Abstract
A problem of optimal stopping for one-dimensional time-homogeneous regular diffusion with the infinite horizon is considered. The diffusion takes values in a finite or infinite interval ]a,b[. The points a and b may be either natural or absorbing or reflecting. The diffusion may have a partial reflection at a finite number of points. A discounting and a cost of observation are allowed. Both can depend on the state of the diffusion. The payoff function g(z) is bounded on any interval [c, d], where a < c < d < b, and twice differentiable with the exception of a finite (may be empty) set of points, where the functions g(z) and \({g}^{{^\prime}}(z)\) may have a discontinuities of the first kind. Let L be an infinitesimal generator of diffusion which includes the terms corresponding to the discounting and the cost of observation. We assume that the set \(\{z : Lg(z) > 0\}\) consists of a finite number of intervals. For such problem we propose a procedure of constructing the value function in a finite number of steps. The procedure is based on a fact that on intervals where Lg(z) > and in neighborhoods of points of partial reflections, points of discontinuities, and points a or b in case of reflection, one can modify the payoff function preserving the value function. Many examples are considered.
Mathematics Subject Classification (2010): 60G40, 60J60, 60J65.
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References
Broadie, M., Detemple, J.: American capped call options on dividend-paying assets. Rev. Financ. Stud. 8(1), 161–191 (1995)
Bronstein, A.L., Hughston, L.P., Pistorius, M.R., Zervos, M.: Discretionary stopping of one-dimensional Ito diffusions with a staircase reward function. J. Appl. Probab. 43, 984–996 (2006)
Dayanik, S., Karatzas, I.: On the optimal stopping problem for one-dimensional diffusions. Stoch. Process. Appl. 107, 173–212 (2003)
Guo, X., Shepp, L.A.: Some optimal stopping problems with nontrivial boundaries for pricing exotic options. J. Appl. Probab. 38(3), 647–658 (2001)
Harrison, J.M., Shepp, L.A.: On skew brownian motion. Ann. Probab. 9(2), 309–313 (1981)
Irle, A.: A forward algorithm for solving optimal stopping problems. J. Appl. Probab. 43, 102–ll3 (2006)
Irle, A.: On forward improvement iteration for stopping problems. In: Proceedings of the Second International Workshop in Sequential Methodologies, Troyes (2009)
Karatzas, I., Ocone, D.: A leavable bounded-velocity stochastic control problem. Stoch. Process. Appl. 99(1), 31–51 (2002)
Karatzas, I., Wang, H.: A barrier option of American type. Appl. Math. Optim. 42(3), 259–279 (2000)
Kolodka, A., Schoenmakers, J.: Iterative construction of the optimal Bermudan stopping time. Finance Stoch. 10, 27–49 (2006)
Minlos, R.A., Zhizhina, E.A.: Limit diffusion process for a non-homogeneous random walk on a one-dimentional lettice. Russ. Math. Surv. 52(2), 327–340 (1997)
Oksendal, B., Reikvam, K.: Viscosity solutions of optimal stopping problems. Stoch. Stoch. Rep. 62(3–4), 285–301 (1998)
Peskir, G., Shiryaev, A.N.: Optimal Stopping and Free-Boundary Problems. Birkhauser, Basel (2006)
Presman, E.L.: On Sonin’s algorithm for solution of the optimal stopping problem. In: Proceedings of the Fourth International Conference on Control Problems, 26–30 Jan 2009, pp. 300–309. Institute of Control Sciences, Moscow (2009)
Presman, E.L.: New approach to the solution of the optimal stopping problem in a discrete time. Stoch. Spec. Issue Dedic. Optim. Stop. Appl. 83, 467–475 (2011)
Presman, E.L.: Solution of Optimal Stopping Problem Based on a Modification of Payoff Function. Springer, Musela Festshrift (2011)
Presman, E.L., Sonin, I.M.: On optimal stopping of random sequences modulated by Markov chain. Theory Probab. Appl. 54(3), 534–542 (2010)
Salminen, P.: Optimal stopping of one-dimensional diffusions. Math. Nachr. 124, 85–101 (1985)
Shiryayev, A.N.: Statistical Sequential Analysis: Optimal Stopping Rules. Nauka, Moscow (1969) (in Russian). English translation of the 2nd edn., Shiryayev A.N.: Optimal Stopping Rules. Springer, New York (1978)
Sonin, I.M.: Two simple theorems in the problems of optimal stopping. In: Proceedings of the 8th INFORMS Applied Probability Conference, Atlanta, p. 27 (1995)
Sonin, I.M.: The elimination algorithm for the problem of optimal stopping. Math. Methods of Oper Res. 49, 111–123 (1999)
Sonin, I.M.: The state reduction and related algorithms and their applications to the study of Markov chains, graph theory and the optimal stopping problem. Adv. Math. 145, 159–188 (1999)
Sonin, I.M.: Optimal stopping of Markov chains and recursive solution of Poisson and Bellman equations. In: Kabanov, Y., Liptser, R., Stoyanov, J. (eds.) From Stochastic Calculus to Mathematical Finance: The Shiryaev Festschrift, pp. 609–621. Springer, Berlin/Heidelberg (2006)
Taylor, H.M.: Optimal stopping in a Markov process. Ann. Math. Stat. 3, 1333–1344 (1968)
Acknowledgements
The author would like to thank A.D. Slastnikov and I.M. Sonin for useful discussions.
This work was partly supported by RFBR grant 10-01-00767-a.
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Presman, E. (2013). Solution of the Optimal Stopping Problem for One-Dimensional Diffusion Based on a Modification of the Payoff Function. In: Shiryaev, A., Varadhan, S., Presman, E. (eds) Prokhorov and Contemporary Probability Theory. Springer Proceedings in Mathematics & Statistics, vol 33. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33549-5_22
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