Infinitesimal generators for two-dimensional Lévy process-driven hypothesis testing

  • Michael Roberts
  • Indranil SenGuptaEmail author
Research Article


In this paper, we present the testing of four hypotheses on two streams of observations that are driven by Lévy processes. This is applicable for sequential decision making on the state of two-sensor systems. In one case, each sensor receives or does not receive a signal obstructed by noise. In another, each sensor receives data driven by Lévy processes with large or small jumps. In either case, these give rise to four possibilities. Infinitesimal generators are presented and analyzed. Bounds for infinitesimal generators in terms of super-solutions and sub-solutions are computed. An application of this procedure for stochastic model is also presented in relation to the financial market.


Lévy process Infinitesimal generator Hypothesis tests Viscosity solution Oil price 

JEL Classification

C12 C13 C62 



The authors would like to thank the anonymous reviewers for their careful reading of the manuscript and for suggesting points to improve the quality of the paper.


  1. Barles, G., Chasseigne, E., Imbert, C.: On the Dirichlet problem for second-order elliptic integro-differential equations. Indiana Univ Math J 57(1), 213–246 (2008)CrossRefGoogle Scholar
  2. Barndorff-Nielsen, O.E.: Superposition of Ornstein–Uhlenbeck type processes. Theory Probab Appl 45, 175–194 (2001)CrossRefGoogle Scholar
  3. Barndorff-Nielsen, O.E., Shephard, N.: Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics. J R Stat Soc Ser B Stat Methodol 63, 167–241 (2001a)CrossRefGoogle Scholar
  4. Barndorff-Nielsen, O.E., Shephard, N.: Modelling by Lévy processes for financial econometrics. In: Barndorff-Nielsen, O.E., Mikosch, T., Resnick, S. (eds.) Lévy Processes : Theory and Applications, pp. 283–318. Basel, Birkhäuser (2001b)CrossRefGoogle Scholar
  5. Barndorff-Nielsen, O.E., Jensen, J.L., Sorensen, M.: Some stationary processes in discrete and continuous time. Adv Appl Probab 30, 989–1007 (1998)CrossRefGoogle Scholar
  6. Baum, C., Veeravalli, V.: A sequential procedure for multihypothesis testing. IEEE Trans Inf Theory 40(6), 1994–1997 (1994)CrossRefGoogle Scholar
  7. Brodsky, B., Darkhovsky, B.: Minimax methods for multihypothesis sequential testing and change-point detection problems. Seq Anal 27(2), 141–173 (2008)CrossRefGoogle Scholar
  8. Carlisle, M., Hadjiliadis, O.: Sequential decision making in two-dimensional hypothesis testing. In: 52nd IEEE Conference on Decision and Control. (2013). Accessed 20 Aug 2018
  9. Chow, Y.S., Robbins, H., Siegmund, D.: The Theory of Optimal Stopping. Dover Publications, Mineola (1991)Google Scholar
  10. Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Hapman and Hall/CRC Financial Mathematics Series, Boca Raton (2003)Google Scholar
  11. Dayanik, S., Poor, V., Sezer, S.: Sequential multi-hypothetis testing for compound Poisson processes. Stochastics 80(1), 19–50 (2008)CrossRefGoogle Scholar
  12. Golubev, G.K., Khas’minski, R.Z.: Sequential testing for several signals in Gaussian white noise. Theory Probab Appl 28, 573–584 (1983)CrossRefGoogle Scholar
  13. Habtemicael, S., SenGupta, I.: Pricing variance and volatility swaps for Barndorff-Nielsen and Shephard process driven financial markets. Int J Financ Eng 03(04), 1650027 (2016)CrossRefGoogle Scholar
  14. Habtemicael, S., SenGupta, I.: Pricing covariance swaps for Barndorff-Nielsen and Shephard process driven financial markets. Ann Financ Econ 11, 1650012 (2016)CrossRefGoogle Scholar
  15. Irle, A.: Transitivity in problems of optimal stopping. Ann Probab 9, 642–647 (1981)CrossRefGoogle Scholar
  16. Issaka, A., SenGupta, I.: Analysis of variance based instruments for Ornstein–Uhlenbeck type models: swap and price index. Ann Finance 13(4), 401–434 (2017)CrossRefGoogle Scholar
  17. Lowther, G.: Lévy Processes, Stochastic Calculus Notes. (2010). Accessed 25 Sept 2018
  18. SenGupta, I., Wilson, W., Nganje, W.: Barndorff-Nielsen and Shephard model: oil hedging with variance swap and option. Math Financ Econ 13(2), 209–226 (2019a)CrossRefGoogle Scholar
  19. SenGupta, I., Nganje, W., Hanson, E.: Refinements of Barndorff-Nielsen and Shephard model: an analysis of crude oil price with machine learning (submitted) (2019b)Google Scholar
  20. Shiryayev, A.N.: Optimal Stopping Rules. Springer, New York (1978)Google Scholar
  21. Wald, A.: Sequential Analysis. Wiley, New York (1947)Google Scholar
  22. Wannenwetsch, J.: Lévy Processes in Finance: The Change of Measure and Non-Linear Dependence, Inaugural-Dissertation zur Erlangung des Grades eines Doktors der Wirtschafts- und Gesellschaftswissenschaften durch die Rechts- und Staatswissenschaftliche Fakultät der Rheinischen Friedrich-Wilhelms-Universität, Bonn. (2005). Accessed 10 Mar 2019

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsNorth Dakota State UniversityFargoUSA

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