zbMATH — the first resource for mathematics

Multidimensional investment problem. (English) Zbl 1404.91255
Summary: In this paper we demonstrate that the Riesz representation of excessive functions is a useful and enlightening tool to study optimal stopping problems. After a short general discussion of the Riesz representation we concretize to geometric Brownian motions. After this, a classical investment problem, also known as exchange-of-baskets-problem, is studied. It is seen that the boundary of the stopping region in this problem can be characterized as a unique solution of an integral equation arising immediately from the Riesz representation of the value function. The two-dimensional case is studied in more detail and a numerical algorithm is presented.

91G20 Derivative securities (option pricing, hedging, etc.)
60G40 Stopping times; optimal stopping problems; gambling theory
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
60G51 Processes with independent increments; Lévy processes
PDF BibTeX Cite
Full Text: DOI
[1] Abramowitz, M., Stegun, I.: Mathematical Functions, 9th Printing. Dover publications Inc., New York (1970)
[2] Alili, L; Kyprianou, AE, Some remarks on first passage of Lévy processes, the American put and pasting principles, Ann. Appl. Probab., 15, 2062-2080, (2005) · Zbl 1083.60034
[3] Beibel, M; Lerche, HR, A note on optimal stopping of regular diffusions under random discounting, Theory Prob. Appl., 45, 657-669, (2000) · Zbl 0994.60046
[4] Blumenthal, R.M., Getoor, R.K.: Markov Processes and Potential Theory Pure and Applied Mathematics, vol. 29. Academic Press, New York (1968) · Zbl 0169.49204
[5] Borodin, A.N., Salminen, P.: Handbook of Brownian Motion-Facts and Formulae, Corrected reprint of the 2nd edition. Birkhäuser Verlag, Basel (2015)
[6] Borwein, J.M., Vanderwerff, J.D.: Convex Functions: Constructions, Characterizations and Counterexamples, vol. 32. Cambridge University Press, Cambridge (2010) · Zbl 1191.26001
[7] Christensen, S, Optimal decision under ambiguity for diffusion processes, Math. Methods Oper. Res., 77, 207-226, (2013) · Zbl 1281.60040
[8] Christensen, S., Crocce, F., Mordecki, E., Salminen, P.: On optimal stopping of multidimensional diffusions (2016). https://arxiv.org/abs/1611.00959 · Zbl 07074619
[9] Christensen, S; Irle, A, A note on pasting conditions for the American perpetual optimal stopping problem, Stat. Probab. Lett., 79, 349-353, (2009) · Zbl 05523240
[10] Christensen, S; Irle, A, A harmonic function technique for the optimal stopping of diffusions, Stochastics, 83, 347-363, (2011) · Zbl 1241.60022
[11] Christensen, S., Salminen, P.: Riesz representation and optimal stopping with two case studies. ArXiv e-prints, arXiv:1309.2469 (2013). Available at arXiv:1309.2469 · Zbl 1114.60034
[12] Christensen, S; Salminen, P; Ta, B, Optimal stopping of strong Markov processes, Stoch. Process. Appl., 123, 1138-1159, (2013) · Zbl 1272.60021
[13] Chung, K.L., Walsh, J.B.: Markov Processes, Brownian Motion, and Time Symmetry Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 249, 2nd edn. Springer, New York (2005)
[14] Crocce, F; Mordecki, E, Explicit solutions in one-sided optimal stopping problems for one-dimensional diffusions, Stochastics, 86, 491-509, (2014) · Zbl 1306.60040
[15] Dayanik, S; Karatzas, I, On the optimal stopping problem for one-dimensional diffusions, Stoch. Process. Appl., 107, 173-212, (2003) · Zbl 1075.60524
[16] De Angelis, T., Federico, S., Ferrari, G.: Optimal boundary surface for irreversible investment with stochastic costs. Math. Opera. Res. (2017) (to appear) · Zbl 1386.93304
[17] Toit, J; Peskir, G, Selling a stock at the ultimate maximum, Ann. Appl. Probab., 19, 983-1014, (2009) · Zbl 1201.60037
[18] Ekström, E; Peskir, G, Optimal stopping games for Markov processes, SIAM J. Control Optim., 47, 684-702, (2008) · Zbl 1163.91011
[19] Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.: Tables of Integral Transforms. McGraw-Hill, New York (1954) · Zbl 0055.36401
[20] Gahungu, J., Smeers, Y.: Optimal time to invest when the price processes are geometric Brownian motions. A tentative based on smooth fit. CORE Discussion Papers 2011034, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE) (2011) · Zbl 1306.60040
[21] Gahungu, J., Smeers, Y.: Sufficient and necessary conditions for perpetual multi-assets exchange options. CORE Discussion Papers 2011035, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE) (2011) · Zbl 0147.16505
[22] Hu, Y; Øksendal, B, Optimal time to invest when the price processes are geometric Brownian motions, Finance Stoch., 2, 295-310, (1998) · Zbl 0904.60030
[23] Johnson, P., Peskir, G.: Quickest detection problems for Bessel processes. Ann. Appl. Probab. 27(2), 1003-1056 (2017) · Zbl 1370.60135
[24] Johnson, P., Peskir, G.: Sequential testing problems for Bessel processes. Trans. Am. Math. Soc (2014) (to appear) · Zbl 1406.60061
[25] Kunita, H; Watanabe, T, Markov processes and martin boundaries. I, Ill. J. Math., 9, 485-526, (1965) · Zbl 0147.16505
[26] McDonald, R; Siegel, D, The value of waiting to invest, Q. J. Econ., 101, 707-727, (1986)
[27] McKean, H, Appendix: A free boundary problem for the heat equation arising from a problem of mathematical economics, Ind. Manag. Rev., 6, 32-39, (1965)
[28] Mordecki, E; Salminen, P, Optimal stopping of hunt and Lévy processes, Stochastics, 79, 233-251, (2007) · Zbl 1114.60034
[29] Nishide, K; Rogers, LCG, Optimal time to exchange two baskets, J. Appl. Probab., 48, 21-30, (2011) · Zbl 1213.60084
[30] Øksendal, B.: Stochastic Differential Equations: An Introduction with Applications, 6th edn. Springer, Berlin (2003). doi:10.1007/978-3-642-14394-6 · Zbl 1025.60026
[31] Øksendal, B., Sulem, A.: Applied Stochastic Control of Jump Diffusions, 2nd edn. Springer, Berlin (2007). doi:10.1007/978-3-540-69826-5 · Zbl 1116.93004
[32] Olsen, T; Stensland, G, On optimal timing of investment when cost components are additive and follow geometric diffusions, J. Econ. Dyn. Control, 16, 39-51, (1992) · Zbl 0751.90021
[33] Paulsen, V, Bounds for the American perpetual put on a stock index, J. Appl. Probab., 38, 55-66, (2001) · Zbl 0979.91042
[34] Peskir, G, On the American option problem, Math. Finance, 15, 169-181, (2005) · Zbl 1109.91028
[35] Peskir, G, Optimal stopping games and Nash equilibrium, Theory Prob. Appl., 53, 558-571, (2009) · Zbl 1209.91052
[36] Peskir, G., Shiryaev, A.N.: Optimal Stopping and Free-Boundary Problems. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2006) · Zbl 1115.60001
[37] Salminen, P.: Optimal stopping of one-dimensional diffusions. Math. Nachr. 124, 85-101 (1985). doi:10.1002/mana.19851240107 · Zbl 0594.60080
[38] Shiryaev, A.N.: Optimal stopping rules. In: Stochastic Modelling and Applied Probability, vol 8. Springer, Berlin (2008). Translated from the 1976 Russian second edition by A. B. Aries, Reprint of the 1978 translation · Zbl 0994.60046
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.