期刊名称：Sankhya. Series A, mathematical statistics and probability
印刷版ISSN：0976-836X
电子版ISSN：0976-8378
出版年度：2005
卷号：67
期号：01
出版社：Indian Statistical Institute
摘要：We show the existence of strong solutions and pathwise uniqueness for two types of one-dimensional stochastic differential equations. The first type allows singular drifts: $$X_t=X_0+ \int_0^t a(X_t) dW_t +\int_{\R} L^w_t(X)\,\mu(dw) \quad \hbox{ for } t\geq 0, $$ where $W$ is a one-dimensional Brownian motion, $a$ is a function that is bounded between two positive constants, $\mu$ is a finite measure with $|\mu (\{w\})|\leq 1$, and $L^w$ is the local time at $w$ for the semimartingale $X$. The second type is the equation $$dX_t= (X_t)^\al dW_t+dL_t,$$ where $L$ is a continuous non-decreasing process that increases only when $X$ is at 0, $\al\in (0,\frac12)$, and $X_t\geq 0$ for all $t$. Although this second equation does not have a unique solution, it does have a unique solution if one restricts attention to those solutions that spend zero time at 0.
