具有相互连接障碍的偏微分方程组的粘性解与多模式切换问题

Systems of fully nonlinear parabolic obstacle problems with Neumann boundary conditions

We prove the existence of a unique viscosity solution to certain systems of fully nonlinear parabolic partial differential equations with interconnected obstacles in the setting of Neumann boundary conditions. The method of proof builds on the classical viscosity solution technique adapted to the setting of interconnected obstacles and construction of explicit viscosity sub- and supersolutions as bounds for Perron's method. Our motivation stems from so called optimal switching problems on bounded domains.

Viscosity Solutions for a System of PDEs and Optimal Switching

In this paper, we study the $m$-states optimal switching problem in finite horizon, when the switching cost functions are arbitrary and can be positive or negative. This has an economic incentive in terms of central evaluation in cases where such organizations or state grants or financial assistance to power plants that promotes green energy in their production activity or what uses less polluting modes in their production. We show existence for optimal strategy via a verification theorem then we show existence and uniqueness of the value processes by using an approximation scheme. In the mark…

Stochastic optimal switching and systems of variational inequalities with interconnected obstacles

This paper studies a system of $m$ variational inequalities with interconnected obstacles in infinite horizon associated to optimal multi-modes switching problems. Our main result is the existence and uniqueness of a continuous solution in viscosity sense, for that system. The proof of the main result strongly relies on the connection between the systems of variational inequalities and reflected backward stochastic differential equations (RBSDEs) with oblique reflection, which will be characterized through a Feynman-Kac's formula. The main feature of our system of infinite horizon RBSDEs is th…

带约束跳的倒向随机微分方程的最优停止问题及相关双障碍偏微分方程

Optimal Stopping of BSDEs with Constrained Jumps and Related Double Obstacle PDEs

本文研究带约束跳的BSDE最优停止问题与双障碍PDE的关系，通过Feynman-Kac表示和压缩论证证明非局部情形下粘性解的存在唯一性，并建立新型非线性Snell包络。

Probabilistic Representation of Viscosity Solutions to Quasi-Variational Inequalities with Non-Local Drivers

We consider quasi-variational inequalities (QVIs) with general non-local drivers and related systems of reflected backward stochastic differential equations (BSDEs) in a Brownian filtration. We show existence and uniqueness of viscosity solutions to the QVIs by first considering the standard (local) setting and then applying a contraction argument. In addition, the contraction argument yields existence and uniqueness of solutions to the related systems of reflected BSDEs and extends the theory of probabilistic representations of PDEs in terms of BSDEs to our specific setting.

具有状态和律依赖共同噪声的Wasserstein PDE的比较原理

A comparison principle for Wasserstein PDEs with state- and law-dependent common noise

针对Wasserstein空间上带有一般共同噪声Hessian的二阶Hamilton-Jacobi-Bellman方程，通过引入非线性测度流并建立空间、测度导数和负Sobolev估计，证明了半连续粘性次解和超解的比较原理，首次为滤波驱动方程提供了此类粘性比较框架，并应用于受控随机滤波问题的值函数唯一性。

Upper envelopes of families of Feller semigroups and viscosity solutions to a class of nonlinear Cauchy problems

In this paper, we consider the (upper) semigroup envelope, i.e. the least upper bound, of a given family of linear Feller semigroups. We explicitly construct the semigroup envelope and show that, under suitable assumptions, it yields viscosity solutions to abstract Hamilton-Jacobi-Bellman-type partial differential equations related to stochastic optimal control problems arising in the field of Robust Finance. We further derive conditions for the existence of a Markov process under a nonlinear expectation related to the semigroup envelope for the case where the state space is locally compact. T…

A Singular Stochastic Control Problem with Interconnected Dynamics

In this paper we study a Markovian two-dimensional bounded-variation stochastic control problem whose state process consists of a diffusive mean-reverting component and of a purely controlled one. The main problem's characteristic lies in the interaction of the two components of the state process: the mean-reversion level of the diffusive component is an affine function of the current value of the purely controlled one. By relying on a combination of techniques from viscosity theory and free-boundary analysis, we provide the structure of the value function and we show that it satisfies a secon…

利用具有时间依赖哈密顿量的Hamilton-Jacobi偏微分方程实现持续科学机器学习

Leveraging Hamilton-Jacobi PDEs with time-dependent Hamiltonians for continual scientific machine learning

本文通过建立SciML优化问题与时间依赖哈密顿量的HJ PDE之间的理论联系，提出基于Riccati的持续学习方法，在避免灾难性遗忘的同时提升可解释性与计算效率。