Karatzas, Ioannis; Yan, Minghan Semimartingales on rays, Walsh diffusions, and related problems of control and stopping. (English) Zbl 1478.60165 Stochastic Processes Appl. 129, No. 6, 1921-1963 (2019). 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 J. B. Walsh [“A diffusion with a discontinuous local time”, in: Temps locaux. Exposés du séminaire J. Azéma–M. Yor, Université Pierre et Marie Curie, Paris, France (1976–1977), Astérisque 52-53. Paris: Societé Mathématique de France (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. Cited in 5 Documents MSC: 60H05 Stochastic integrals 60G17 Sample path properties 60G40 Stopping times; optimal stopping problems; gambling theory 60H30 Applications of stochastic analysis (to PDEs, etc.) 60J60 Diffusion processes 93E20 Optimal stochastic control Keywords:semimartingales on rays; tree-topology; Walsh semimartingales and diffusions; Skorokhod reflection; local time; stochastic calculus; explosion times; Feller’s test; stochastic control; optimal stopping PDFBibTeX XMLCite \textit{I. Karatzas} and \textit{M. Yan}, Stochastic Processes Appl. 129, No. 6, 1921--1963 (2019; Zbl 1478.60165) Full Text: DOI arXiv References: [1] R. Atar, A. Cohen, Serve the shortest queue and Walsh Brownian motion. Preprint, arXiv:1802.02748v1; R. Atar, A. Cohen, Serve the shortest queue and Walsh Brownian motion. Preprint, arXiv:1802.02748v1 · Zbl 1409.60037 [2] Barlow, M. T.; Pitman, J. W.; Yor, M., On Walsh’s Brownian motions, (Séminaire de Probabilités XXIII. Séminaire de Probabilités XXIII, Lecture Notes in Mathematics, vol. 1372 (1989), Springer-Verlag: Springer-Verlag New York), 275-293 · Zbl 0747.60072 [3] Davis, M. H.A.; Zervos, M., A problem of singular control with discretionary stopping, Ann. Appl. Probab., 4, 226-240 (1994) · Zbl 0796.93111 [4] Dayanik, S.; Karatzas, I., On the optimal stopping problem for one-dimensional diffusions, Stochastic Process. Appl., 107, 173-212 (2003) · Zbl 1075.60524 [5] Dubins, L. E.; Savage, L. J., How to Gamble if You Must: Inequalities for Stochastic Processes (1965, 1976), Dover: Dover New York [6] Dynkin, E. B.; Yushkevich, A. A., Markov Processes: Theorems and Problems (1969), Plenum Press: Plenum Press New York [7] Engelbert, H. J.; Schmidt, W., (On the Behaviour of Certain Functionals of the Wiener Process and Applications to Stochastic Differential Equations. On the Behaviour of Certain Functionals of the Wiener Process and Applications to Stochastic Differential Equations, Lecture Notes in Control and Information Sciences, vol. 36 (1981), Springer-Verlag: Springer-Verlag Berlin), 47-55 · Zbl 0468.60077 [8] Engelbert, H. J.; Schmidt, W., (On One-Dimensional Stochastic Differential Equations with Generalized Drift. On One-Dimensional Stochastic Differential Equations with Generalized Drift, Lecture Notes in Control and Information Sciences, vol. 69 (1984), Springer-Verlag: Springer-Verlag Berlin), 143-155 · Zbl 0545.60060 [9] Engelbert, H. J.; Schmidt, W., On solutions of stochastic differential equations without drift, Z. Wahrscheinlichkeitstheor. Verwandte Geb., 68, 287-317 (1985) · Zbl 0535.60049 [10] Ernst, Ph., Exercising control when confronted by a (Brownian) spider, Oper. Res. Lett., 44, 487-490 (2016) · Zbl 1380.90275 [11] Fitzsimmons, P. J.; Kuter, K. E., Harmonic functions on Walsh’s Brownian motion, Stochastic Process. Appl., 124, 2228-2248 (2014) · Zbl 1320.60138 [12] Freidlin, M.; Sheu, S., Diffusion processes on graphs: stochastic differential equations, large deviation principle, Probab. Theory Related Fields, 116, 181-220 (2000) · Zbl 0957.60088 [13] Hajri, H.; Touhami, W., Itô’s formula for Walsh’s Brownian motion and applications, Statist. Probab. Lett., 87, 48-53 (2014) · Zbl 1315.60066 [14] Ichiba, T.; Karatzas, I.; Prokaj, V.; Yan, M., Stochastic integral equations for Walsh semimartingales, Ann. Inst. Henri Poincaré B, 54, 2, 726-756 (2018) · Zbl 1391.60090 [15] Karatzas, I.; Ocone, D. L., A leavable bounded-velocity stochastic control problem, Stochastic Process. Appl., 99, 31-51 (2002) · Zbl 1064.93049 [16] Karatzas, I.; Shreve, S. E., Brownian Motion and Stochastic Calculus (1991), Springer-Verlag: Springer-Verlag NY · Zbl 0734.60060 [17] Karatzas, I.; Sudderth, W. D., Control and stopping of a diffusion process on an interval, Ann. Appl. Probab., 9, 188-196 (1999) · Zbl 0938.93067 [18] Karatzas, I.; Sudderth, W. D., The controller-and-stopper game for a linear diffusion, Ann. Probab., 29, 1111-1127 (2001) · Zbl 1039.60043 [19] Pestien, V. C.; Sudderth, W. D., Continuous-time red and black: how to control a diffusion to a goal, Math. Oper. Res., 10, 599-611 (1985) · Zbl 0596.93052 [20] Picard, J., Stochastic calculus and martingales on trees, Ann. Inst. Henri Poincaré B, 41, 631-683 (2005) · Zbl 1077.60030 [21] Revuz, D.; Yor, M., Continuous Martingales and Brownian Motion (1999), Springer-Verlag: Springer-Verlag New York · Zbl 0917.60006 [22] Salminen, P., Optimal stopping of one-dimensional diffusions, Math. Nachr., 124, 85-101 (1985) · Zbl 0594.60080 [23] Touzi, N., With chapter 13 by Angès Tourin, (Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE. Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE, Fields Institute Monographs, vol. 29 (2013), Springer: Springer New York) [24] Walsh, J. B., A diffusion with a discontinuous local time, Astérisque, 52-53, 37-45 (1978) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.