Abstract
We consider a degenerate stochastic differential equation that has a sticky point in the Markov process sense. We prove that weak existence and weak uniqueness hold, but that pathwise uniqueness does not hold nor does a strong solution exist.
Highlights
The one-dimensional stochastic differential equation dXt = σ(Xt) dWt (1.1)has been the subject of intensive study for well over half a century
Regular continuous strong Markov processes on the line which are on natural scale and have speed measure given by (1.3) are known as sticky Brownian motions
We show weak uniqueness, that is, if (X, W, P) and (X, W, P) are two weak solutions to (5.2) with X and X starting at x0 and in addition X and X have speed measure m, the joint law of (X, W ) under P equals the joint law of (X, W ) under P
Summary
Has been the subject of intensive study for well over half a century. What can one say about pathwise uniqueness when σ is allowed to be zero at certain points? a large amount is known, but there are many unanswered questions remaining. Regular continuous strong Markov processes on the line which are on natural scale and have speed measure given by (1.3) are known as sticky Brownian motions. These were first studied by Feller in the 1950’s and Itô and McKean in the 1960’s. (Local times in the Markov process sense can be different in general.) Engelbert and Peskir proved weak uniqueness of the joint law of (X, W ) and proved that no strong solution exists They considered a one-sided version of this equation, where X ≥ 0, and showed that it is equivalent to (1.4). Farnsworth for suggesting a mathematical finance interpretation of a sticky point
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