Anthony Lecavil, Anthony Le Cavil, Nadia Oudjane, Francesco Russo
We propose a nonlinear forward Feynman-Kac type equation, which represents the solution of a non-conservative semilinear parabolic Partial Differential Equations (PDE). We show in particular existence and uniqueness. The solution of that type of equation can be approached via a weighted particle system.
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The paper is devoted to the construction of a probabilistic particle algorithm. This is related to nonlin-ear forward Feynman-Kac type equation, which represents the solution of a nonconservative semilinear parabolic Partial Differential Equations (PDE). Illustrations of the efficiency of the algorithm are provided by numerical experiments.
On nonlinear Feynman-Kac formulas for viscosity solutions of semilinear parabolic partial differential equations with gradient-dependent nonlinearities
本文将Feynman-Kac表示推广到非线性项依赖解梯度的半线性抛物型偏微分方程，通过随机不动点方程建立粘性解的表示公式。
On the well-posedness of a class of McKean Feynman-Kac equations
该研究分析带Feynman-Kac扰动的McKean型方程（MFKE）的适定性，在系数无强正则性假设下给出弱、强存在性及轨道唯一性条件，借助关联MFKE与非保守半线性抛物PDE解的表示定理完成证明。
The classical Feynman-Kac identity builds a bridge between stochastic analysis and partial differential equations (PDEs) by providing stochastic representations for classical solutions of linear Kolmogorov PDEs. This opens the door for the derivation of sampling based Monte Carlo approximation methods, which can be meshfree and thereby stand a chance to approximate solutions of PDEs without suffering from the curse of dimensionality. In this article we extend the classical Feynman-Kac formula to certain semilinear Kolmogorov PDEs. More specifically, we identify suitable solutions of stochastic…
Semilinear Feynman-Kac Formulae for $B$-Continuous Viscosity Solutions
本文用概率方法证明无穷维半线性 PDE 的 $B$-连续粘性解存在且唯一，并给出基于倒向随机微分方程的首个非线性 Feynman–Kac 表示框架。
We study a class of backward doubly stochastic differential equations (BDSDEs) involving martingales with spatial parameters, and show that they provide probabilistic interpretations (Feynman-Kac formulae) for certain semilinear stochastic partial differential equations (SPDEs) with space-time noise. As an application of the Feynman-Kac formulae, random periodic solutions and stationary solutions to certain SPDEs are obtained.
It is known that Markovian forward-backward stochastic differential equations provide nonlinear Feynman-Kac representation formulae for semilinear parabolic PDEs. We show that non-Markovian forward-backward stochastic differential equations provide nonlinear Feynman-Kac formulae for semilinear path-dependent PDEs. This extends the result proved in Ekren, Keller, Touzi, and Zhang [4] to the case with a possibly degenerate diffusion coefficient in the forward dynamics.
This paper presents a probabilistic interpretation for the weak Sobolev solution of the obstacle problem for semilinear parabolic partial integro-differential equations (PIDEs). The results of Leandre (1985) concerning the homeomorphic property for the solution of SDEs with jumps are used to construct random test functions for the variational equation for such PIDEs. This results in the natural connection with the associated Reflected Backward Stochastic Differential Equations with jumps (RBSDEs), namely Feynman Kac's formula for the solution of the PIDEs. Moreover it gives an application to t…
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…
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。
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.
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…
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,…
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。
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。
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