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Weak convergence of semimartingales and discretisation methods

Author

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  • Platen, Eckhard
  • Rebolledo, Rolando

Abstract

Given a semimartingale one can construct a system ([lambda], A, B, C) where [lambda] is the distribution of the initial value and (A, B, C) is the triple of global characteristics. Thus, given a process X and a system ([lambda], A, B, C) one can look for all probability measures P such that X is a P-semimartingale with initial distribution [lambda] and global characteristics (A, B, C). We say that such a measure P is a solution to the semimartingale problem ([lambda], A, B, C). The paper is devoted to the study of a special type of semimartingale problem. We look for sufficient conditions to insure the existence of solutions and we develop a method to construct them by means of time-discretised schemes, using weak topology for probability measures.

Suggested Citation

  • Platen, Eckhard & Rebolledo, Rolando, 1985. "Weak convergence of semimartingales and discretisation methods," Stochastic Processes and their Applications, Elsevier, vol. 20(1), pages 41-58, July.
    • Handle: RePEc:eee:spapps:v:20:y:1985:i:1:p:41-58
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    Cited by:

    1. Nicolas Merener & Paul Glasserman, 2003. "Numerical solution of jump-diffusion LIBOR market models," Finance and Stochastics, Springer, vol. 7(1), pages 1-27.
    2. Chernov, Mikhail & Graveline, Jeremy & Zviadadze, Irina, 2018. "Crash Risk in Currency Returns," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 53(1), pages 137-170, February.
    3. Nicola Bruti-Liberati, 2007. "Numerical Solution of Stochastic Differential Equations with Jumps in Finance," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1, July-Dece.
    4. Chernov, Mikhail & Ronald Gallant, A. & Ghysels, Eric & Tauchen, George, 2003. "Alternative models for stock price dynamics," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 225-257.
    5. Michael S. Johannes & Nicholas G. Polson & Jonathan R. Stroud, 2009. "Optimal Filtering of Jump Diffusions: Extracting Latent States from Asset Prices," The Review of Financial Studies, Society for Financial Studies, vol. 22(7), pages 2559-2599, July.
    6. Casella, Bruno & Roberts, Gareth O., 2011. "Exact Simulation of Jump-Diffusion Processes with Monte Carlo Applications," MPRA Paper 95217, University Library of Munich, Germany.
    7. Nicola Bruti-Liberati, 2007. "Numerical Solution of Stochastic Differential Equations with Jumps in Finance," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1-2007, January-A.
    8. Chernov, Mikhail, 2007. "On the Role of Risk Premia in Volatility Forecasting," Journal of Business & Economic Statistics, American Statistical Association, vol. 25, pages 411-426, October.
    9. Bruno Casella & Gareth O. Roberts, 2011. "Exact Simulation of Jump-Diffusion Processes with Monte Carlo Applications," Methodology and Computing in Applied Probability, Springer, vol. 13(3), pages 449-473, September.

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