BSDE，建立解的存在性、比较、稳定性及相关 PDE 粘性解唯一性，并应用于稳健控制中的效用与确定性等价问题。