an:07039131
Zbl 1409.60066
Peskir, Goran
Continuity of the optimal stopping boundary for two-dimensional diffusions
EN
Ann. Appl. Probab. 29, No. 1, 505-530 (2019).
1050-5164 2168-8737
2019
j
60G40 60J60 60H30 35K20 35J25 35R35
optimal stopping; free boundary; continuity of the optimal stopping boundary; two-dimensional diffusion process; smooth fit; second-order parabolic/elliptic PDE; local time-space calculus
Summary: We first show that a smooth fit between the value function and the gain function at the optimal stoppnig boundary for a two-dimensional diffusion process implies the absence of boundary's discontinuities of the first kind (the right-hand and left-hand limits exist but differ). We then show that the smooth fit itself is satisfied over the flat portion of the optimal stopping boundary arising from any of its hypothesised jumps. Combining the two facts we obtain that the optimal stopping boundary is continuous whenever it has no discontinuities of the second kind. The derived fact holds both in the parabolic and elliptic case under the sole hypothesis of HÃ¶lder continuous coefficients, thus improving upon all known results in the parabolic case, and establishing the fact for the first time in elliptic case. The method of proof relies upon regularity results for the second-order parabolic/elliptic PDEs and makes use of the local time-space calculus techniques.