On the one-sided tanaka equation with drift

Ioannis Karatzas, Albert N. Shiryaev, Mykhaylo Shkolnikov

Research output: Contribution to journalArticlepeer-review

13 Scopus citations

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 ≤ 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”.

Original languageEnglish (US)
Pages (from-to)664-677
Number of pages14
JournalElectronic Communications in Probability
Volume16
DOIs
StatePublished - Jan 1 2011

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Keywords

  • Comparison theorems for diffusions
  • Skew Brownian motion
  • Sticky Brownian motion
  • Stochastic differential equation
  • Strong existence
  • Strong uniqueness
  • Tanaka equation
  • Weak existence
  • Weak uniqueness

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