Skip to main content
SpringerLink
Log in

An Optimal Double Stopping Rule for a Buying-Selling Problem

  • Published:
Methodology and Computing in Applied Probability Aims and scope Submit manuscript

Abstract

We consider a buying-selling problem with a finite time horizon when two stops of a sequence of dependent observations can be made. The aim is to find an optimal sequential procedure which maximizes the total expected revenue. In this paper, we obtain an optimal double stopping rule and apply it for a geometric random walk and an autoregressive sequence.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Abramowitz M, Stegun IA (eds) (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover Publications, New York

  • Albright SC (1974) Optimal sequential assignments with random arrival times. Manag Sci 21(1):60–67

    Article  MathSciNet  Google Scholar 

  • Albright SC (1977) A bayesian approach to a generalized housing selling problem. Manag Sci 24(4):432–440

    Article  Google Scholar 

  • Blocker AW (2014) FastGHQuad: Fast Rcpp implementation of Gauss-Hermite quadrature. https://CRAN.R-project.org/package=fastGHQuad. R package version 0.2

  • Bruss FT, Ferguson TS (1997) Multiple buying or selling with vector offers. J Appl Probab 34(4):959–973

    Article  MathSciNet  Google Scholar 

  • Chow YS, Robbins H, Siegmund D (1971) Great expectations: the theory of optimal stopping. Houghton Mifflin, Boston

    MATH  Google Scholar 

  • Chun YH, Plante RD, Schneider H (2002) Buying and selling an asset over the finite time horizon: a non-parametric approach. Eur J Oper Res 136(1):106–120

    Article  MathSciNet  Google Scholar 

  • David I (1998) Explicit results for a class of asset-selling problems. Eur J Oper Res 110(3):576–584

    Article  Google Scholar 

  • David I, Levi O (2001) Asset-selling problems with holding costs. Int J Prod Econ 71:317–321

    Article  Google Scholar 

  • Dombrovsky Y, Perrin N (1994) On adaptive search and optimal stopping mate choice. Am Nat 144(2):355–361

    Article  Google Scholar 

  • Ferguson TS (1989) Who solved the secretary problem? Stat Sci 4(3):282–296

    Article  MathSciNet  Google Scholar 

  • Haggstrom GW (1967) Optimal sequential procedures when more than one stop is required. Ann Math Stat 38(6):1618–1626

    Article  MathSciNet  Google Scholar 

  • Hutchinson J, Halupka K (2004) Mate choice when males are in patches: optimal strategies and good rules of thumb. J Theor Biol 231(1):129–151

    Article  MathSciNet  Google Scholar 

  • Karpowicz A, Szajowski K (2007) Double optimal stopping of a risk process. Stochastics 79(1-2):155–167

    Article  MathSciNet  Google Scholar 

  • Kösters H (2004) A note on multiple stopping rules. Optimization 53(1):69–75

    Article  MathSciNet  Google Scholar 

  • Krasnosielska-Kobos A (2016) Construction of Nash equilibrium based on multiple stopping problem in multi-person game. Math Meth Oper Res 83(1):53–70

    Article  MathSciNet  Google Scholar 

  • Lippman SA, McCall JJ (1976) The economics of job search: a survey. Econ Inq 14(2):155–189

    Article  Google Scholar 

  • MacQueen J, Miller RG (1960) Optimal persistence policies. Oper Res 8(3):362–380

    Article  MathSciNet  Google Scholar 

  • Moser L (1956) On a problem of Cayley. Scr Mater 22:289–292

    MATH  Google Scholar 

  • Nagaraja CH, Brown LD, Zhao LH (2011) An autoregressive approach to house price modeling. Ann Appl Stat 5(1):124–149

    Article  MathSciNet  Google Scholar 

  • Nikolaev ML (1979) Generalized sequential procedures. Lith Math J 19(3):318–325

    Article  MathSciNet  Google Scholar 

  • Nikolaev ML (1981) Test for optimality of a generalized sequence of procedures. Lith Math J 21(3):253–258

    Article  MathSciNet  Google Scholar 

  • Nikolaev ML (1999) On optimal multiple stopping of Markov sequences. Theory of Probability and Its Applications 43(2):298–306

    Article  MathSciNet  Google Scholar 

  • Nikolaev ML, Sofronov GY (2007) A multiple optimal stopping rule for sums of independent random variables. Discret Math Appl 17(5):463–473

    Article  Google Scholar 

  • Papastavrou JD, Rajagopalan S, Kleywegt AJ (1996) The dynamic and stochastic knapsack problem with deadlines. Manag Sci 42(12):1706–1718

    Article  Google Scholar 

  • R Core Team (2018) R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org/

  • Real L (1990) Search theory and mate choice. I. Models of single sex-discrimination. Am Nat 136(3):376–405

    Article  Google Scholar 

  • Real L (1991) Search theory and mate choice. II. Mutual interaction, assortative mating, and equilibrium variation in male and female fitness. Am Nat 138(4):901–917

    Article  Google Scholar 

  • Righter R (1987) The stochastic sequential assignment problem with random deadlines. Probab Eng Inf Sci 1:189–202

    Article  Google Scholar 

  • Rosenfield DB, Shapiro RD, Butler DA (1983) Optimal strategies for selling an asset. Manag Sci 29(9):1051–1061

    Article  Google Scholar 

  • Shiryaev AN (1999) Essentials of stochastic finance: facts, models, theory, vol 3 World scientific

  • Sofronov G (2013) An optimal sequential procedure for a multiple selling problem with independent observations. Eur J Oper Res 225(2):332–336

    Article  MathSciNet  Google Scholar 

  • Sofronov G, Keith JM, Kroese DP (2006) An optimal sequential procedure for a buying-selling problem with independent observations. J Appl Probab 43(2):454–462

    Article  MathSciNet  Google Scholar 

  • Sofronov GY (2016) A multiple optimal stopping rule for a buying–selling problem with a deterministic trend. Stat Pap 57(4):1107–1119

    Article  MathSciNet  Google Scholar 

  • Stadje W (1985) On multiple stopping rules. Optimization 16(3):401–418

    Article  MathSciNet  Google Scholar 

  • Stein WE, Seale DA, Rapoport A (2003) Analysis of heuristic solutions to the best choice problem. Eur J Oper Res 151(1):140–152

    Article  MathSciNet  Google Scholar 

  • Targino RS, Peters GW, Sofronov G, Shevchenko PV (2017) Optimal exercise strategies for operational risk insurance via multiple stopping times. Methodol Comput Appl Probab 19(2):487–518

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author is grateful to the referees for their helpful remarks on the manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Macquarie University, Sydney, NSW, 2109, Australia

    Georgy Yu. Sofronov

Authors
  1. Georgy Yu. Sofronov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Georgy Yu. Sofronov.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sofronov, G.Y. An Optimal Double Stopping Rule for a Buying-Selling Problem. Methodol Comput Appl Probab 22, 1–12 (2020). https://doi.org/10.1007/s11009-018-9684-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11009-018-9684-6

Keywords

Mathematics Subject Classification (2010)

Search

Navigation