Filtration-Consistent Dynamic Operator with a Floor and Associated Reflected Backward Stochastic Differential Equations

随机Lipschitz系数下带违约时间的双重反射倒向随机微分方程：滤链与广义Dynkin博弈

Doubly reflected BSDEs with default time under stochastic Lipschitz coefficients: Filtration links and generalized Dynkin games

该研究在渐进扩大滤子下，针对带违约时间的随机Lipschitz系数DRBSDEs，建立解的存在唯一性，探讨其与参考滤子中方程的关系，并刻画了相关非线性Dynkin博弈的价值过程与鞍点。

Reflected BSDEs on Filtered Probability Spaces

We study the problem of existence and uniqueness of solutions of backward stochastic differential equations with two reflecting irregular barriers, $L^p$ data and generators satisfying weak integrability conditions. We deal with equations on general filtered probability spaces. In case the generator does not depend on the $z$ variable, we first consider the case $p=1$ and we only assume that the underlying filtration satisfies the usual conditions of right-continuity and completeness. Additional integrability properties of solutions are established if $p\in(1,2]$ and the filtration is quasi-le…

Generalized BSDEs with random time horizon in a progressively enlarged filtration

We study generalized backward stochastic differential equations (BSDEs) up to a random time horizon $\vartheta$, which is not a stopping time, under minimal assumptions regarding the properties of $\vartheta$. In contrast to existing works in this area, we do not impose specific assumptions on the random time $\vartheta$ and we study the existence of solutions to BSDEs and reflected BSDEs with a random time horizon through the method of reduction. In addition, we also examine BSDEs and reflected BSDEs with a làdlàg driver where the driver is allowed to have a finite number of common jumps with…

Doubly Reflected BSDEs in the predictable setting

In this paper, we introduce a specific kind of doubly reflected Backward Stochastic Differential Equations (in short DRBSDEs), defined on probability spaces equipped with general filtration that is essentially non quasi-left continuous, where the barriers are assumed to be predictable processes. We call these equations predictable DRBSDEs. Under a general type of Mokobodzki's condition, we show the existence of the solution (in consideration of the driver's nature) through a Picard iteration method and a Banach fixed point theorem. By using an appropriate generalization of Itô's formula due to…

$\mathbb{L}^p$-解 $(1<p<2)$：具有一般跳跃和随机单调生成元的反射倒向随机微分方程

$\mathbb{L}^p$-solutions $(1 <p< 2)$ for reflected BSDEs with general jumps and stochastic monotone generators

本文研究具有一般跳跃和随机单调生成元的反射倒向随机微分方程，通过惩罚方法证明了$\mathbb{L}^p$（$1<p<2$）解的存在唯一性。

由非齐次简单列维过程驱动的双重反射倒向随机微分方程：在广义 Dynkin 博弈中的应用

Doubly reflected BSDEs driven by Inhomogeneous simple Levy processes: Applications to generalized Dynkin games

研究由非齐次简单列维过程驱动的双重反射倒向随机微分方程，用惩罚方法建立存在唯一性结果与比较原理，给出美式博弈期权估值和广义 Dynkin 博弈等应用，并在合适假设下建立博弈鞍点存在性。

Backward stochastic differential equations with conditional reflection and related recursive optimal control problems

We introduce a new type of reflected backward stochastic differential equations (BSDEs) for which the reflection constraint is imposed on its main solution component, denoted as $Y$ by convention, but in terms of its conditional expectation $\mathbb{E}[Y_t|\mathcal{G}_{t}]$ on a general sub-filtration $\{\mathcal{G}_{t}\}.$ We thus term such equation as conditionally reflected BSDE (for short, conditional RBSDE). Conditional RBSDE subsumes classical RBSDE with a pointwise reflection barrier, and the recent developed BSDE with a mean reflection constraint, as its two special and extreme cases: …

Reflected backward stochastic differential equations under stopping with an arbitrary random time

This paper addresses reflected backward stochastic differential equations (RBSDE hereafter) that take the form of \begin{eqnarray*} \begin{cases} dY_t=f(t,Y_t, Z_t)d(t\wedgeτ)+Z_tdW_t^τ+dM_t-dK_t,\quad Y_τ=ξ, Y\geq S\quad\mbox{on}\quad \Lbrack0,τ\Lbrack,\quad \displaystyle\int_0^τ(Y_{s-}-S_{s-})dK_s=0\quad P\mbox{-a.s..}\end{cases} \end{eqnarray*} Here $τ$ is an arbitrary random time that might not be a stopping time for the filtration $\mathbb F$ generated by the Brownian motion $W$. We consider the filtration $\mathbb G$ resulting from the progressive enlargement of $\mathbb F$ with $τ$ wher…

Well-posedness and penalization schemes for generalized BSDEs and reflected generalized BSDEs

The paper is directly motivated by the pricing of vulnerable European and American options in a general hazard process setup and a related study of the corresponding pre-default backward stochastic differential equations (BSDE) and pre-default reflected backward stochastic differential equations (RBSDE). We work with a generic filtration $\FF$ for which the martingale representation property is assumed to hold with respect to a square-integrable martingale $M$ and the goal of this work is of twofold. First, we aim to establish the well-posedness results and comparison theorems for a generalize…

On controller-stopper problems with jumps and their applications to indifference pricing of American options

We consider controller-stopper problems in which the controlled processes can have jumps. The global filtration is represented by the Brownian filtration, enlarged by the filtration generated by the jump process. We assume that there exists a conditional probability density function for the jump times and marks given the filtration of the Brownian motion and decompose the global controller-stopper problem into controller-stopper problems with respect to the Brownian filtration, which are determined by a backward induction. We apply our decomposition method to indifference pricing of American o…

一般滤子空间上由右连左极增过程驱动的带不规则障碍的广义反射倒向随机微分方程

Generalized reflected BSDEs with irregular obstacles driven by RCLL increasing processes on general filtered space

该研究在一般滤子空间上，针对由RCLL增过程驱动的GBSDEs和GRBSDEs，建立了存在唯一性等结果，提出两种方法求解特定GRBSDEs并推广到一般利普希茨情形。

Dual Representation as Stochastic Differential Games of Backward Stochastic Differential Equations and Dynamic Evaluations

In this Note, assuming that the generator is uniform Lipschitz in the unknown variables, we relate the solution of a one dimensional backward stochastic differential equation with the value process of a stochastic differential game. Under a domination condition, a filtration-consistent evaluations is also related to a stochastic differential game. This relation comes out of a min-max representation for uniform Lipschitz functions as affine functions. The extension to reflected backward stochastic differential equations is also included.