论文标题
在漂移系数不连续的情况下，具有超线系数的SDE溶液的存在，独特性和近似
Existence, uniqueness and approximation of solutions of SDEs with superlinear coefficients in the presence of discontinuities of the drift coefficient
论文作者
论文摘要
通过考虑以下情况下，向标量随机微分方程（SDE）提出了存在，唯一性和$ L_P $ - APPROXIMATION结果，该情况考虑到漂移系数具有有限的许多空间不连续性，而两个系数都可以在空间上生长（在空间可变）。这些不连续性通过分段局部Lipschitz的连续性和分段单调型条件来描述，而在漂移系数的不连续点处，扩散系数被认为是局部Lipschitz的连续和非分类。此外，系数的超线性质取决于系数的（局部）Lipschitz常数的多项式生长决定。获得了此类SDE的强溶液的存在和独特性。此外，对于适用的$ p $值范围的经典$ L_P $ -Error Rate $ 1/2 $，用于驯服的Euler方案，用于近似这些解决方案。据作者所知，这些是该类别的SDE的第一个存在，独特性和近似结果。
Existence, uniqueness, and $L_p$-approximation results are presented for scalar stochastic differential equations (SDEs) by considering the case where, the drift coefficient has finitely many spatial discontinuities while both coefficients can grow superlinearly (in the space variable). These discontinuities are described by a piecewise local Lipschitz continuity and a piecewise monotone-type condition while the diffusion coefficient is assumed to be locally Lipschitz continuous and non-degenerate at the discontinuity points of the drift coefficient. Moreover, the superlinear nature of the coefficients is dictated by a suitable coercivity condition and a polynomial growth of the (local) Lipschitz constants of the coefficients. Existence and uniqueness of strong solutions of such SDEs are obtained. Furthermore, the classical $L_p$-error rate $1/2$, for a suitable range of values of $p$, is recovered for a tamed Euler scheme which is used for approximating these solutions. To the best of the authors' knowledge, these are the first existence, uniqueness and approximation results for this class of SDEs.
