尖点系数的严格SDE比较与反例
Strict SDE Comparison for Cusp Coefficients and Counterexamples
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中文总结 AI 辅助
本文针对一维随机微分方程，利用二维Lyapunov论证给出严格比较的充分条件，并证明对幂函数系数成立当且仅当指数在[1/2,1)内，同时构造反例表明光滑性或Hölder连续性不足以保证严格比较。
中文摘要 AI 辅助
我们为一维随机微分方程\\[ d X_t^x=\sigma(X_t^x)\\,d B_t, \quad X_0^x=x\in I=(\ell,r), \\]（其中$\sigma>0$且连续）的解的严格比较提供了一个易于处理的充分条件。我们的证明基于二维Lyapunov论证，这使得我们能够对$W^{1,p}_{\text{loc}}(I)$（$1\le p<2$）中的某些系数证明严格比较。我们以$\sigma(x):=1+|x|^\beta$（$x\in\mathbb{R}$，$\beta \in (0,1)$）为例进行说明，并表明严格比较成立当且仅当$\beta\in[\frac12,1)$。我们给出例子表明，即使结合有界性、一致椭圆性、全局强存在性和路径唯一性，$\sigma\in W^{1,p}_{\text{loc}}(\mathbb R)$（$p\in(1,2)$）或$\sigma\in C^\beta(\mathbb R)$（$\beta\in[\frac12,1)$）都不足以保证严格比较。
英文摘要
We provide a tractable sufficient condition for strict comparison for solutions of the one-dimensional stochastic differential equation \[ d X_t^x=σ(X_t^x)\,d B_t, \quad X_0^x=x\in I=(\ell,r), \] for $σ>0$ and continuous. Our proof is based on a two-dimensional Lyapunov argument, which allows us to prove strict comparison for some coefficients in $W^{1,p}_{\text{loc}}(I)$, $1\le p<2$. We illustrate using $σ(x):=1+|x|^β$ for $x\in\mathbb{R}$, $β\in (0,1)$, and show that strict comparison holds if and only if $β\in[\frac12,1)$. We give examples showing that neither $σ\in W^{1,p}_{\text{loc}}(\mathbb R)$, $p\in(1,2)$, nor $σ\in C^β(\mathbb R)$, $β\in[\frac12,1)$, is sufficient for strict comparison, even when combined with boundedness, uniform ellipticity, global strong existence, and pathwise uniqueness.
