Advances in Applied Probability

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

Giorgio Ferrari and Paavo Salminen

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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.

Article information

Adv. in Appl. Probab., Volume 48, Number 1 (2016), 298-314.

First available in Project Euclid: 8 March 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 93E20: Optimal stochastic control
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60G51: Processes with independent increments; Lévy processes 91B70: Stochastic models

Free-boundary irreversible investment singular stochastic control optimal stopping Lévy process Bank and El Karoui's representation theorem base capacity


Ferrari, Giorgio; Salminen, Paavo. Irreversible investment under Lévy uncertainty: an equation for the optimal boundary. Adv. in Appl. Probab. 48 (2016), no. 1, 298--314.

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  • Abel, A. B. and Eberly, J. C. (1996). Optimal investment with costly reversibility. Rev. Econom. Stud. 63, 581–593.
  • 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.
  • Baldursson, F. M. and Karatzas, I. (1996). Irreversible investment and industry equilibrium. Finance Stoch. 1, 69–89.
  • Bank, P. (2005). Optimal control under a dynamic fuel constraint. SIAM J. Control Optimization 44, 1529–1541.
  • Bank, P. and El Karoui, N. (2004). A stochastic representation theorem with applications to optimization and obstacle problems. Ann. Prob. 32, 1030–1067.
  • 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..
  • Bank, P. and Riedel, F. (2001). Optimal consumption choice with intertemporal substitution. Ann. Appl. Prob. 11, 750–788.
  • Bentolila, S. and Bertola, G. (1990). Firing costs and labour demand: how bad is Eurosclerosis? Rev. Econom. Stud. 57, 381–402.
  • Bertoin, J. (1996). Lévy Processes. Cambridge University Press.
  • Bertola, G. (1998). Irreversible investment. Res. Econom. 52, 3–37.
  • Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion–-Facts and Formulae, 2nd edn. Birkhäuser, Basel.
  • Boyarchenko, S. (2004). Irreversible decisions and record-setting news principles. Amer. Econom. Rev. 94, 557–568.
  • Boyarchenko, S. I. and Levendorskiĭ, S. Z. (2002). Perpetual American options under Lévy processes. SIAM J. Control Optimization 40, 1663-1696.
  • 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.
  • Chiarolla, M. B. and Haussmann, U. G. (2009). On a stochastic irreversible investment problem. SIAM J. Control Optimization 48, 438–462.
  • Christensen, S., Salminen, P. and Ta, B. Q. (2013). Optimal stopping of strong Markov processes. Stoch. Process. Appl. 123, 1138–1159.
  • 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.
  • Deligiannidis, G., Le, H. and Utev, S. (2009). Optimal stopping for processes with independent increments, and applications. J. Appl. Prob. 46, 1130–1145.
  • Dellacherie, C. and Meyer, P.-A. (1978). Probabilities and Potential (North-Holland Math. Stud. 29), North-Holland, Amsterdam.
  • Dixit, A. K. and Pindyck, R. S. (1994). Investment Under Uncertainty. Princeton University Press.
  • 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.)
  • Ferrari, G. (2015). On an integral equation for the free-boundary of stochastic, irreversible investment problems. Ann. Appl. Prob. 25, 150–176.
  • Karatzas, I. (1981). The monotone follower problem in stochastic decision theory. Appl. Math. Optimization 7, 175–189.
  • 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.
  • Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.
  • 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
  • McDonald, R. and Siegel, D. (1986). The value of waiting to invest. Quart. J. Econom. 101, 707–727.
  • Mordecki, E. (2002). Optimal stopping and perpetual options for Lévy processes. Finance Stoch. 6, 473–493.
  • Mordecki, E. and Salminen, P. (2007). Optimal stopping of Hunt and Lévy processes. Stochastics 79, 233–251.
  • Øksendal, A. (2000). Irreversible investment problems. Finance Stoch. 4, 223–250.
  • Peskir, G. and Shiryaev, A. N. (2000). Sequential testing problems for Poisson processes. Ann. Statist. 28, 837–859.
  • Peskir, G. and Shiryaev, A. (2006). Optimal stopping and free-boundary problems. Birkhäuser, Basel.
  • 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.
  • Pindyck, R. S. (1988). Irreversible investment, capacity choice, and the value of the firm. Amer. Econom. Rev. 78, 969–985.
  • Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin.
  • Riedel, F. and Su, X. (2011). On irreversible investment. Finance Stoch. 15, 607–633.
  • Salminen, P. (2011). Optimal stopping, Appell polynomials, and Wiener–Hopf factorization. Stochastics 83, 611–622.
  • Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions (Camb. Stud. Adv. Math. 68). Cambridge University Press.
  • Steg, J.-H. (2012). Irreversible investment in oligopoly. Finance Stoch. 16, 207–224.
  • Topkis, D. M. (1978). Minimizing a submodular function on a lattice. Operat. Res. 26, 305–321.