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Stackelberg equilibria in a continuous-time vertical contracting model with uncertain demand and delayed information

Part of: Stochastic analysis Game theory

Published online by Cambridge University Press:  30 March 2016

Bernt Øksendal ,
Leif Sandal  and
Jan Ubøe
Bernt Øksendal
Affiliation:
Department of Mathematics, University of Oslo, PO Box 1053 Blindern, 0316 Oslo, Norway
Leif Sandal
Affiliation:
Norwegian School of Economics, Helleveien 30, 5045 Bergen, Norway
Jan Ubøe
Affiliation:
Norwegian School of Economics, Helleveien 30, 5045 Bergen, Norway. Email address: jan.uboe@nhh.no.
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Abstract

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We consider explicit formulae for equilibrium prices in a continuous-time vertical contracting model. A manufacturer sells goods to a retailer, and the objective of both parties is to maximize expected profits. Demand is an Itô-Lévy process, and to increase realism, information is delayed. We provide complete existence and uniqueness proofs for a series of special cases, including geometric Brownian motion and the Ornstein-Uhlenbeck process, both with time-variable coefficients. Moreover, explicit solution formulae are given, so these results are operational. An interesting finding is that information that is more precise may be a considerable disadvantage for the retailer.

Keywords

Vertical contractingstochastic differential gamedelayed informationItô-Lévy process

MSC classification

Primary: 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 91A15: Stochastic games
Type
Part 5. Finance and econometrics
Information
Journal of Applied Probability , Volume 51 , Issue A: Celebrating 50 Years of The Applied Probability Trust , December 2014 , pp. 213 - 226
Copyright
Copyright © Applied Probability Trust 2014 

References

Barndorff-Nielsen, O. E. (1998). Processes of normal inverse Gaussian type. Finance Stoch. 2, 41–68.CrossRefGoogle Scholar
Barndorff-Nielsen, O. E., and Shephard, N. (2001). Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. J. R. Statist. Soc. B 63, 167–241.Google Scholar
Bensoussan, A., Cakanyildirim, M., Feng, Q., and Sethi, S. P. (2009). Optimal ordering policies for stochastic inventory problems with observed information delays. Production Operat. Manag. 18, 546–559.Google Scholar
Cachón, G. P. (2003). Supply chain coordination with contracts. In Handbooks of Operations Research and Management Science, Vol. 11, eds Graves, S. C. and de Kok, A. G., Amsterdam, Elsevier, pp. 227–339.Google Scholar
Calzolari, A., Florchinger, P., and Nappo, G. (2011). Nonlinear filtering for stochastic systems with fixed delay: approximation by a modified Milstein scheme. Comput. Math. Appl. 61, 2498–2509.CrossRefGoogle Scholar
Eberlein, E. (2009). Jump type Lévy processes. In Handbook of Financial Time Series, eds Andersen, T. G. et al., Springer, Berlin, pp. 439–455.Google Scholar
Edgeworth, F. Y. (1888). The mathematical theory of banking. J. R. Statist. Soc. 51, 11–127.Google Scholar
Kaplan, R. S. (1970). A dynamic inventory model with stochastic lead times. Manag. Sci. 16, 491–507.Google Scholar
Lariviere, M. A., and Porteus, E. L. (2001). Selling to the newsvendor: an analysis of price-only contracts. Manufacturing Service Operat. Manag. 3, 293–305.Google Scholar
Maller, R. A., Müller, G., and Szimayer, A. (2009). Ornstein-Uhlenbeck processes and extensions. In Handbook of Financial Time Series, eds Anderson, T. G. et al., Springer, Berlin, pp. 421–437.Google Scholar
Merton, R. C. (1969). Lifetime portfolio selection under uncertainty: the continuous time case. Rev. Econom. Statist. 51, 247–257.Google Scholar
Nicolato, E., and Venardos, E. (2003). Option pricing in stochastic volatility models of the Ornstein-Uhlenbeck type. Math. Finance 13, 445–466.Google Scholar
Oksendal, B., and Sulem, A. (2007). Applied Stochastic Control of Jump Diffusions, 2nd edn. Springer, Berlin.Google Scholar
Oksendal, B., Sandal, L. and Uboe, J. (2013). Stochastic Stackelberg equilibria with applications to time-dependent newsvendor models, J. Econom. Dynam. Control 37, 1284–1299.CrossRefGoogle Scholar
Qin, Q. et al.(2011). The newsvendor problem: review and directions for future research. Europ. J. Operat. Res. 213, 361–374.Google Scholar
Samuelson, P. A. (1965). Rational theory of warrant pricing. Indust. Manag. Rev. 6, 13–31.Google Scholar
Song, J.-S., and Zipkin, P. H. (1996). The joint effect of leadtime variance and lot size in a parallel processing environment. Manag. Sci. 42, 1352–1363.Google Scholar
Taylor, T. A., and Xiao, W. (2010). Does a manufacturer benefit from selling to a better-forecasting retailer? Manag. Sci. 56, 1584–1598.Google Scholar
Vasicek, O. (1977). An equilibrium characterisation of the term structure. J. Financial Econom. 5, 177–188.Google Scholar