Pathwise uniqueness for a degenerate stochastic differential equation

Details Pathwise uniqueness for a degenerate stochastic differential equation Pathwise uniqueness for a degenerate stochastic differential equation

2006-01-20

arxiv.org/abs/math/0601505v2

Annals of Probability 2007, Vol. 35, No. 6, 2385-2418

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/009117907000000033 the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117907000000033

We introduce a new method of proving pathwise uniqueness, and we apply it to the degenerate stochastic differential equation \[dX_t=|X_t|^{\alpha} dW_t,\] where $W_t$ is a one-dimensional Brownian motion and $\alpha\in(0,1/2)$. Weak uniqueness does not hold for the solution to this equation. If one restricts attention, however, to those solutions that spend zero time at 0, then pathwise uniqueness does hold and a strong solution exists. We also consider a class of stochastic differential equations with reflection.

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