Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-22T17:24:50.301Z Has data issue: false hasContentIssue false
  • English
  • Français

Bounded Variation Control of Itô Diffusions with Exogenously Restricted Intervention Times

Part of: Stochastic processes Miscellaneous topics in calculus of variations and optimal control Stochastic systems and control Markov processes

Published online by Cambridge University Press:  22 February 2016

Jukka Lempa
Jukka Lempa*
Affiliation:
Oslo and Akershus University College
*
∗ Postal address: School of Business, Faculty of Social Sciences, Oslo and Akershus University College, PO Box 4, St. Olavs Plass, 0130 Oslo, Norway. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, bounded variation control of one-dimensional diffusion processes is considered. We assume that the agent is allowed to control the diffusion only at the jump times of an observable, independent Poisson process. The agent's objective is to maximize the expected present value of the cumulative payoff generated by the controlled diffusion over its lifetime. We propose a relatively weak set of assumptions on the underlying diffusion and the instantaneous payoff structure, under which we solve the problem in closed form. Moreover, we illustrate the main results with an explicit example.

Keywords

Bounded variation controlItô diffusionPoisson processresolvent operatorsingular stochastic control

MSC classification

Primary: 93E20: Optimal stochastic control
Secondary: 60J60: Diffusion processes 49N25: Impulsive optimal control problems 60G40: Stopping times; optimal stopping problems; gambling theory
Type
Research Article
Information
Advances in Applied Probability , Volume 46 , Issue 1 , March 2014 , pp. 102 - 120
Copyright
© Applied Probability Trust 

References

Alvarez, L. H. R. (1999). A class of solvable singlular stochastic control problems. Stoch. Reports 67, 83–122.Google Scholar
Alvarez, L. H. R. and Koskela, E. (2007). The forest rotation problem with stochastic harvest and amenity value. Natur. Resource Modeling 20, 477–509.Google Scholar
Alvarez, L. H. R. and Lempa, J. (2008). On the optimal stochastic impulse control of linear diffusions. SIAM J. Control Optimization 47, 703–732.Google Scholar
Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion—Facts and Formulae, 2nd edn. Birkhäuser, Basel.Google Scholar
Dixit, A. K. and Pindyck, R. S. (1994). Investment Under Uncertainty. Princeton University Press.Google Scholar
Dupuis, P. and Wang, H. (2002). Optimal stopping with random intervention times. Adv. Appl. Prob. 34, 141–157.Google Scholar
Elliott, R. J. (1982). Stochastic Calculus and Applications (Appl. Math. (New York) 18). Springer, New York.Google Scholar
Guo, X. and Liu, J. (2005). Stopping at the maximum of geometric Brownian motion when signals are received. J. Appl. Prob. 42, 826–838.Google Scholar
Karatzas, I. (1983). A class of singular stochastic control problems. Adv. Appl. Prob. 15, 225–254.Google Scholar
Kobila, T. Ø. (1993). A class of solvable stochastic investment problems involving singular controls. Stoch. Reports 43, 29–63.Google Scholar
Korn, R. (1999). Some applications of impulse control in mathematical finance. Math. Methods Operat. Res. 50, 493–518.CrossRefGoogle Scholar
Lempa, J. (2012). Optimal stopping with information constraint. Appl. Math. Optimization 66, 147–173.CrossRefGoogle Scholar
Matsumoto, K. (2006). Optimal portfolio of low liquid assets with a log-utility function. Finance Stoch. 10, 121–145.CrossRefGoogle Scholar
Ohnishi, M. and Tsujimura, M. (2006). An impulse control of a geometric Brownian motion with quadratic costs. Europ. J. Operat. Res. 168, 311–321.CrossRefGoogle Scholar
Øksendal, A. (2000). Irreversible investment problems. Finance Stoch. 4, 223–250.Google Scholar
Øksendal, B. (1999). Stochastic control problems where small intervention costs have big effects. Appl. Math. Optimization 40, 355–375.Google Scholar
Øksendal, B. (2000). Stochastic Differential Equations. An Introduction with Applications, 5th edn. Springer, Berlin.Google Scholar
Pham, N. and Tankov, P. (2008). A model of optimal consumption under liqiudity risk with random trading times. Math. Finance 18, 613–627.Google Scholar
Rogers, L. C. G. and Zane, O. (2002). A simple model of liquidity effects. In Advances in Finance and Stochastics, Springer, Berlin, pp. 161–176.Google Scholar
Wang, H. (2001). Some control problems with random intervention times. Adv. Appl. Prob. 33, 404–422.Google Scholar
Weerasinghe, A. (2005). A bounded variation control problem for diffusion processes. SIAM J. Control Optimization 4, 389–417.Google Scholar
Willassen, Y. (1998). The stochastic rotation problem: a generalization of faustmann's formula to stochastic forest growth. J. Econom. Dynam. Control 22, 573–596.Google Scholar