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Drift and volatility estimation in discrete time. (English) Zbl 0895.90047

Summary: In discrete time the increment of the logarithm of the price of a risky asset is supposed to involve two parameters which may be thought of as the ‘drift’ and ‘volatility’. It is assumed these parameters take finitely many values, and that they change value like a Markov chain on this state space. Filtering and parameter estimation techniques from Hidden Markov Models are then applied to obtain recursive estimates of the ‘drift’ and ‘volatility’. Further, all parameters in the model can be estimated. The method is illustrated by applying the results to two series of prices.

MSC:

91B82 Statistical methods; economic indices and measures
62P20 Applications of statistics to economics
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References:

[1] Elliott, R. J., Exact adaptive filters for Markov chains observed in Gaussian noise, Automatica, 30, 1399-1408 (1994) · Zbl 0823.93061
[2] Elliott, R. J.; Aggoun, L.; Moore, J. B., Hidden Markov models: estimation and control, (Applications of Mathematics, vol. 29 (1994), Springer: Springer New York) · Zbl 0902.93066
[3] Elliott, R. J.; Rishel, R. W., Estimating the implicit interest rate of a risky asset, Stochastic Processes and Applications, 49, 199-206 (1994) · Zbl 0797.60036
[4] Fama, E. F.; Gibbons, M. R., A comparison of inflation forecasts, Journal of Monetary Economics, 3, 327-348 (1984)
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