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Continuity of the optimal stopping boundary for two-dimensional diffusions. (English) Zbl 1409.60066
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 Hölder 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.

60G40 Stopping times; optimal stopping problems; gambling theory
60J60 Diffusion processes
60H30 Applications of stochastic analysis (to PDEs, etc.)
35K20 Initial-boundary value problems for second-order parabolic equations
35J25 Boundary value problems for second-order elliptic equations
35R35 Free boundary problems for PDEs
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[1] Barlow, M. T. and Yor, M. (1981). (Semi-) martingale inequalities and local times. Z. Wahrsch. Verw. Gebiete55 237–254. · Zbl 0451.60050
[2] Blumenthal, R. M. and Getoor, R. K. (1968). Markov Processes and Potential Theory. Pure and Applied Mathematics, 29. Academic Press, New York. · Zbl 0169.49204
[3] Caffarelli, L. A. (1977). The regularity of free boundaries in higher dimensions. Acta Math.139 155–184. · Zbl 0386.35046
[4] Caffarelli, L. and Salsa, S. (2005). A Geometric Approach to Free Boundary Problems. Graduate Studies in Mathematics68. Amer. Math. Soc., Providence, RI. · Zbl 1083.35001
[5] Christensen, S. and Salminen, P. (2018). Multidimensional investment problem. Math. Financ. Econ.12 75–95. · Zbl 1404.91255
[6] Cox, A. M. G. and Peskir, G. (2015). Embedding laws in diffusions by functions of time. Ann. Probab.43 2481–2510. · Zbl 1335.60150
[7] De Angelis, T. (2015). A note on the continuity of free-boundaries in finite-horizon optimal stopping problems for one-dimensional diffusions. SIAM J. Control Optim.53 167–184. · Zbl 1338.60117
[8] De Angelis, T., Federico, S. and Ferrari, G. (2017). Optimal boundary surface for irreversible investment with stochastic costs. Math. Oper. Res.42 1135–1161. · Zbl 1386.93304
[9] De Angelis, T. and Peskir, G. (2016). Global \(C^{1}\) regularity of the value function in optimal stopping problems. Probab. Statist. Group Manchester. Research Report No. 13 (to appear). · Zbl 1396.60041
[10] De Angelis, T. and Stabile, G. (2017). On Lipschitz continuous optimal stopping boundaries. Preprint. Available at arXiv:1701.07491. · Zbl 1442.60048
[11] du Toit, J. and Peskir, G. (2009). Selling a stock at the ultimate maximum. Ann. Appl. Probab.19 983–1014. · Zbl 1201.60037
[12] Ferrari, G. (2018). On the optimal management of public debt: A singular stochastic control problem. SIAM J. Control Optim.56 1938–1975.
[13] Friedman, A. (1964). Partial Differential Equations of Parabolic Type. Prentice Hall, Englewood Cliffs, NJ. · Zbl 0144.34903
[14] Friedman, A. (1975). Parabolic variational inequalities in one space dimension and smoothness of the free boundary. J. Funct. Anal.18 151–176. · Zbl 0295.35045
[15] Gilbarg, D. and Trudinger, N. S. (2001). Elliptic Partial Differential Equations of Second Order. Springer, Berlin. · Zbl 1042.35002
[16] Johnson, P. and Peskir, G. (2017). Quickest detection problems for Bessel processes. Ann. Appl. Probab.27 1003–1056. · Zbl 1370.60135
[17] Johnson, P. and Peskir, G. (2018). Sequential testing problems for Bessel processes. Trans. Amer. Math. Soc.370 2085–2113. · Zbl 1406.60061
[18] Novikov, A. A. (1971). The moment inequalities for stochastic integrals. Theory Probab. Appl.16 538–541. · Zbl 0246.60047
[19] Peskir, G. (2005). On the American option problem. Math. Finance15 169–181. · Zbl 1109.91028
[20] Peskir, G. (2005). A change-of-variable formula with local time on curves. J. Theoret. Probab.18 499–535. · Zbl 1085.60033
[21] Peskir, G. (2007). A change-of-variable formula with local time on surfaces. In Séminaire de Probabilités XL. Lecture Notes in Math.1899 69–96. Springer, Berlin. · Zbl 1141.60035
[22] Peskir, G. and Shiryaev, A. (2006). Optimal Stopping and Free-Boundary Problems. Birkhäuser, Basel. · Zbl 1115.60001
[23] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin. · Zbl 0917.60006
[24] Shiryayev, A.
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