A class of remarkable submartingales

亚martingale条件用于半马尔可夫过程的弱收敛

Submartingale Condition for Weak Convergence for Semi-Markov Processes

本文提出了一种针对半马尔可夫过程弱收敛的亚martingale条件修改，指出需额外条件以保证收敛性。

集值下鞅的表示定理

A representation theorem for set-valued submartingales

本文提出并证明了一类集值下鞅的一般表示定理，扩展了非平凡初始集值鞅的随机积分表示，并证明该结果涵盖了文献中关于退化与非退化集值鞅的现有结论。

On the study of processes of classes $\sum(H)$ and $\sum_{s}(H)$

In papers by Yor, a remarkable class $(Σ)$ of submartingales is introduced, which, up to technicalities, are submartingales $(X_{t})_{t\geq0}$ whose increasing process is carried by the times $t$ such that $X_{t}=0$. These submartingales have several applications in stochastic analysis: for example, the resolution of Skorokhod embedding problem, the study of Brownian local times and the study of zeros of continuous martingales. The submartingales of class $(Σ)$ have been extensively studied in a series of articles by Nikeghbali (part of them in collaboration with Najnudel, some others with Che…

平均剩余寿命过程及相关子鞅

Mean residual life processes and associated submartingales

本文利用Madan-Yor方法为递增凸支配的双参数过程构造子鞅，证明积分生存函数MTP$_2$等价于逐对TP$_2$，并展示MRL过程的应用。

The submartingale problem for a class of degenerate elliptic operators

We consider the degenerate elliptic operator acting on $C^2$ functions on $[0,\infty)^d$: \[ L f(x)=\sum_{i=1}^d a_i(x) x_i^{α_i} \frac{\partial^2 f}{\partial x_i^2} (x) +\sum_{i=1}^d b_i(x) \frac{\partial f}{\partial x_i}(x), \] where the $a_i$ are continuous functions that are bounded above and below by positive constants, the $b_i$ are bounded and measurable, and the $α_i\in (0,1)$. We impose Neumann boundary conditions on the boundary of $[0,\infty)^d$. There will not be uniqueness for the submartingale problem corresponding to $L$. If we consider, however, only those solutions to the subm…

A short Proof of the Doob-Meyer Theorem

Every submartingale S of class D has a unique Doob-Meyer decomposition S=M+A, where M is a martingale and A is a predictable increasing process starting at 0. We provide a short and elementary prove of the Doob-Meyer decomposition theorem. Several previously known arguments are included to keep the paper self-contained.

On some universal sigma finite measures and some extensions of Doob's optional stopping theorem

In this paper, we associate, to any submartingale of class $(Σ)$, defined on a filtered probability space $(Ω, \mathcal{F}, \mathbb{P}, (\mathcal{F}_t)_{t \geq 0})$, which satisfies some technical conditions, a $σ$-finite measure $\mathcal{Q}$ on $(Ω, \mathcal{F})$, such that for all $t \geq 0$, and for all events $Λ_t \in \mathcal{F}_t$: $$ \mathcal{Q} [Λ_t, g\leq t] = \mathbb{E}_{\mathbb{P}} [\mathds{1}_{Λ_t} X_t]$$ where $g$ is the last hitting time of zero of the process $X$. This measure $\mathcal{Q}$ has already been defined in several particular cases, some of them are involved in the s…

通过多元虚假平稳仿射Volterra动力学视角探讨均值-方差问题

On explicit solutions to a class of quadratic BSDEJs driven by affine Volterra processes with jumps and applications

本文研究了连续时间Markowitz均值-方差投资组合选择问题，采用多元虚假平稳仿射Volterra模型，通过求解Riccati反向随机微分方程的随机因子解，推导出最优反馈控制的显式解，并分析了多维Riccati-Volterra系统的解对最优投资策略和均值-方差有效前沿的影响。