Electronic Communications in Probability

On the Novikov-Shiryaev Optimal Stopping Problems in Continuous Time

Andreas Kyprianou and Budhi Surya

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Novikov and Shiryaev (2004) give explicit solutions to a class of optimal stopping problems for random walks based on other similar examples given in Darling et al. (1972). We give the analogue of their results when the random walk is replaced by a Lévy process. Further we show that the solutions show no contradiction with the conjecture given in Alili and Kyprianou (2004) that there is smooth pasting at the optimal boundary if and only if the boundary of the stopping reigion is irregular for the interior of the stopping region.

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Electron. Commun. Probab., Volume 10 (2005), paper no. 15, 146-154.

Accepted: 22 July 2005
First available in Project Euclid: 4 June 2016

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Kyprianou, Andreas; Surya, Budhi. On the Novikov-Shiryaev Optimal Stopping Problems in Continuous Time. Electron. Commun. Probab. 10 (2005), paper no. 15, 146--154. doi:10.1214/ECP.v10-1144. https://projecteuclid.org/euclid.ecp/1465058080

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  • Alili, L. and Kyprianou, A.E. (2004) Some remarks on first passage of Lévy processes, the American put and smooth pasting. Ann. Appl. Probab. 15, 2062–2080.
  • Bertoin, J. (1997) Regularity of the half-line for Lévy processes. Bull. Sci. Math. 121, 345–354.
  • Boyarchenko, S.I. and Levendorskiǐ, S.Z. (2002) Non-Gaussian Merton-Black-Scholes theory. Advanced Series on Statistical Science & Applied Probability, 9. World Scientific Publishing Co., Inc., River Edge, NJ.
  • Darling, D. A., Liggett, T. and Taylor, H. M. (1972) Optimal stopping for partial sums. Ann. Math. Statist. 43, 1363–1368.
  • Lukacs, E. (1970) Characteristic functions. Second edition, revised and enlarged. Hafner Publishing Co., New York.
  • Mordecki, E. (2002) Optimal stopping and perpetual options for Lévy processes. Fin. Stoch. 6, 473–493.
  • Novikov, A. and Shiryaev, A. N. (2004) On an effective solution of the optimal stopping problem for random walks. To appear in Theory of Probability and Their Applications.
  • Rogozin, B. A. (1968) The local behavior of processes with independent increments. Theory Probab. Appl. 13 507–512.
  • Sato, K. (1999) Lévy processes and infinitely divisible distributions. Cambridge University Press.
  • Schoutens, W. (2000) Stochastic processes and orthogonal polynomials. Lecture Notes in Mathematics, nr. 146. Springer.
  • Shtatland, E.S. (1965) On local properties of processes with independent incerements. Theory Probab. Appl. 10, 317–322.