Some thoughts on multiparameter stochastic processes

多参数随机缝合引理及与高斯 sheet 相关的局部时的正则性

A multiparameter Stochastic Sewing lemma and the regularity of local times associated to Gaussian sheets

该研究建立多参数随机缝合引理，推导局部非确定性高斯场局部时的正则性估计，借此得到平面SDEs的噪声正则化结果，并发现加性利用随机场参数的正则化效应多源于驱动场的边界项。

A multiparameter Garsia-Rodemich-Rumsey inequality and some applications

We extend the classical Garsia-Rodemich-Rumsey inequality to the multiparameter situation. The new inequality is applied to obtain some joint Hölder continuity along the rectangles for fractional Brownian fields $W(t, x)$ and for the solution $u(t, y)$ of stochastic heat equation with additive white noise.

多参数Lévy白噪声理论及其应用

Multiparameter Lévy white noise theory and applications

该研究构建了多参数Lévy sheet及其补偿Poisson随机测度的白噪声理论与演算体系，可应用于受Lévy噪声影响的随机偏微分方程研究。

Moments of $k$-hop counts in the random-connection model

We derive moment identities for the stochastic integrals of multiparameter processes in a random-connection model based on a point process admitting a Papangelou intensity. Those identities are written using sums over partitions, and they reduce to sums over non-flat partition diagrams in case the multiparameter processes vanish on diagonals. As an application, we obtain general identities for the moments of $k$-hop counts in the random-connection model, which simplify the derivations available in the literature.

Multivariate stochastic delay differential equations and CAR representations of CARMA processes

In this study we show how to represent a continuous time autoregressive moving average (CARMA) as a higher order stochastic delay differential equation, which may be thought of as a continuous-time equivalent of the AR($\infty$) representation. Furthermore, we show how this representation gives rise to a prediction formula for CARMA processes. To be used in the above mentioned results we develop a general theory for multivariate stochastic delay differential equations, which will be of independent interest, and which will have particular focus on existence, uniqueness and representations.

Stochastic viscosity solutions of reflected stochastic partial differential equations with non-Lipschitz coefficients

This paper, is an attempt to extend the notion of stochastic viscosity solution to reflected semi-linear stochastic partial differential equations (RSPDEs, in short) with non-Lipschitz condition on the coefficients. Our method is fully probabilistic and use the recently developed theory on reflected backward doubly stochastic differential equations (RBDSDEs, in short). Among other, we prove the existence of the stochastic viscosity solution, and further extend the nonlinear Feynman-Kac formula to reflected SPDEs, like one appear in \cite{2}. Indeed, in their recent work, Aman and Mrhardy \cite…