Reduction and reconstruction of SDEs via Girsanov and quasi Doob symmetries

A Wong-Zakai theorem for SDEs with singular drift

We study stochastic differential equations (SDEs) with multiplicative Stratonovich-type noise of the form $ dX_t = b(X_t) dt + σ(X_t)\circ d W_t, X_0=x_0\in\mathbb{R}^d, t\geq0,$ with a possibly singular drift $b\in L^{p}(\mathbb{R}^d)$, $p>d$ and $p\geq 2$, and show that such SDEs can be approximated by random ordinary differential equations by smoothing the noise and the singular drift at the same time. We further prove a support theorem for this class of SDEs in a rather simple way using the Girsanov theorem.

On a maximal inequality and its application to SDEs with singular drift

In this paper we present a Doob type maximal inequality for stochastic processes satisfying the conditional increment control condition. If we assume, in addition, that the margins of the process have uniform exponential tail decay, we prove that the supremum of the process decays exponentially in the same manner. Then we apply this result to the construction of the almost everywhere stochastic flow to stochastic differential equations with singular time dependent divergence-free drift.

由$G$-布朗运动驱动的具有二次生成函数的双反射反向随机微分方程

Doubly Reflected Backward SDEs Driven by $G$-Brownian Motion with Quadratic Generator

本文研究由$G$-布朗运动驱动的双反射反向随机微分方程在二次生成函数下的性质，证明了解的存在性和唯一性，并通过罚函数反射建立其与具有双障碍的非线性偏微分方程的关系。

Weak Existence and Uniqueness for McKean-Vlasov SDEs with Common Noise

This paper concerns the McKean-Vlasov stochastic differential equation (SDE) with common noise. An appropriate definition of a weak solution to such an equation is developed. The importance of the notion of compatibility in this definition is highlighted by a demonstration of its rôle in connecting weak solutions to McKean-Vlasov SDEs with common noise and solutions to corresponding stochastic partial differential equations (SPDEs). By keeping track of the dependence structure between all components in a sequence of approximating processes, a compactness argument is employed to prove the exist…

由相关分数布朗运动驱动的混合 SDE 的高斯型密度估计

Gaussian-type density estimates for mixed SDEs driven by correlated fractional Brownian motions

本文研究由相关分数布朗运动混合驱动的多维 SDE 弱解的类高斯密度，在短程与长程相关情形下分别施加奇异与 Hölder 连续漂移，基于 Girsanov 定理与指数 Orlicz 空间框架证明密度存在性并导出高斯型上下界。

结构化神经随机微分方程用于泛函校准

Structured Neural SDEs for Functional Calibration

本文提出SLiSDE，一种基于结构化线性随机层的神经SDE模型，通过门控堆叠和Girsanov倾斜实现高效并行模拟与稳定重要性采样，在功能校准任务中优于全神经基线。

缓增分布空间中随机偏微分方程的弱解

Weak Solutions of SPDEs in the space of Tempered distributions

本文利用Girsanov定理，在缓增分布空间中为系数满足单调型不等式的随机偏微分方程构造弱解，并借助相关随机微分方程及Novikov条件实现。

SDE到时变阈值的首达时间的精确模拟

Exact simulation of the first-passage time of SDEs to time-dependent thresholds

本文扩展了精确模拟方法以处理随机过程中时变阈值的首达时间问题，利用Brownian motion和Girsanov变换直接模拟首达时间，避免路径模拟和离散化误差，并验证了其有效性和应用价值。

A trajectorial interpretation of the dissipations of entropy and Fisher information for stochastic differential equations

The dissipation of general convex entropies for continuous time Markov processes can be described in terms of backward martingales with respect to the tail filtration. The relative entropy is the expected value of a backward submartingale. In the case of (non necessarily reversible) Markov diffusion processes, we use Girsanov theory to explicit the Doob-Meyer decomposition of this submartingale. We deduce a stochastic analogue of the well known entropy dissipation formula, which is valid for general convex entropies, including the total variation distance. Under additional regularity assumptio…

Fréchet derivatives of expected functionals of solutions to stochastic differential equations

In the analysis of stochastic dynamical systems described by stochastic differential equations (SDEs), it is often of interest to analyse the sensitivity of the expected value of a functional of the solution of the SDE with respect to perturbations in the SDE parameters. In this paper, we consider path functionals that depend on the solution of the SDE up to a stopping time. We derive formulas for Fréchet derivatives of the expected values of these functionals with respect to bounded perturbations of the drift, using the Cameron-Martin-Girsanov theorem for the change of measure. Using these de…

从单条数据轨迹估计SDE的噪声

Noise estimation of SDE from a single data trajectory

本文提出了一种无需假设遍历或平稳性的方法，从单条数据轨迹中估计SDE的噪声并识别其动力学。