Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-24T09:25:43.026Z Has data issue: false hasContentIssue false
  • English
  • Français

Discrete-time risk-aware optimal switching with non-adapted costs

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

Published online by Cambridge University Press:  06 June 2022

Randall Martyr Open the ORCID record for Randall Martyr [Opens in a new window] ,
John Moriarty Open the ORCID record for John Moriarty [Opens in a new window]  and
Magnus Perninge
Randall Martyr*
Affiliation:
Kingston University London
John Moriarty*
Affiliation:
Queen Mary University of London
Magnus Perninge*
Affiliation:
Linnaeus University
*
*Postal address: River House, 5357 High Street, Surrey, UK. Email address: r.martyr@kingston.ac.uk
**Postal address: School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London, UK. Email address: jmoriarty@qmul.ac.uk
***Postal address: PG Vejdes väg 7, 352 52 Växjö, Sweden.
Get access Rights & Permissions [Opens in a new window]

Abstract

We solve non-Markovian optimal switching problems in discrete time on an infinite horizon, when the decision-maker is risk-aware and the filtration is general, and establish existence and uniqueness of solutions for the associated reflected backward stochastic difference equations. An example application to hydropower planning is provided.

Keywords

Infinite horizonoptimal switchingrisk measuresreflected backward stochastic difference equationshydropower planning

MSC classification

Primary: 93E20: Optimal stochastic control
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 91B08: Individual preferences 49N30: Problems with incomplete information
Type
Original Article
Information
Advances in Applied Probability , Volume 54 , Issue 2 , June 2022 , pp. 625 - 655
Check for updates
Copyright
© The Author(s), 2022. 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

An, L., Cohen, S. N. and Ji, S. (2013). Reflected backward stochastic difference equations and optimal stopping problems under g-expectation. Preprint. Available at https://arxiv.org/abs/1305.0887.Google Scholar
Bäuerle, N. and JaŚkiewicz, A. (2018). Stochastic optimal growth model with risk sensitive preferences. J. Econom. Theory 173, 181–200.CrossRefGoogle Scholar
Carmona, R. and Ludkovski, M. (2010). Valuation of energy storage: an optimal switching approach. Quant. Finance 10, 359–374.CrossRefGoogle Scholar
Cheridito, P., Delbaen, F. and Kupper, M. (2006). Dynamic monetary risk measures for bounded discrete-time processes. Electron. J. Prob. 11, 57–106.CrossRefGoogle Scholar
Cheridito, P. and Kupper, M. (2011). Composition of time-consistent dynamic monetary risk measures in discrete time. Internat. J. Theoret. Appl. Finance 14, 137–162.CrossRefGoogle Scholar
Cohen, S. N. and Elliott, R. J. (2011). Backward stochastic difference equations and nearly time-consistent nonlinear expectations. SIAM J. Control Optimization 49, 125–139.CrossRefGoogle Scholar
Detlefsen, K. and Scandolo, G. (2005). Conditional and dynamic convex risk measures. Finance Stoch. 9, 539–561.CrossRefGoogle Scholar
Follmer, H. and Schied, A. (2016). Stochastic Finance: an Introduction in Discrete Time, 4th edn. De Gruyter, Berlin.CrossRefGoogle Scholar
Frittelli, M. and Rosazza Gianin, E. (2002). Putting order in risk measures. J. Banking Finance 26, 1473–1486.CrossRefGoogle Scholar
Kose, U. and Ruszczynski, A. (2020). Risk-averse learning by temporal difference methods. Preprint. Available at https://arxiv.org/abs/2003.00780v1.Google Scholar
Krätschmer, V. and Schoenmakers, J. (2010). Representations for optimal stopping under dynamic monetary utility functionals. SIAM J. Financial Math. 1, 811–832.CrossRefGoogle Scholar
Lundström, N. L. P., Olofsson, M. and Önskog, T. (2020). Management strategies for run-of-river hydropower plants—an optimal switching approach. Preprint. Available at https://arxiv.org/abs/2009.10554.Google Scholar
Pflug, G. C. and Pichler, A. (2014). Multistage Stochastic Optimization. Springer, Cham.CrossRefGoogle Scholar
Pichler, A. and Schlotter, R. (2020). Martingale characterizations of risk-averse stochastic optimization problems. Math. Program. 181, 377–403.CrossRefGoogle Scholar
Pichler, A. and Shapiro, A. (2019). Risk averse stochastic programming: time consistency and optimal stopping. Preprint. Available at https://arxiv.org/abs/1808.10807.Google Scholar
Rieder, U. (1976). On optimal policies and martingales in dynamic programming. J. Appl. Prob. 13, 507–518.CrossRefGoogle Scholar
Ruszczyński, A. (2010). Risk-averse dynamic programming for Markov decision processes. Math. Program. 125, 235–261.CrossRefGoogle Scholar
Ruszczyński, A. and Shapiro, A. (2006). Conditional risk mappings. Math. Operat. Res. 31, 544–561.CrossRefGoogle Scholar
Shapiro, A., Dentcheva, D. and Ruszczynski, A. (2014). Lectures on Stochastic Programming, 2nd edn. Society for Industrial and Applied Mathematics/Mathematical Optimization Society, Philadelphia.Google Scholar
Shen, Y., Stannat, W. and Obermayer, K. (2013). Risk-sensitive Markov control processes. SIAM J. Control Optimization 51, 3652–3672.CrossRefGoogle Scholar
Ugurlu, K. (2018). Robust optimal control using conditional risk mappings in infinite horizon. J. Comput. Appl. Math. 344, 275–287.CrossRefGoogle Scholar