Stochastic differential equtions with non-lipschitz coefficients:II. Dependence with respect to initial values

Stochastic invariance of closed sets for jump-diffusions with non-Lipschitz coefficients

We provide necessary and sufficient first order geometric conditions for the stochastic invariance of a closed subset of R^d with respect to a jump-diffusion under weak regularity assumptions on the coefficients. Our main result extends the recent characterization proved in Abi Jaber, Bouchard and Illand (2016) to jump-diffusions. We also derive an equivalent formulation in the semimartingale framework.

Jump stochastic differential equations with non-Lipschitz and superlinearly growing coefficients

In the paper, we consider the no-explosion condition and pathwise uniqueness for SDEs driven by a Poisson random measure with coefficients that are super-linear and non-Lipschitz. We give a comparison theorem in the one-dimensional case under some additional condition. Moreover, we study the non-contact property and the continuity with respect to the space variable of the stochastic flow. As an application, we will show that there exists a unique strong solution for SDEs with coefficients like $x\log|x|$.

Stochastic invariance of closed sets with non-Lipschitz coefficients

This paper provides a new characterization of the stochastic invariance of a closed subset of R^d with respect to a diffusion. We extend the well-known inward pointing Stratonovich drift condition to the case where the diffusion matrix can fail to be differentiable: we only assume that the covariance matrix is. In particular, our result can be directly applied to construct affine diffusions and polynomial preserving diffusions on any arbitrary closed set.

Averaging principle for slow-fast stochastic differential equations with time dependent locally Lipschitz coefficients

This paper is devoted to studying the averaging principle for stochastic differential equations with slow and fast time-scales, where the drift coefficients satisfy local Lipschitz conditions with respect to the slow and fast variables, and the coefficients in the slow equation depend on time $t$ and $ω$. Making use of the techniques of time discretization and truncation, we prove that the slow component strongly converges to the solution of the corresponding averaged equation.

G-SVIEs的一些性质

Some properties of G-SVIEs

本文研究了G-SVIEs在不同系数条件下的解的存在性、唯一性和连续性。

On perturbations of an ODE with non-Lipschitz coefficients by a small self-similar noise

We study the limit behavior of differential equations with non-Lipschitz coefficients that are perturbed by a small self-similar noise. It is proved that the limiting process is equal to the maximal solution or minimal solution with certain probabilities $p_+$ and $p_-=1-p_+$, respectively. We propose a space-time transformation that reduces the investigation of the original problem to the study of the exact growth rate of a solution to a certain SDE with self-similar noise. This problem is interesting in itself. Moreover, the probabilities $p_+$ and $p_-$ coincide with probabilities that the …

分支粗糙微分方程解的Lipschitz连续性

The Lipschitz continuity of the solution to branched rough differential equations

本文利用Grossman Larson Hopf代数与张量Hopf代数的同构，将次黎曼几何技术用于分支粗糙微分方程，证明了解对初值、向量场和驱动粗糙路径的显式Lipschitz连续性。

具有局部Lipschitz系数的SDEs密度的Malliavin可微性与光滑性

Malliavian differentiablity and smoothness of density for SDES with locally Lipschitz coefficients

本文研究了具有超线性增长漂移的随机微分方程解的Malliavin可微性，证明了在单调多项式增长漂移条件下的任意阶可微性，并在Hörmander假设下证明了密度的无限次可微性。

Lévy驱动的McKean-Vlasov随机微分方程在Lipschitz条件下的适定性与混沌传播

Well-posedness and propagation of chaos for L{é}vy-driven McKean-Vlasov SDEs under Lipschitz assumptions

本文证明Lévy驱动的McKean-Vlasov SDE在Lipschitz条件下的强适定性，并给出路径级定量混沌传播结果，同时改进线性相互作用及α-稳定噪声情形的收敛速率。

Lyapunov条件下非Lipschitz系数Kolmogorov方程的正则性保持

Regularity preservation in Kolmogorov equations for non-Lipschitz coefficients under Lyapunov conditions

本文针对非Lipschitz系数的Kolmogorov方程，在Lyapunov条件下证明漂移和扩散系数的Lipschitz常数分别为$o(\log V)$和$o(\sqrt{\log V})$时，可替代全局Lipschitz连续性以保持正则性。

Approximation of the invariant distribution for a class of ergodic SDEs with one-sided Lipschitz continuous drift coefficient using an explicit tamed Euler scheme

We consider the long-time behavior of an explicit tamed Euler scheme applied to a class of stochastic differential equations driven by additive noise, under a one-sided Lipschitz continuity condition. The setting encompasses drift nonlinearities with polynomial growth. First, we prove that moment bounds for the numerical scheme hold, with at most polynomial dependence with respect to the time horizon. Second, we apply this result to obtain error estimates, in the weak sense, in terms of the time-step size and of the time horizon, to quantify the error to approximate averages with respect to th…

Stochastic averaging for the non-autonomous mixed stochastic differential equations with locally Lipschitz coefficients

This paper investigates a non-autonomous slow-fast system, which is generalized by stochastic differential equations (SDEs) with locally Lipschitz coefficients, subjected to standard Brownian motion (Bm) and fractional Brownian motion (fBm) with Hurst parameter 1/2<H<1. We concentrate on how to handle both types of integrals with respect to Bm and fBm and the locally Lispchitz continuity. The pathwise approach and the Ito stochastic calculus are combined with the technique of stopping time to establish the averaging principle where the averaged equation is defined. Then, the slow component of …

The Quantum Wasserstein Distance of Order 1

We propose a generalization of the Wasserstein distance of order 1 to the quantum states of $n$ qudits. The proposal recovers the Hamming distance for the vectors of the canonical basis, and more generally the classical Wasserstein distance for quantum states diagonal in the canonical basis. The proposed distance is invariant with respect to permutations of the qudits and unitary operations acting on one qudit and is additive with respect to the tensor product. Our main result is a continuity bound for the von Neumann entropy with respect to the proposed distance, which significantly strengthe…