Sankhya: The Indian Journal of Statistics

2005, Volume 67, Pt. 1, 19--45

One-Dimensional Stochastic Differential Equations with Singular and Degenerate Coefficients


Richard F. Bass, University of Connecticut, Storrs, USA
Zhen-Qing Chen, University of Washington, Seattle,  USA

SUMMARY. 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.

AMS (1991) subject classification. Primary 60H10; secondary 60J55, 60J60.

Key words and phrases. Stochastic differential equations, pathwise uniqueness, strong solution, singular drift, degenerate coefficients, comparison principle.

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