Splitting: Tanaka's SDE revisited

On flows associated to Tanaka's SDE and related works

We review the construction of flows associated to Tanaka's SDE from [9] and give an easy proof of the classification of these flows by means of probability measures on [0, 1]. Our arguments also simplify some proofs in the subsequent papers [2, 3, 7, 4].

Discrete approximation to solution flows of Tanaka SDE related to Walsh Brownian motion

In a previous work, we have defined a Tanaka SDE related to Walsh Brownian motion which depends on kernels. It was shown that there are only one Wiener solution and only one flow of mappings solving this equation. In the terminology of Le Jan and Raimond, these are respectively the stronger and the weaker among all solutions. In this paper, we obtain these solutions as limits of discrete models.

由分数布朗运动驱动的随机微分方程的 Tanaka 公式

Tanaka formula for SDEs driven by fractional Brownian motion

本文为分数布朗运动驱动的非线性随机微分方程推导出 Tanaka 型公式，包含 Skorokhod 积分与 Malliavin 迹修正，并通过新方法建立 L2 收敛，推广至凸泛函。

Stratonovich SDE with irregular coefficients: Girsanov's example revisited

In this paper we study the Stratonovich stochastic differential equation $\mathrm{d} X=|X|^α\circ\mathrm{d} B$, $α\in(-1,1)$, which has been introduced by Cherstvy et al. [New Journal of Physics 15:083039 (2013)] in the context of analysis of anomalous diffusions in heterogeneous media. We determine its weak and strong solutions, which are homogeneous strong Markov processes \chng{spending zero time at $0$: for $α\in (0,1)$, these solutions have the form $$ X_t^θ=\bigl((1-α)B_t^θ\bigr)^{1/(1-α)}, $$ where $B^θ$ is the $θ$-skew Brownian motion driven by $B$ and starting at $\frac{1}{1-α}(X_0)^{…

McKean--Vlasov随机微分方程的弱同步化

Weak synchronisation for McKean--Vlasov SDEs

本文提出转移原理，将噪声同步化从冻结扩散SDE推广至McKean--Vlasov SDE，并首次应用于集合卡尔曼采样，同时改进乘性噪声同步化准则至非紧致控制设定。