Classical solutions to reaction-diffusion systems for hedging problems with interacting Ito and point processes

Pathwise integrals and Ito-Tanaka Formula for Gaussian processes

We prove the Ito-Tanaka formula and the existence of pathwise stochastic integrals for a wide class of Gaussian processes. Motivated by financial applications, we define the stochastic integrals as forward-type pathwise integrals introduced by Föllmer and as pathwise generalized Lebesgue-Stieltjes integrals introduced by Zähle. As an application, we illustrate the importance of Ito-Tanaka formula for pricing and hedging of financial derivatives.

Martingale representation for Poisson processes with applications to minimal variance hedging

We consider a Poisson process $η$ on a measurable space $(\BY,\mathcal{Y})$ equipped with a partial ordering, assumed to be strict almost everwhwere with respect to the intensity measure $λ$ of $η$. We give a Clark-Ocone type formula providing an explicit representation of square integrable martingales (defined with respect to the natural filtration associated with $η$), which was previously known only in the special case, when $λ$ is the product of Lebesgue measure on $\R_+$ and a $σ$-finite measure on another space $\BX$. Our proof is new and based on only a few basic properties of Poisson p…

Stochastic equations with delay: optimal control via BSDEs and regular solutions of Hamilton-Jacobi-Bellman equations

We consider an Ito stochastic differential equation with delay, driven by brownian motion, whose solution, by an appropriate reformulation, defines a Markov process $X$ with values in a space of continuous functions $\mathbf C$, with generator $\mathcal L$. We then consider a backward stochastic differential equation depending on $X$, with unknown processes $(Y,Z)$, and we study properties of the resulting system, in particular we identify the process $Z$ as a deterministic functional of $X$. We next prove that the forward-backward system provides a suitable solution to a class of parabolic pa…

硫酸化反应的混合模型：随机连续介质环境中的随机粒子

A hybrid model of sulphation reactions: stochastic particles in a random continuum environment

提出一种混合随机-连续模型，结合随机微分方程与连续场，研究碳酸钙硫酸化导致的大理石腐蚀，发现随机性引发非均匀腐蚀。

随机扰动反应扩散系统中平面行波的多维稳定性

Multidimensional Stability of Planar Travelling Waves for Stochastically Perturbed Reaction-Diffusion Systems

本文研究二维及以上空间中带乘性噪声的反应扩散系统，通过随机相位跟踪机制在指数长时间尺度上建立平面波的多维稳定性，并扩展前向积分理论以处理非 Itô 型随机积分。

带输运噪声与临界超线性扩散的反应扩散方程：弱耗散系统的全局适定性

Reaction-diffusion equations with transport noise and critical superlinear diffusion: Global well-posedness of weakly dissipative systems

该研究针对d维环面上带输运噪声的反应扩散系统，结合极大正则性、随机分析等工具，提出新的强制性条件与L^p(L^q)框架，证明了多类标量方程、耗散及弱耗散系统的全局适定性，部分无噪声情形结果也为新结论。