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 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. (C) 2018 Elsevier B.V. All rights reserved.