Semimartingales on rays, Walsh diffusions, and related problems of control and stopping

We introduce a class of continuous planar processes, called “semimartingales on rays”, and develop for them a change-of-variable formula involving quite general classes of test functions. Special cases of such processes are diffusions which choose, once at the origin, the rays for their subsequent v...

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Bibliographic Details
Published in:Stochastic processes and their applications 2019-06, Vol.129 (6), p.1921-1963
Main Authors: Karatzas, Ioannis, Yan, Minghan
Format: Article
Language:English
Subjects:
Explosion times
Feller’s test
Local time
Optimal stopping
Semimartingales on rays
Skorokhod reflection
Stochastic calculus
Stochastic control
Tree-topology
Walsh semimartingales and diffusions
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Summary:We introduce a class of continuous planar processes, called “semimartingales on rays”, and develop for them a change-of-variable formula involving quite general classes of test functions. Special cases of such processes are diffusions which choose, once at the origin, the rays for their subsequent voyage according to a fixed probability measure in the manner of Walsh (1978). We develop existence and uniqueness results for these “Walsh diffusions”, study their asymptotic behavior, and develop tests for explosions in finite time. We use these results to find an optimal strategy, in a problem of stochastic control with discretionary stopping involving Walsh diffusions.
ISSN:0304-4149
1879-209X
DOI:10.1016/j.spa.2018.06.012