Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T19:44:26.705Z Has data issue: false hasContentIssue false
  • English
  • Français

Irreversible investment under Lévy uncertainty: an equation for the optimal boundary

Part of: Stochastic processes Mathematical economics Stochastic systems and control

Published online by Cambridge University Press:  24 March 2016

Giorgio Ferrari  and
Paavo Salminen
Giorgio Ferrari*
Affiliation:
Bielefeld University
Paavo Salminen*
Affiliation:
Åbo Akademi University
*
* Postal address: Faculty of Science and Engineering, Åbo Akademi University, Fänriksgatan 3 B, FIN-20500 Åbo, Finland. Email address: [email protected]" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink">[email protected]">[email protected]
** Postal address: Center for Mathematical Economics, Bielefeld University, Universitätstrasse 25, 33615 Bielefeld, Germany. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We derive a new equation for the optimal investment boundary of a general irreversible investment problem under exponential Lévy uncertainty. The problem is set as an infinite time-horizon, two-dimensional degenerate singular stochastic control problem. In line with the results recently obtained in a diffusive setting, we show that the optimal boundary is intimately linked to the unique optional solution of an appropriate Bank–El Karoui representation problem. Such a relation and the Wiener–Hopf factorization allow us to derive an integral equation for the optimal investment boundary. In case the underlying Lévy process hits any point in R with positive probability we show that the integral equation for the investment boundary is uniquely satisfied by the unique solution of another equation which is easier to handle. As a remarkable by-product we prove the continuity of the optimal investment boundary. The paper is concluded with explicit results for profit functions of Cobb–Douglas type and CES type. In the former case the function is separable and in the latter case nonseparable.

Keywords

Free-boundaryirreversible investmentsingular stochastic controloptimal stoppingLévy processBank and El Karoui's representation theorembase capacity

MSC classification

Primary: 93E20: Optimal stochastic control
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 60G51: Processes with independent increments; L'evy processes 91B70: Stochastic models
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 1 , March 2016 , pp. 298 - 314
Copyright
Copyright © Applied Probability Trust 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Abel, A. B. and Eberly, J. C. (1996). Optimal investment with costly reversibility. Rev. Econom. Stud. 63, 581–593. CrossRefGoogle Scholar
[2]Alili, L. and Kyprianou, A. E. (2005). Some remarks on first passage of Lévy processes, the American put and pasting principles. Ann. Appl. Prob. 15, 2062–2080. CrossRefGoogle Scholar
[3]Baldursson, F. M. and Karatzas, I. (1996). Irreversible investment and industry equilibrium. Finance Stoch. 1, 69–89. Google Scholar
[4]Bank, P. (2005). Optimal control under a dynamic fuel constraint. SIAM J. Control Optimization 44, 1529–1541. CrossRefGoogle Scholar
[5]Bank, P. and El Karoui, N. (2004). A stochastic representation theorem with applications to optimization and obstacle problems. Ann. Prob. 32, 1030–1067. Google Scholar
[6]Bank, P. and Föllmer, H. (2003). American options, multi-armed bandits, and optimal consumption plans: a unifying view. In Paris-Princeton Lectures on Mathematical Finance (Lecture Notes Math. 1814), Springer, Berlin, pp. 1–42. Google Scholar
[7]Bank, P. and Riedel, F. (2001). Optimal consumption choice with intertemporal substitution. Ann. Appl. Prob. 11, 750–788. Google Scholar
[8]Bentolila, S. and Bertola, G. (1990). Firing costs and labour demand: how bad is Eurosclerosis? Rev. Econom. Stud. 57, 381–402. CrossRefGoogle Scholar
[9]Bertoin, J. (1996). Lévy Processes. Cambridge University Press. Google Scholar
[10]Bertola, G. (1998). Irreversible investment. Res. Econom. 52, 3–37. Google Scholar
[11]Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion—Facts and Formulae, 2nd edn. Birkhäuser, Basel. CrossRefGoogle Scholar
[12]Boyarchenko, S. (2004). Irreversible decisions and record-setting news principles. Amer. Econom. Rev. 94, 557–568. Google Scholar
[13]Boyarchenko, S. I. and Levendorskiĭ, S. Z. (2002). Perpetual American options under Lévy processes. SIAM J. Control Optimization 40, 1663–1696. Google Scholar
[14]Chiarolla, M. B. and Ferrari, G. (2014). Identifying the free boundary of a stochastic, irreversible investment problem via the Bank–El Karoui representation theorem. SIAM J. Control Optimization 52, 1048–1070. Google Scholar
[15]Chiarolla, M. B. and Haussmann, U. G. (2009). On a stochastic irreversible investment problem. SIAM J. Control Optimization 48, 438–462. CrossRefGoogle Scholar
[16]Christensen, S., Salminen, P. and Ta, B. Q. (2013). Optimal stopping of strong Markov processes. Stoch. Process. Appl. 123, 1138–1159. Google Scholar
[17]Csáki, E., Földes, A. and Salminen, P. (1987). On the joint distribution of the maximum and its location for a linear diffusion. Ann. Inst. H. Poincaré Prob. Statist. 23, 179–194. Google Scholar
[18]Deligiannidis, G., Le, H. and Utev, S. (2009). Optimal stopping for processes with independent increments, and applications. J. Appl. Prob. 46, 1130–1145. Google Scholar
[19]Dellacherie, C. and Meyer, P.-A. (1978). Probabilities and Potential (North-Holland Math. Stud. 29). North-Holland, Amsterdam. Google Scholar
[20]Dixit, A. K. and Pindyck, R. S. (1994). Investment Under Uncertainty. ceton University Press. CrossRefGoogle Scholar
[21]El Karoui, N. and Karatzas, I. (1991). A new approach to the Skorohod problem, and its applications. Stoch. Stoch. Reports 34, 57–82. (Correction: 36 (1991), 265.) Google Scholar
[22]Ferrari, G. (2015). On an integral equation for the free-boundary of stochastic, irreversible investment problems. Ann. Appl. Prob. 25, 150–176. Google Scholar
[23]Karatzas, I. (1981). The monotone follower problem in stochastic decision theory. Appl. Math. Optimization 7, 175–189. CrossRefGoogle Scholar
[24]Karatzas, I. and Shreve, S. E. (1984). Connections between optimal stopping and singular stochastic control I. Monotone follower problems. SIAM J. Control Optimization 22, 856–877. Google Scholar
[25]Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin. Google Scholar
[26]Liang, J., Yang, M. and Jiang, L. (2013). A closed-form solution for the exercise strategy in real options model with a jump-diffusion process. SIAM J. Appl. Math. 73, 549–571. CrossRefGoogle Scholar
[27]McDonald, R. and Siegel, D. (1986). The value of waiting to invest. Quart. J. Econom. 101, 707–727. CrossRefGoogle Scholar
[28]Mordecki, E. (2002). Optimal stopping and perpetual options for Lévy processes. Finance Stoch. 6, 473–493. Google Scholar
[29]Mordecki, E. and Salminen, P. (2007). Optimal stopping of Hunt and Lévy processes. Stochastics 79, 233–251. CrossRefGoogle Scholar
[30]Øksendal, A. (2000). Irreversible investment problems. Finance Stoch. 4, 223–250. Google Scholar
[31]Peskir, G. and Shiryaev, A. N. (2000). Sequential testing problems for Poisson processes. Ann. Statist. 28, 837–859. Google Scholar
[32]Peskir, G. and Shiryaev, A. (2006). Optimal Stopping and Free-Boundary Problems. Birkhäuser, Basel. Google Scholar
[33]Pham, H., (2006). Explicit solution to an irreversible investment model with a stochastic production capacity. In From Stochastic Calculus to Mathematical Finance, Springer, Berlin, pp. 547–566. Google Scholar
[34]Pindyck, R. S. (1988). Irreversible investment, capacity choice, and the value of the firm. Amer. Econom. Rev. 78, 969–985. Google Scholar
[35]Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin. Google Scholar
[36]Riedel, F. and Su, X. (2011). On irreversible investment. Finance Stoch. 15, 607–633. CrossRefGoogle Scholar
[37]Salminen, P. (2011). Optimal stopping, Appell polynomials, and Wiener–Hopf factorization. Stochastics 83, 611–622. CrossRefGoogle Scholar
[38]Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions (Camb. Stud. Adv. Math. 68). Cambridge University Press. Google Scholar
[39]Steg, J.-H. (2012). Irreversible investment in oligopoly. Finance Stoch. 16, 207–224. CrossRefGoogle Scholar
[40]Topkis, D. M. (1978). Minimizing a submodular function on a lattice. Operat. Res. 26, 305–321. CrossRefGoogle Scholar