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Research Papers

Numerical methods for dynamic Bertrand oligopoly and American options under regime switching

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Pages 1741-1762 | Received 05 Aug 2015, Accepted 03 Mar 2016, Published online: 28 Apr 2016

References

  • Amarala, S. and Wan, J.W., Multigrid methods for systems of hyperbolic conservation laws. SIAM J. Multiscale Model. Simul., 2013, 11, 586–614.
  • Ayache, E., Equity-credit problem. In Encyclopedia of Quantitative Finance, edited by R. Cont, pp. 571–575, 2010 (Wiley: New York).
  • Barles, G., Convergence of numerical schemes for degenerate parabolic equations arising in finance. In Numerical Methods in Finance, edited by L.C.G Rogers and D. Talay, pp. 1–21, 1997 (Cambridge University Press: Cambridge).
  • Barles, G. and Rouy, E., A strong comparison result for the Bellman equation arising in stochastic exit time control problems and applications. Commun. Partial Differ. Equ., 1998, 23, 1945–2033.
  • Barles, G. and Souganidis, P., Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Anal., 1991, 4, 271–283.
  • Barles, G. and Jakobsen, E.R., Error bounds for monotone approximation schemes for Hamilton--Jacobi--Bellman equations. SIAM J. Numer. Anal., 2005, 43, 540–558.
  • Bellman, R., Introduction to the Mathematical Theory of Control Processes: Nonlinear processes. Mathematics in Science and Engineering, 1971 (Academic Press: New York).
  • Bergman, Y., Option pricing with differential interest rates. Rev. Financ. Stud., 1995, 8, 475–500.
  • Bertrand, J., Théorie mathématique de la richesse sociale [Mathematical theory of social wealth]. J. Savants, 1883, 67, 499–508.
  • Bokanowski, O., Maroso, S. and Zidani, H., Some convergence results for Howard’s algorithm. SIAM J. Numer. Anal., 2009, 47, 3001–3026.
  • Brandt, A., Multi-level adaptive solutions to boundary-value problems. Math. Comput., 1977, 31, 333–390.
  • Chaumont, S., A strong comparison result for viscosity solutions to Hamilton--Jacobi--Bellman equations with Dirichlet conditions on a non-smooth boundary. Preprint, Institute Elie Cartan, Universitè Nancy I, 2004.
  • Clift, S.S. and Forsyth, P.A., Numerical solution of two asset jump diffusion models for option valuation. Appl. Numer. Math., 2008, 58, 743–782.
  • Cournot, A., Recherches sur les Principes Mathématique de la Théorie des Richesses [Researches into the Mathematical Principles of the Theory of Wealth], 1838 (Hachette: Paris). English translation by N.T. Bacon, published in Economic Classics, Macmillan, 1897, and reprinted in 1960 by Augustus M. Kelley, 1838.
  • Crepey, S., About pricing equations in finance. In Proceedings of the Paris-Princeton Lectures on Mathematical Finance, edited by A. Carmona, pp. 63–203, 2010 (Berlin: Springer).
  • Dockner, E., Jorgensen, S., Long, N.V. and Sorger, G., Differential Games in Economics and Management Science, 2000 (Cambridge University Press: New York).
  • Forsyth, P.A., Numerical computation for financial modelling. Course notes, University of Waterloo, 200 University Avenue West, Waterloo, ON, Canada, 2012.
  • Forsyth, P.A. and Labahn, G., Numerical methods for controlled Hamilton--Jacobi--Bellman PDEs in finance. J. Comput. Finance, 2007, 11, 1–44.
  • Forsyth, P.A. and Vetzal, K.R., Quadratic convergence of a penalty method for valuing American options. SIAM J. Sci. Comput., 2002, 23, 2095–2122.
  • Han, D. and Wan, J.W., Multigrid methods for second order Hamilton--Jacobi--Bellman and Hamilton--Jacobi--Bellman-Isaacs equations. SIAM J. Sci. Comput., 2013, 35, S323–S344.
  • Holtz, M. and Kunoth, A., B-Spline based monotone multigrid methods. SIAM J. Numer. Anal., 2007, 45, 1175–1199.
  • Hoppe, R.H., Multi-grid methods for Hamilton--Jacobi--Bellman equations. Numer. Math., 1986, 49, 239–254.
  • Howard, R.A., Dynamic Programming and Markov Process, 1960 (MIT Press: Cambridge, MA).
  • Huang, Y., Forsyth, P. and Labahn, G., Methods for pricing American options under regime switching. SIAM J. Sci. Comput., 2011, 33, 2144–2168.
  • Huang, Y. and Forsyth, P.A., Analysis of a penalty method for pricing a guaranteed minimum withdrawal benefit (GMWB). IMA J. Numer. Anal., 2012, 32, 320–351.
  • Ishii, H. and Koike, S., Viscosity solutions for monotone systems of second order elliptic PDEs. Commun. Partial Differ. Equ., 1991, 16, 1095–1128.
  • Kennedy, J., Hedging contingent claims in markets with jumps. PhD Thesis, University of Waterloo, 200 University Avenue West, Waterloo, ON, Canada, 2007.
  • Ledvina, A. and Sircar, R., Dynamic Bertrand oligopoly. Appl. Math. Optim., 2011, 63, 11–44.
  • Lions, P. and Mercier, B., Approximation numerique des equations Hamilton--Jacobi--Bellman [Numerical approximation of the Hamilton-Jacobi-Bellman equations]. RAIRO -- Anal. Numer., 1980, 14, 369–393.
  • Oosterlee, C.W., On multigrid for linear complimentarity problems with application to American-style options. Electron. Trans. Numer. Anal., 2003, 15, 165–185.
  • Pooley, D.M., Forsyth, P.A. and Vetzal, K.R., Numerical convergence properties of option pricing PDEs with uncertain volatility. IMA J. Numer. Anal., 2003, 23, 241–267.
  • Reisinger, C. and Forsyth, P.A., Piecewise constant policy approximations to Hamilton--Jacobi--Bellman equations. Techinical Report, 2015.
  • Trottenberg, U., Oosterlee, C.W. and Schuller, A., Multigrid, 2001 (Academic Press: San Diego, CA).
  • Tse, S.T., Numerical methods for optimal trade execution. PhD Thesis, University of Waterloo, 200 University Avenue West, Waterloo, ON, Canada, 2012.
  • Wal, J., Discounted Markov games: Generalized policy iteration method. J. Optim. Theory Appl., 1978, 25, 125–138.
  • Wang, J. and Forsyth, P.A., Maximal use of central differencing for Hamilton--Jacobi--Bellman PDEs in finance. SIAM J. Numer. Anal., 2008, 46, 1580–1601.
  • Yao, D., Zhang, Q. and Zhou, X.Y., A regime switching model for European options. In Stochastic Processes, Optimization and Control Theory: Applications in Financial Engineering, Queueing Networks and Manufacturing Systems, edited by H. Yan, G. Yin and Q. Zhang, Vol. 94, pp. 281–300, 2006 (Springer International Series in Operations Research and Management Science: New York).
  • Zvan, R., Forsyth, P.A. and Vetzal, K.R., A finite volume approach for contingent claims valuation. IMA J. Numer. Anal., 2001, 21, 703–731.

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