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- Abramowicz, Konrad, 1983-, et al.
(författare)
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On the error of the Monte Carlo pricing method for Asian option
- 2008
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Ingår i: Journal of Numerical and Applied Mathematics. - 0868-6912. ; 96:1, s. 1-10
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Tidskriftsartikel (refereegranskat)abstract
- We consider a Monte Carlo method to price a continuous arithmetic Asian option with a given precision. Piecewise constant approximation and plain simulation are used for a wide class of models based on L\'{e}vy processes. We give bounds of the possible discretization and simulation errors. The sufficient numbers of discretization points and simulations to obtain requested accuracy are derived. To demonstrate the general approach, the Black-Scholes model is studied in more detail. We undertake the case of continuous averaging and starting time zero, but the obtained results can be applied to the discrete case and generalized for any time before an execution date. Some numerical experiments and comparison to the PDE based method are also presented.
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| 5. |
- Shykula, Mykola, 1978-
(författare)
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Quantization of Random Processes and Related Statistical Problems
- 2006
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In this thesis we study a scalar uniform and non-uniform quantization of random processes (or signals) in average case setting. Quantization (or discretization) of a signal is a standard task in all nalog/digital devices (e.g., digital recorders, remote sensors etc.). We evaluate the necessary memory capacity (or quantization rate) needed for quantized process realizations by exploiting the correlation structure of the model random process. The thesis consists of an introductory survey of the subject and related theory followed by four included papers (A-D).In Paper A we develop a quantization coding method when quantization levels crossings by a process realization are used for its coding. Asymptotical behavior of mean quantization rate is investigated in terms of the correlation structure of the original process. For uniform and non-uniform quantization, we assume that the quantization cellwidth tends to zero and the number of quantization levels tends to infinity, respectively.In Papers B and C we focus on an additive noise model for a quantized random process. Stochastic structures of asymptotic quantization errors are derived for some bounded and unbounded non-uniform quantizers when the number of quantization levels tends to infinity. The obtained results can be applied, for instance, to some optimization design problems for quantization levels.Random signals are quantized at sampling points with further compression. In Paper D the concern is statistical inference for run-length encoding (RLE) method, one of the compression techniques, applied to quantized stationary Gaussian sequences. This compression method is widely used, for instance, in digital signal and image processing. First, we deal with mean RLE quantization rates for various probabilistic models. For a time series with unknown stochastic structure, we investigate asymptotic properties (e.g., asymptotic normality) of two estimates for the mean RLE quantization rate based on an observed sample when the sample size tends to infinity.These results can be used in communication theory, signal processing, coding, and compression applications. Some examples and numerical experiments demonstrating applications of the obtained results for synthetic and real data are presented.
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| 7. |
- Shykula, Mykola, et al.
(författare)
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Stochastic structure of asymptotic quantization errors
- 2006
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Ingår i: Statistics and Probability Letters. - : Elsevier. - 0167-7152 .- 1879-2103. ; 76:5, s. 453-464
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Tidskriftsartikel (refereegranskat)abstract
- We consider quantization of continuous-valued random variables and processes in a probabilistic framework. Stochastic structure for non-uniform quantization errors is studied for a wide class of random variables. Asymptotic properties of the additive quantization noise model for a random process are derived for uniform and non-uniform quantizers. Some examples and numerical experiments demonstrating the rate of convergence in the obtained asymptotic results are presented.
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