Topics in Walsh Semimartingales and Diffusions

This dissertation is devoted to theories of processes we call ``Walsh semimartingales" and ``Walsh diffusions", as well as to related optimization problems of control and stopping. These processes move on the plane along rays emanating from the origin; and when at the origin, the processes choose the rays of their subsequent voyage according to a fixed probability measure---in a manner described by Walsh (1978) as a direct generalization of the skew Brownian motion. We first review in Chapter 1 some key results regarding the celebrated skew Brownian motions and Walsh Brownian motions. These results include the characterization of skew Brownian motions via stochastic equations in Harrison & Shepp (1981), the construction of Walsh Brownian motions in Barlow, Pitman & Yor (1989), and the important result of Tsirel'son (1997) regarding the nature of the filtration generated by the Walsh Brownian motion. Various generalizations of Walsh Brownian motions are described in detail in Chapter 2. We formally define there Walsh semimartingales as a subclass of planar processes we call ``semimartingales on rays". We derive for such processes Freidlin-Sheu-type change-of-variable formulas, as well as two-dimensional versions of the Harrison-Shepp equations. The actual construction of Walsh semimartingales is given next. Walsh diffusions are then defined as a subclass of Walsh semimartingales, described by stochastic equations which involve local drift and dispersion characteristics. The associated local submartingale problems, strong Markov properties, existence, uniqueness, asymptotic behavior, and tests for explosions in finite time, are studied in turn. Finally, with Walsh semimartingales as state-processes, we study in Chapter 3 succesively a pure optimal stopping problem, a stochastic control problem with discretionary stopping, and a stochastic game between a controller and a stopper. We derive for these problems optimal strategies in surprisingly explicit from. Crucial for the analysis underpinning these results, are the change-of-variable formulas derived in Chapter 2. Most of the results in Chapters 2 and 3 are based on two papers, [21] and [31], both cowritten by the author of this dissertation. Some results and proofs are rearranged and rewritten here.