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arXiv · 1108.4069

On the one-sided Tanaka equation with drift

Ioannis Karatzas·Albert N. Shiryaev·Mykhaylo Shkolnikov
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Abstract

We study questions of existence and uniqueness of weak and strong solutions for a one-sided Tanaka equation with constant drift λ. We observe a dichotomy in terms of the values of the drift parameter: for λ\leq 0, there exists a strong solution which is pathwise unique, thus also unique in distribution; whereas for λ>0, the equation has a unique in distribution weak solution, but no strong solution (and not even a weak solution that spends zero time at the origin). We also show that strength and pathwise uniqueness are restored to the equation via suitable "Brownian perturbations".

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Ioannis Karatzas, Albert N. Shiryaev, Mykhaylo Shkolnikov. 2011-08-19. On the one-sided Tanaka equation with drift. https://arxiv.org/abs/1108.4069

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