We develop an intrinsic well-posedness theory for nonlinear backward stochastic differential equations (BSDEs) driven simultaneously by Brownian motion and a (level-$2$) rough path of finite $p$-variation. Unlike earlier approaches based on smooth approximation or transformation methods, we formulate the equation directly by viewing its solution as a rough semimartingale (in the sense of \cite{friz2023rough}). This framework is particularly suited to BSDEs, whose Brownian martingale component is only implicitly defined and lacks the \emph{a priori} time regularity required by stochastic-sewing-based controlled rough path methods \cite{fhl21,allan2024rough}. We establish comparison, existence, uniqueness, and stability under the natural regularity condition $H\in C_b^\gamma$, $\gamma>p$. The main analytical difficulty, namely, the loss of integrability arising from nonlinear composition, is overcome through conditional $p$-variation norms with BMO-type properties. Finally, by randomizing the rough driver as a Brownian rough path, we establish a direct correspondence between rough BSDEs and backward doubly stochastic differential equations (BDSDEs).
In this paper, we establish the existence and uniqueness of solutions for mean-field stochastic differential equations (MF-SDEs in short) and backward stochastic differential equations (MF-BSDE) driven by anomalous sub-diffusions $\{B_{L_t}; t\geq 0\}$ with random coefficients, respectively. Here $B$ is a Brownian moti...
We study the backward-Euler scheme for a class of stochastic functional differential equations (SFDEs) with memory driven by a fractional Brownian motion with Hurst parameter H>1/2. We first establish the local consistency of the method, with a local truncation error of order H -$\rho$- $\beta$ + 1, and then, as the ma...
Alexander Abreu, Lisandro Fermin, E. Mordecki et al.· 0 citations
This paper is concerned with a stochastic optimal control problem under inside information. The control process depends on an $\mathcal{F}_{T_0}$-measurable random variable $Y$, representing the static inside information, and is adapted to the enlarged filtration generated by the underlying Brownian motion and the rand...
This paper investigates the averaging behavior of a class of slow–fast fractional stochastic differential equations that incorporate two distinct types of memory effects. The slow variable is governed by a Caputo fractional derivative of order α∈(0,1), and is driven by fractional Brownian motion with a Hurst index H∈(1...
We study the stochastic differential equation $$d X_t=b(t,X_t)d t+\sqrt{2}d W_t$$ on $\mathbb R^d$, where $b$ is a time-dependent, divergence-free distributional drift of critical H\"older--Besov regularity $-1$, strengthened by an iterated-logarithmic correction. For every initial probability law, we construct a weak...
For weak solutions to quasilinear degenerate parabolic equations of $p$-Laplace type, a central obstacle in applying the method of intrinsic scaling to prove their H\"{o}lder regularity is the derivation of forward-in-time propagation estimate for the spatial measure of level sets. In this paper, we revisit and overcom...
Xia Hao, Yan Li, Zhi-Wen Zhao· 0 citations
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