Abstract
We consider the diffusive limit of a typical pure-jump Markovian control problem as the intensity of the driving Poisson process tends to infinity. We show that the convergence speed is provided by the Hölder exponent of the Hessian of the limit problem, and explain how correction terms can be constructed. This provides an alternative efficient method for the numerical approximation of the optimal control of a pure-jump problem in situations with very high intensity of jumps. We illustrate this approach in the context of a display advertising auction problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Note that their Assumption (L) is not required since we are considering a finite time interval [0, T], this can be easily seen from the proof of this theorem.
Recall that the competition here models the distribution of the maximum bid of all other participants, so this is the average of the maximum of other participants’ bids.
References
Andrews, B.: Fully nonlinear parabolic equations in two space variables (2004). arXiv:math/0402235
Barles, G.: An Introduction to the Theory of Viscosity Solutions for First-Order Hamilton-Jacobi Equations and Applications, in Hamilton-Jacobi Equations: Approximations. Numerical Analysis and Applications. Springer, New York (2013)
Bäuerle, N.: Approximation of optimal reinsurance and dividend payout policies. Math. Financ.: Int. J. Math., Stat. Financ. Econ. 14(1), 99–113 (2004)
Bertsekas, D.P., Shreve, S.E.: Stochastic Optimal Control: The Discrete-Time Case. Academic Press, New York (1978)
Blum, A., Kumar, V., Rudra, A., Wu, F.: Online learning in online auctions. Theor. Computer Sci. 324(2–3), 137–146 (2004)
Bouchard, B.: Stochastic targets with mixed diffusion processes and viscosity solutions. Stoch. Processes Appl. 101(2), 273–302 (2002)
Bouchard, B., Geiss, S., Gobet, E.: First time to exit of a continuous Itô process: general moment estimates and \(L^{1}\)-convergence rate for discrete time approximations. Bernoulli 23(3), 1631–1662 (2017)
Bouchard, B., Touzi, N.: Weak dynamic programming principle for viscosity solutions. SIAM J. Control Optim. 49(3), 948–962 (2011)
Brémaud, P.: Point Processes and Queues: Martingale Dynamics. Springer, Cham (1981)
Bubeck, S., Devanur, N.R., Huang, Z., Niazadeh, R.: Multi-scale online learning: theory and applications to online auctions and pricing. J. Mach. Learn. Res. 20(1), 2248–2284 (2019)
Bulow, J., Klemperer, P.: Auctions versus negotiations. Am. Econ. Rev. 86(1), 180–94 (1996)
Cesa-Bianchi, N., Gentile, C., Mansour, Y.: Regret minimization for reserve prices in second-price auctions. IEEE Trans. Inf. Theor. 61(1), 549–564 (2014)
Chen, H., Yao, D.: Fundamentals of queueing networks. Stoch Model Appl Probabil 2, 871 (2001)
Cohen, A., Young, V.R.: Rate of convergence of the probability of ruin in the Cramér–Lundberg model to its diffusion approximation. Insurance: Math. Econ. 93, 333–340 (2020)
Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992)
Croissant, L., Abeille, M., Calauzenes, C.: Real-time optimisation for online learning in auctions. In Hal Daumé III and Aarti Singh, editors, Proceedings of the 37th International Conference on Machine Learning, volume 119 of Proceedings of Machine Learning Research, pp 2217–2226. PMLR, 13–18 (Jul 2020)
Fernandez-Tapia, J., Guéant, O., Lasry, J.-M.: Optimal real-time bidding strategies. Appl. Math. Res. Exp 2017(1), 142–183 (2016)
Fleming, W.H., Souganidis, P.E.: On the existence of value functions of two-player, zero-sum stochastic differential games. Indiana Univ. Math. J. 38(2), 293–314 (1989)
Jacod, J., Shiryaev, A.: Limit theorems for stochastic processes, vol. 288. Springer Science & Business Media, Cham (2013)
Ladyzhenskaia, O.A., Solonnikov, V., Ural’tseva, N.N.: Linear and quasi-linear equations of parabolic type, volume 23. American Mathematical Soc., (1988)
Lieberman, G.M.: Second order parabolic differential equations. World scientific, Singapore (1996)
Myerson, R.B.: Optimal auction design. Math. Op. Res. 6(1), 58–73 (1981)
Ostrovsky, M., Schwarz, M.: Reserve prices in internet advertising auctions: A field experiment. In Proceedings of the 12th ACM Conference on Electronic Commerce, EC ’11, pp 59–60, New York, NY, USA (2011). Association for Computing Machinery
Paes Leme, R., Pal, M., Vassilvitskii, S.: A field guide to personalized reserve prices. In Proceedings of the 25th International Conference on World Wide Web, WWW ’16, pp 1093–1102, Republic and Canton of Geneva, CHE, (2016). International World Wide Web Conferences Steering Committee
Vickrey, W.: Counterspeculation, auctions, and competitive sealed tenders. J. Financ 16(1), 8–37 (1961)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Mihai Sirbu.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Abeille, M., Bouchard, B. & Croissant, L. Diffusive Limit Approximation of Pure-Jump Optimal Stochastic Control Problems. J Optim Theory Appl 196, 147–176 (2023). https://doi.org/10.1007/s10957-022-02135-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-022-02135-7