On the one-sided tanaka equation with drift

Ioannis Karatzas; Albert N. Shiryaev; Mykhaylo Shkolnikov

doi:10.1214/ECP.v16-1665

On the one-sided tanaka equation with drift

Ioannis Karatzas
, Albert N. Shiryaev
, Mykhaylo Shkolnikov

Research output: Contribution to journal › Article › peer-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 language	English (US)
Pages (from-to)	664-677
Number of pages	14
Journal	Electronic Communications in Probability
Volume	16
DOIs	https://doi.org/10.1214/ECP.v16-1665
State	Published - Jan 1 2011
Externally published	Yes

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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10.1214/ECP.v16-1665

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title = "On the one-sided tanaka equation with drift",

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”.",

keywords = "Comparison theorems for diffusions, Skew Brownian motion, Sticky Brownian motion, Stochastic differential equation, Strong existence, Strong uniqueness, Tanaka equation, Weak existence, Weak uniqueness",

author = "Ioannis Karatzas and Shiryaev, \{Albert N.\} and Mykhaylo Shkolnikov",

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doi = "10.1214/ECP.v16-1665",

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T1 - On the one-sided tanaka equation with drift

AU - Karatzas, Ioannis

AU - Shiryaev, Albert N.

AU - Shkolnikov, Mykhaylo

PY - 2011/1/1

Y1 - 2011/1/1

N2 - 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”.

AB - 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”.

KW - Comparison theorems for diffusions

KW - Skew Brownian motion

KW - Sticky Brownian motion

KW - Stochastic differential equation

KW - Strong existence

KW - Strong uniqueness

KW - Tanaka equation

KW - Weak existence

KW - Weak uniqueness

UR - https://www.scopus.com/pages/publications/80755143187

UR - https://www.scopus.com/pages/publications/80755143187#tab=citedBy

U2 - 10.1214/ECP.v16-1665

DO - 10.1214/ECP.v16-1665

M3 - Article

AN - SCOPUS:80755143187

SN - 1083-589X

VL - 16

SP - 664

EP - 677

JO - Electronic Communications in Probability

JF - Electronic Communications in Probability

ER -

On the one-sided tanaka equation with drift

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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