Counterexamples to local Lipschitz and local Hölder continuity with respect to the initial values for additive noise driven SDEs with smooth drift coefficient functions with at most polynomially growing derivatives

Drift reduction method for SDEs driven by inhomogeneous singular L{é}vy noise

We study SDE $$ d X_t = b(X_t) \, dt + A(X_{t-}) \, d Z_t, \quad X_{0} = x \in \mathbb{R}^d, \quad t \geq 0 $$ where $Z=(Z^1, \dots, Z^d)^T$, with $Z^i, i=1,\dots, d$ being independent one-dimensional symmetric jump Lévy processes, not necessarily identically distributed. In particular, we cover the case when each $Z^i$ is one-dimensional symmetric $α_i$-stable process ($α_i \in (0,2)$ and they are not necessarily equal). Under certain assumptions on $b$, $A$ and $Z$ we show that the weak solution to the SDE is uniquely defined and Markov, we provide a representation of the transition probabil…

Heat kernel of supercritical SDEs with unbounded drifts

Let $α\in(0,2)$ and $d\in{\mathbb N}$. Consider the following SDE in ${\mathbb R}^d$:$${\rm d}X_t=b(t,X_t){\rm d} t+a(t,X_{t-}){\rm d} L^{(α)}_t,\ \ X_0=x,$$where $L^{(α)}$ is a $d$-dimensional rotationally invariant $α$-stable process, $b:{\mathbb R}_+\times{\mathbb R}^d\to{\mathbb R}^d$ and $a:{\mathbb R}_+\times{\mathbb R}^d\to{\mathbb R}^d\otimes{\mathbb R}^d$ are H{ö}lder continuous functions in space, with respective order $β,γ\in (0,1)$ such that $(β\wedge γ)+α>1$, uniformly in $t$. Here $b$ may be unbounded.When $a$ is bounded and uniformly elliptic, we show that the unique solution $X…

局部Lipschitz系数随机微分方程的基本均方收敛定理及其应用

A fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications

针对系数多项式增长且满足单边Lipschitz条件的随机微分方程，证明了基本均方收敛定理，并应用于平衡格式和全隐式方法。

稳定过程驱动的McKean-Vlasov随机微分方程的混沌定量弱传播

Quantitative weak propagation of chaos for stable-driven McKean-Vlasov SDEs

本文研究旋转不变α-稳定过程驱动的McKean-Vlasov随机微分方程，利用对应半群的正则性与倒向Kolmogorov方程描述的动力学，在半群层面证明了平均场相互作用粒子系统的新混沌传播估计。

On the strong regularity of degenerate additive noise driven stochastic differential equations with respect to their initial values

Recently in [M. Hairer, M. Hutzenthaler, and A. Jentzen, Ann. Probab. 43, 2 (2015), 468--527] and [A. Jentzen, T. Müller-Gronbach, and L. Yaroslavtseva, Commun. Math. Sci. 14, 6 (2016), 1477--1500] stochastic differential equations (SDEs) with smooth coefficient functions have been constructed which have an arbitrarily slowly converging modulus of continuity in the initial value. In these SDEs it is crucial that some of the first order partial derivatives of the drift coefficient functions grow at least exponentially and, in particular, quicker than any polynomial. However, in applications SDE…

Gradient estimates for SDEs without monotonicity type conditions

We prove gradient estimates for transition Markov semigroups $(P_t)$ associated to SDEs driven by multiplicative Brownian noise having possibly unbounded $C^1$-coefficients, without requiring any monotonicity type condition. In particular, first derivatives of coefficients can grow polynomially and even exponentially. We establish pointwise estimates with weights for $D_x P_tφ$ of the form \[ {\sqrt{t}} \, |D_x P_t φ(x) | \le c \, (1+ |x|^k) \, \| φ\|_{\infty} \] $t \in (0,1]$, $φ\in C_b ({\mathbb R}^d)$, $x \in {\mathbb R}^d.$ To prove the result we use two main tools. First, we consider a Fe…

A splitting method for SDEs with locally Lipschitz drift: Illustration on the FitzHugh-Nagumo model

In this article, we construct and analyse an explicit numerical splitting method for a class of semi-linear stochastic differential equations (SDEs) with additive noise, where the drift is allowed to grow polynomially and satisfies a global one-sided Lipschitz condition. The method is proved to be mean-square convergent of order 1 and to preserve important structural properties of the SDE. First, it is hypoelliptic in every iteration step. Second, it is geometrically ergodic and has an asymptotically bounded second moment. Third, it preserves oscillatory dynamics, such as amplitudes, frequenci…

多级皮卡近似法用于具有非常数扩散系数的麦肯-瓦尔萨夫随机微分方程

Multilevel Picard approximations for McKean-Vlasov stochastic differential equations with nonconstant diffusion

本文提出了一种多级皮卡近似法用于求解具有非常数扩散系数的麦肯-瓦尔萨夫随机微分方程，在标准Lipschitz条件下，证明该方法在L²意义下近似解，且计算成本随维度和误差容忍度的倒数呈多项式增长。