Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-28T15:59:35.287Z Has data issue: false hasContentIssue false
  • English
  • Français

Ergodic control of diffusions with random intervention times

Part of: Miscellaneous topics in calculus of variations and optimal control Hamilton-Jacobi theories, including dynamic programming Markov processes

Published online by Cambridge University Press:  25 February 2021

Harto Saarinen  and
Jukka Lempa
Harto Saarinen*
Affiliation:
University of Turku
Jukka Lempa*
Affiliation:
University of Turku
*
*Postal address: Department of Mathematics and Statistics, FI - 20014 Turun Yliopisto, Finland.
*Postal address: Department of Mathematics and Statistics, FI - 20014 Turun Yliopisto, Finland.
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

Bounded variation controlergodic controldiffusion processresolvent operatorPoisson processsingular stochastic control

MSC classification

Primary: 49N25: Impulsive optimal control problems
Secondary: 49L20: Dynamic programming method 60J60: Diffusion processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 58 , Issue 1 , March 2021 , pp. 1 - 21
Check for updates
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. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alvarez, L. H. R. (1999). A class of solvable singular stochastic control problems. Stoch. Stoch. Reports 67, 83–122.CrossRefGoogle Scholar
Alvarez, L. H. R. (2004). A class of solvable impulse control problems. Appl. Math. Optimization 49, 265–295.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, 703–732.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, 71–97.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, 173–212.CrossRefGoogle Scholar
Dupuis, P. and Wang, H. (2002). Optimal stopping with random intervention times. Adv. Appl. Prob. 34, 141–157.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, 325–359.CrossRefGoogle Scholar
Fleming, W. H. (1968). Optimal control of partially observable diffusions. SIAM J. Control Optimization 6, 194–214.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, 407–446.CrossRefGoogle Scholar
Fleming, W. H. and McEneaey, W. (1995). Risk sensitive control on an infinite time horizon. SIAM J. Control Optimization 33, 1881–1915.CrossRefGoogle Scholar
Guo, X. and Zhang, Q. (2004). Closed-form solutions for perpetual American put options with regime switching. SIAM J. Appl. Math. 64, 2034–2049.Google Scholar
Harrison, J. M., Sellke, T. M. and Taylor, A. J. (1983). Impulse control of Brownian motion. Math. Operat. Res. 8, 454–466.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, 331–355.CrossRefGoogle Scholar
Karatzas, I. (1983). A class of singular stochastic control problems. Adv. Appl. Prob. 15, 225–254.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, 147–173.CrossRefGoogle Scholar
Lempa, J. (2012). Optimal stopping with random exercise lag. Math. Operat. Res. 75, 273–286.CrossRefGoogle Scholar
Lempa, J. (2014). Bounded variation control of Itô diffusion with exogenously restricted intervention times. Adv. Appl. Prob. 46, 102–120.CrossRefGoogle Scholar
Lempa, J. (2017). A class of solvable multiple entry problems with forced exits. Appl. Math. Optimization 76, 1–27.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, 239–271.CrossRefGoogle Scholar
Menaldi, J. L. and Robin, M. (2016). On some optimal stopping problems with constraint. SIAM J. Control Optimization 54, 2650–2671.CrossRefGoogle Scholar
Menaldi, J. L. and Robin, M. (2017). On some impulse control problems with constraint. SIAM J. Control Optimization 55, 3204–3225.CrossRefGoogle Scholar
Menaldi, J. L. and Robin, M. (2018). On some ergodic impulse control problems with constraint. SIAM J. Control Optimization 56, 2690–2711.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, 1098–1125.CrossRefGoogle Scholar
Robin, M. (1983). Long-term average cost control problems for continuous time Markov processes: A survey. Acta Appl. Math. 1, 281–299.CrossRefGoogle Scholar
Rogers, L. C. G. (2001). The relaxed investor and parameter uncertainty. Finance Stoch. 5, 131–154.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. 161–176.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, 404–422.CrossRefGoogle Scholar