IDEAS home Printed from https://ideas.repec.org/a/eee/ejores/v282y2020i2p602-613.html
   My bibliography  Save this article

On double-boundary non-crossing probability for a class of compound processes with applications

Author

Listed:
  • Dimitrova, Dimitrina S.
  • Ignatov, Zvetan G.
  • Kaishev, Vladimir K.
  • Tan, Senren

Abstract

We develop an efficient method for computing the probability that a non-decreasing, pure jump (compound) stochastic process stays between arbitrary upper and lower boundaries (i.e., deterministic functions, possibly discontinuous) within a finite time period. The compound process is composed of a process modelling the arrivals of certain events (e.g., demands for a product in inventory systems, customers in queuing, or claims/capital gains in insurance/dual risk models), and a sequence of independent and identically distributed random variables modelling the sizes of the events. The events arrival process is assumed to belong to the wide class of point processes with conditional stationary independent increments which includes (non-)homogeneous Poisson, binomial, negative binomial, mixed Poisson and doubly stochastic Poisson (i.e., Cox) processes as special cases. The proposed method is based on expressing the non-exit probability through Chapman–Kolmogorov equations, re-expressing them in terms of a circular convolution of two vectors which is then computed applying fast Fourier transform (FFT). We further demonstrate that our FFT-based method is computationally efficient and can be successfully applied in the context of inventory management (to determine an optimal replenishment policy), ruin theory (to evaluate ruin probabilities and related quantities) and double-barrier option pricing or simply computing non-exit probabilities for Brownian motion with general boundaries.

Suggested Citation

  • Dimitrova, Dimitrina S. & Ignatov, Zvetan G. & Kaishev, Vladimir K. & Tan, Senren, 2020. "On double-boundary non-crossing probability for a class of compound processes with applications," European Journal of Operational Research, Elsevier, vol. 282(2), pages 602-613.
  • Handle: RePEc:eee:ejores:v:282:y:2020:i:2:p:602-613
    DOI: 10.1016/j.ejor.2019.09.058
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S037722171930815X
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.ejor.2019.09.058?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Mirko Kremer & Brent Moritz & Enno Siemsen, 2011. "Demand Forecasting Behavior: System Neglect and Change Detection," Management Science, INFORMS, vol. 57(10), pages 1827-1843, October.
    2. Dimitrina S. Dimitrova & Zvetan G. Ignatov & Vladimir K. Kaishev, 2017. "On the First Crossing of Two Boundaries by an Order Statistics Risk Process," Risks, MDPI, vol. 5(3), pages 1-14, August.
    3. Goffard, Pierre-Olivier & Lefèvre, Claude, 2018. "Duality in ruin problems for ordered risk models," Insurance: Mathematics and Economics, Elsevier, vol. 78(C), pages 44-52.
    4. Jing-Sheng Song & Paul H. Zipkin, 1996. "The Joint Effect of Leadtime Variance and Lot Size in a Parallel Processing Environment," Management Science, INFORMS, vol. 42(9), pages 1352-1363, September.
    5. Fusai, Gianluca & Germano, Guido & Marazzina, Daniele, 2016. "Spitzer identity, Wiener-Hopf factorization and pricing of discretely monitored exotic options," European Journal of Operational Research, Elsevier, vol. 251(1), pages 124-134.
    6. de Kok, Ton & Grob, Christopher & Laumanns, Marco & Minner, Stefan & Rambau, Jörg & Schade, Konrad, 2018. "A typology and literature review on stochastic multi-echelon inventory models," European Journal of Operational Research, Elsevier, vol. 269(3), pages 955-983.
    7. Wolfgang Bischoff & Enkelejd Hashorva & Jürg Hüsler & Frank Miller, 2003. "Exact asymptotics for Boundary crossings of the brownian bridge with trend with application to the Kolmogorov test," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 55(4), pages 849-864, December.
    8. Lengu, D. & Syntetos, A.A. & Babai, M.Z., 2014. "Spare parts management: Linking distributional assumptions to demand classification," European Journal of Operational Research, Elsevier, vol. 235(3), pages 624-635.
    9. Hasan Arslan & Stephen C. Graves & Thomas A. Roemer, 2007. "A Single-Product Inventory Model for Multiple Demand Classes," Management Science, INFORMS, vol. 53(9), pages 1486-1500, September.
    10. Moscovich, Amit & Nadler, Boaz, 2017. "Fast calculation of boundary crossing probabilities for Poisson processes," Statistics & Probability Letters, Elsevier, vol. 123(C), pages 177-182.
    11. Willemain, Thomas R. & Smart, Charles N. & Schwarz, Henry F., 2004. "A new approach to forecasting intermittent demand for service parts inventories," International Journal of Forecasting, Elsevier, vol. 20(3), pages 375-387.
    12. Gordon K. C. Chen & Peter R. Winters, 1966. "Forecasting Peak Demand for an Electric Utility with a Hybrid Exponential Model," Management Science, INFORMS, vol. 12(12), pages 531-537, August.
    13. Dassios, Angelos & Jang, Jiwook, 2003. "Pricing of catastrophe reinsurance and derivatives using the Cox process with shot noise intensity," LSE Research Online Documents on Economics 2849, London School of Economics and Political Science, LSE Library.
    14. Berling, Peter & Marklund, Johan, 2013. "A model for heuristic coordination of real life distribution inventory systems with lumpy demand," European Journal of Operational Research, Elsevier, vol. 230(3), pages 515-526.
    15. Dassios, Angelos & Jang, Jiwook & Zhao, Hongbiao, 2015. "A risk model with renewal shot-noise Cox process," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 55-65.
    16. Johansson, Lina & Sonntag, Danja R. & Marklund, Johan & Kiesmüller, Gudrun P., 2020. "Controlling distribution inventory systems with shipment consolidation and compound Poisson demand," European Journal of Operational Research, Elsevier, vol. 280(1), pages 90-101.
    17. Gutierrez, Rafael S. & Solis, Adriano O. & Mukhopadhyay, Somnath, 2008. "Lumpy demand forecasting using neural networks," International Journal of Production Economics, Elsevier, vol. 111(2), pages 409-420, February.
    18. Kramer, Walter & Ploberger, Werner & Alt, Raimund, 1988. "Testing for Structural Change in Dynamic Models," Econometrica, Econometric Society, vol. 56(6), pages 1355-1369, November.
    19. Dimitrina S. Dimitrova & Vladimir K. Kaishev & Shouqi Zhao, 2015. "Modeling Finite‐Time Failure Probabilities in Risk Analysis Applications," Risk Analysis, John Wiley & Sons, vol. 35(10), pages 1919-1939, October.
    20. Dassios, Angelos & Jang, Jiwook & Zhao, Hongbiao, 2015. "A risk model with renewal shot-noise Cox process," LSE Research Online Documents on Economics 64051, London School of Economics and Political Science, LSE Library.
    21. Lefèvre, Claude & Picard, Philippe, 2011. "A new look at the homogeneous risk model," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 512-519.
    22. Bijvank, Marco & Johansen, Søren Glud, 2012. "Periodic review lost-sales inventory models with compound Poisson demand and constant lead times of any length," European Journal of Operational Research, Elsevier, vol. 220(1), pages 106-114.
    23. Willemain, Thomas R. & Smart, Charles N. & Shockor, Joseph H. & DeSautels, Philip A., 1994. "Forecasting intermittent demand in manufacturing: a comparative evaluation of Croston's method," International Journal of Forecasting, Elsevier, vol. 10(4), pages 529-538, December.
    24. Teunter, Ruud H. & Syntetos, Aris A. & Zied Babai, M., 2011. "Intermittent demand: Linking forecasting to inventory obsolescence," European Journal of Operational Research, Elsevier, vol. 214(3), pages 606-615, November.
    25. R H Teunter & L Duncan, 2009. "Forecasting intermittent demand: a comparative study," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 60(3), pages 321-329, March.
    26. Yu-Sheng Zheng, 1992. "On Properties of Stochastic Inventory Systems," Management Science, INFORMS, vol. 38(1), pages 87-103, January.
    27. Olof Stenius & Ayşe Gönül Karaarslan & Johan Marklund & A. G. de Kok, 2016. "Exact Analysis of Divergent Inventory Systems with Time-Based Shipment Consolidation and Compound Poisson Demand," Operations Research, INFORMS, vol. 64(4), pages 906-921, August.
    28. N. K. Kwak & Walter A. Garrett, Jr. & Sam Barone, 1977. "A Stochastic Model of Demand Forecasting for Technical Manpower Planning," Management Science, INFORMS, vol. 23(10), pages 1089-1098, June.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Pinçe, Çerağ & Turrini, Laura & Meissner, Joern, 2021. "Intermittent demand forecasting for spare parts: A Critical review," Omega, Elsevier, vol. 105(C).
    2. Zhu, Sha & Dekker, Rommert & van Jaarsveld, Willem & Renjie, Rex Wang & Koning, Alex J., 2017. "An improved method for forecasting spare parts demand using extreme value theory," European Journal of Operational Research, Elsevier, vol. 261(1), pages 169-181.
    3. Jože Martin Rožanec & Blaž Fortuna & Dunja Mladenić, 2022. "Reframing Demand Forecasting: A Two-Fold Approach for Lumpy and Intermittent Demand," Sustainability, MDPI, vol. 14(15), pages 1-21, July.
    4. Prak, Dennis & Rogetzer, Patricia, 2022. "Timing intermittent demand with time-varying order-up-to levels," European Journal of Operational Research, Elsevier, vol. 303(3), pages 1126-1136.
    5. Syntetos, Aris A. & Zied Babai, M. & Gardner, Everette S., 2015. "Forecasting intermittent inventory demands: simple parametric methods vs. bootstrapping," Journal of Business Research, Elsevier, vol. 68(8), pages 1746-1752.
    6. Li, Chongshou & Lim, Andrew, 2018. "A greedy aggregation–decomposition method for intermittent demand forecasting in fashion retailing," European Journal of Operational Research, Elsevier, vol. 269(3), pages 860-869.
    7. Prak, Derk & Teunter, Rudolf & Babai, M. Z. & Syntetos, A. A. & Boylan, D, 2018. "Forecasting and Inventory Control with Compound Poisson Demand Using Periodic Demand Data," Research Report 2018010, University of Groningen, Research Institute SOM (Systems, Organisations and Management).
    8. Lolli, F. & Gamberini, R. & Regattieri, A. & Balugani, E. & Gatos, T. & Gucci, S., 2017. "Single-hidden layer neural networks for forecasting intermittent demand," International Journal of Production Economics, Elsevier, vol. 183(PA), pages 116-128.
    9. Evangelos Spiliotis & Spyros Makridakis & Artemios-Anargyros Semenoglou & Vassilios Assimakopoulos, 2022. "Comparison of statistical and machine learning methods for daily SKU demand forecasting," Operational Research, Springer, vol. 22(3), pages 3037-3061, July.
    10. Teunter, Ruud H. & Syntetos, Aris A. & Zied Babai, M., 2011. "Intermittent demand: Linking forecasting to inventory obsolescence," European Journal of Operational Research, Elsevier, vol. 214(3), pages 606-615, November.
    11. Kourentzes, Nikolaos, 2013. "Intermittent demand forecasts with neural networks," International Journal of Production Economics, Elsevier, vol. 143(1), pages 198-206.
    12. Turrini, Laura & Meissner, Joern, 2019. "Spare parts inventory management: New evidence from distribution fitting," European Journal of Operational Research, Elsevier, vol. 273(1), pages 118-130.
    13. Hu, Qiwei & Boylan, John E. & Chen, Huijing & Labib, Ashraf, 2018. "OR in spare parts management: A review," European Journal of Operational Research, Elsevier, vol. 266(2), pages 395-414.
    14. Pennings, Clint L.P. & van Dalen, Jan & van der Laan, Erwin A., 2017. "Exploiting elapsed time for managing intermittent demand for spare parts," European Journal of Operational Research, Elsevier, vol. 258(3), pages 958-969.
    15. Petropoulos, Fotios & Apiletti, Daniele & Assimakopoulos, Vassilios & Babai, Mohamed Zied & Barrow, Devon K. & Ben Taieb, Souhaib & Bergmeir, Christoph & Bessa, Ricardo J. & Bijak, Jakub & Boylan, Joh, 2022. "Forecasting: theory and practice," International Journal of Forecasting, Elsevier, vol. 38(3), pages 705-871.
      • Fotios Petropoulos & Daniele Apiletti & Vassilios Assimakopoulos & Mohamed Zied Babai & Devon K. Barrow & Souhaib Ben Taieb & Christoph Bergmeir & Ricardo J. Bessa & Jakub Bijak & John E. Boylan & Jet, 2020. "Forecasting: theory and practice," Papers 2012.03854, arXiv.org, revised Jan 2022.
    16. Bacchetti, Andrea & Saccani, Nicola, 2012. "Spare parts classification and demand forecasting for stock control: Investigating the gap between research and practice," Omega, Elsevier, vol. 40(6), pages 722-737.
    17. Berling, Peter & Johansson, Lina & Marklund, Johan, 2023. "Controlling inventories in omni/multi-channel distribution systems with variable customer order-sizes," Omega, Elsevier, vol. 114(C).
    18. Mariusz Doszyn, 2020. "Accuracy of Intermittent Demand Forecasting Systems in the Enterprise," European Research Studies Journal, European Research Studies Journal, vol. 0(4), pages 912-930.
    19. Romeijnders, Ward & Teunter, Ruud & van Jaarsveld, Willem, 2012. "A two-step method for forecasting spare parts demand using information on component repairs," European Journal of Operational Research, Elsevier, vol. 220(2), pages 386-393.
    20. Sarlo, Rodrigo & Fernandes, Cristiano & Borenstein, Denis, 2023. "Lumpy and intermittent retail demand forecasts with score-driven models," European Journal of Operational Research, Elsevier, vol. 307(3), pages 1146-1160.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:ejores:v:282:y:2020:i:2:p:602-613. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/eor .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.