Skip to main content Accessibility help
×
Home
Hostname: page-component-5cfd469876-tkzrn Total loading time: 0.202 Render date: 2021-06-24T10:45:36.077Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true }

Ergodic control of diffusions with random intervention times

Published online by Cambridge University Press:  25 February 2021

Harto Saarinen
Affiliation:
University of Turku
Jukka Lempa
Affiliation:
University of Turku
Corresponding
E-mail address:

Abstract

We study an ergodic singular control problem with constraint of a regular one-dimensional linear diffusion. The constraint allows the agent to control the diffusion only at the jump times of an independent Poisson process. Under relatively weak assumptions, we characterize the optimal solution as an impulse-type control policy, where it is optimal to exert the exact amount of control needed to push the process to a unique threshold. Moreover, we discuss the connection of the present problem to ergodic singular control problems, and illustrate the results with different well-known cost and diffusion structures.

Type
Research Papers
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below.

References

Alvarez, L. H. R. (1999). A class of solvable singular stochastic control problems. Stoch. Stoch. Reports 67, 83122.CrossRefGoogle Scholar
Alvarez, L. H. R. (2004). A class of solvable impulse control problems. Appl. Math. Optimization 49, 265295.CrossRefGoogle Scholar
Alvarez, L. H. R. (2018). A class of solvable stationary singular stochastic control problems. Preprint, arXiv:1803.03464.Google Scholar
Alvarez, L. H. R. and Hening, A. (2018). Optimal sustainable harvesting of populations in random environments. Preprint, arXiv:1807.02464.Google Scholar
Alvarez, L. H. R. and Lempa, J. (2008). On the optimal stochastic impulse control of linear diffusions. SIAM J. Control Optimization 47, 703732.CrossRefGoogle Scholar
Alvarez, L. H. R. and Stenbacka, R. (2001). Adoption of uncertain multi-stage technology projects: A real options approach. J. Math. Econ. 35, 7197.CrossRefGoogle Scholar
Arapostathis, A., Borkar, V. S. and Ghosh, M. K. (2012). Ergodic Control of Diffusion Processes. Cambridge University Press.Google Scholar
Bain, A. and Crisan, D. (2009). Fundamentals of Stochastic Filtering. Springer, New York.CrossRefGoogle Scholar
Bass, R. (1998). Diffusions and Elliptic Operators. Springer, New York.Google Scholar
Borodin, A. N. and Salminen, P. (2015). Handbook of Brownian Motion – Facts and Formulae, 2nd edn. Birkhäuser, Basel.Google Scholar
Dayanik, S. and Karatzas, I. (2003). On the optimal stopping problem for one-dimensional diffusions. Stoch. Process. Appl. 107, 173212.CrossRefGoogle Scholar
Dupuis, P. and Wang, H. (2002). Optimal stopping with random intervention times. Adv. Appl. Prob. 34, 141157.CrossRefGoogle Scholar
Evans, S. N., Hening, A. and Schreiber, S. J. (2015). Protected polymorphisms and evolutionary stability of patch-selection strategies in stochastic environments. J. Math. Biol. 71, 325359.CrossRefGoogle Scholar
Fleming, W. H. (1968). Optimal control of partially observable diffusions. SIAM J. Control Optimization 6, 194214.CrossRefGoogle Scholar
Fleming, W. H. (1999). Controlled Markov processes and mathematical finance. In Nonlinear Analysis, Differential Equations and Control, eds F. H. Clarke, R. J. Stern and G. Sabidussi. Kluwer, Dordrecht, 407446.CrossRefGoogle Scholar
Fleming, W. H. and McEneaey, W. (1995). Risk sensitive control on an infinite time horizon. SIAM J. Control Optimization 33, 18811915.CrossRefGoogle Scholar
Guo, X. and Zhang, Q. (2004). Closed-form solutions for perpetual American put options with regime switching. SIAM J. Appl. Math. 64, 20342049.Google Scholar
Harrison, J. M., Sellke, T. M. and Taylor, A. J. (1983). Impulse control of Brownian motion. Math. Operat. Res. 8, 454466.CrossRefGoogle Scholar
Jack, A. and Zervos, M. (2006). A singular control problem with an expected and pathwise ergodic performance criterion. J. Appl. Math. Stoch. Anal. 2006, 82538.CrossRefGoogle Scholar
Jiang, Z. and Pistorius, M. R. (2008). On perpetual American put valuation and first-passage in a regime-switching model with jumps. Finance Stoch. 12, 331355.CrossRefGoogle Scholar
Karatzas, I. (1983). A class of singular stochastic control problems. Adv. Appl. Prob. 15, 225254.CrossRefGoogle Scholar
Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus. Springer, New York.Google Scholar
Kushner, H. J. and Dupuis, P. G. (1992). Numerical Methods for Stochastic Control Problems in Continuous Time. Springer, New York.CrossRefGoogle Scholar
Lempa, J. (2012). Optimal stopping with information constraint. Appl. Math. Optimization 66, 147173.CrossRefGoogle Scholar
Lempa, J. (2012). Optimal stopping with random exercise lag. Math. Operat. Res. 75, 273286.CrossRefGoogle Scholar
Lempa, J. (2014). Bounded variation control of Itô diffusion with exogenously restricted intervention times. Adv. Appl. Prob. 46, 102120.CrossRefGoogle Scholar
Lempa, J. (2017). A class of solvable multiple entry problems with forced exits. Appl. Math. Optimization 76, 127.Google Scholar
Lempa, J. and Saarinen, H. (2019). On asymptotic relations between singular and constrained control problems of one-dimensional diffusions. Preprint, arXiv:1909.06115.Google Scholar
Matomäki, P. (2012). On solvability of a two-sided singular control problem. Math. Operat. Res. 76, 239271.CrossRefGoogle Scholar
Menaldi, J. L. and Robin, M. (2016). On some optimal stopping problems with constraint. SIAM J. Control Optimization 54, 26502671.CrossRefGoogle Scholar
Menaldi, J. L. and Robin, M. (2017). On some impulse control problems with constraint. SIAM J. Control Optimization 55, 32043225.CrossRefGoogle Scholar
Menaldi, J. L. and Robin, M. (2018). On some ergodic impulse control problems with constraint. SIAM J. Control Optimization 56, 26902711.CrossRefGoogle Scholar
Picard, J. (1986). Nonlinear filtering of one-dimensional diffusions in the case of a high signal-to-noise ratio. SIAM J. Appl. Math. 46, 10981125.CrossRefGoogle Scholar
Robin, M. (1983). Long-term average cost control problems for continuous time Markov processes: A survey. Acta Appl. Math. 1, 281299.CrossRefGoogle Scholar
Rogers, L. C. G. (2001). The relaxed investor and parameter uncertainty. Finance Stoch. 5, 131154.CrossRefGoogle Scholar
Rogers, L. C. G. and Zane, O. (2002). A simple model of liquidity effects. In Advances in Finance and Stochastics. Springer, Berlin, pp. 161176.CrossRefGoogle Scholar
Sethi, S. P., Zhang, H. and Zhang, Q. (2005). Average-Cost Control of Stochastic Manufacturing Systems. Springer, New York.Google Scholar
Revuz, D and Yor, M (1991). Continuous Martingales and Brownian Motion. Springer, Berlin.CrossRefGoogle Scholar
Wang, H. (2001). Some control problems with random intervention times. Adv. Appl. Prob. 33, 404422.CrossRefGoogle Scholar

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Ergodic control of diffusions with random intervention times
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Ergodic control of diffusions with random intervention times
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Ergodic control of diffusions with random intervention times
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *