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August  2014, 19(6): 1549-1562. doi: 10.3934/dcdsb.2014.19.1549

## On strong causal binomial approximation for stochastic processes

 1 Department of Mathematics and Statistics, Curtin University, GPO Box U1987, Perth, Western Australia, 6845

Received  November 2013 Revised  March 2014 Published  June 2014

This paper considers binomial approximation of continuous time stochastic processes. It is shown that, under some mild integrability conditions, a process can be approximated in mean square sense and in other strong metrics by binomial processes, i.e., by processes with fixed size binary increments at sampling points. Moreover, this approximation can be causal, i.e., at every time it requires only past historical values of the underlying process. In addition, possibility of approximation of solutions of stochastic differential equations by solutions of ordinary equations with binary noise is established. Some consequences for the financial modelling and options pricing models are discussed.
Citation: Nikolai Dokuchaev. On strong causal binomial approximation for stochastic processes. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1549-1562. doi: 10.3934/dcdsb.2014.19.1549
##### References:
 [1] V. Abramov, F. Klebaner and R. Liptser, The Euler-Maruyama approximations for the CEV model, Discrete and Continuous Dynamical Systems. Series B, 16 (2011), 1-14. doi: 10.3934/dcdsb.2011.16.1. [2] E. Akyildirim, Y. Dolinsky and H. M. Soner, Approximating stochastic volatility by recombinant trees, preprint, arXiv:1205.3555, 2012. [3] K. I. Amin, On the computation of continuous time option prices using discrete approximations, Journal of Financial and Quantitative Analysis, 26 (1991), 477-495. doi: 10.2307/2331407. [4] P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968. [5] S. Borovkova, R. Burton and H. Dehling, Limit theorems for functionals of mixing processes with application to U-statistics and dimension estimation, Trans. Amer. Math. Soc., 353 (2001), 4261-4318. doi: 10.1090/S0002-9947-01-02819-7. [6] J. Dedecker and C. Prieur, New dependence coefficients. Examples and applications to statistics, Probab. Theory and Relat. Fields, 132 (2005), 203-236. doi: 10.1007/s00440-004-0394-3. [7] N. G. Dokuchaev, Mathematical Finance: Core Theory, Problems, and Statistical Algorithms, Routledge, London and New York, 2007. doi: 10.4324/9780203964729. [8] N. Dokuchaev, Discrete time market with serial correlations and optimal myopic strategies, European Journal of Operational Research, 177 (2007), 1090-1104. doi: 10.1016/j.ejor.2006.01.004. [9] N. Dokuchaev, On statistical indistinguishability of the complete and incomplete markets, preprint, arXiv:1209.4695, 2012. doi: 10.2139/ssrn.2149951. [10] M. D. Donsker, Justification and extension of Doob's heuristic approach to the Kolmogorov-Smirnov theorems, Annals of Mathematical Statistics, 23 (1952), 277-281. doi: 10.1214/aoms/1177729445. [11] D. Heath, R. Jarrow and A. Morton, Bond pricing and the term structure of interest rates: A discrete time approximation, Journal of Financial and Quantitative Analysis, 25 (1990), 419-440. doi: 10.2307/2331009. [12] D. J. Higham, X. Mao and A. M. Stuart, Strong convergence of numerical methods for nonlinear stochastic differential equations, SIAM J. Num. Anal., 40 (2002), 1041-1063. doi: 10.1137/S0036142901389530. [13] I. A. Ibragimov, Some limit theorems for stationary processes, Theory of probability and its applications, 7 (1962), 361-392. [14] I. A. Ibragimov, Properties of sample functions of stochastic processes and embedding theorems, Theory of probability and its applications, 18 (1973), 442-453. doi: 10.1137/1118059. [15] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992. doi: 10.1007/978-3-662-12616-5. [16] D. B. Nelson and K. Ramaswamy, Simple binomial processes as diffusion approximations in financial models, Review of Financial Studies, 3 (1990), 393-430. doi: 10.1093/rfs/3.3.393. [17] R. Nickl, M. Reiş, J. Söhl and M. Trabs, High-frequency Donsker theorems for Lévy measures, preprint, arXiv:1310.2523, 2013. [18] A. Rodkina and N. Dokuchaev, Instability and stability of solutions of systems of nonlinear stochastic difference equations with diagonal noise, Journal of Difference Equations and Applications, 20 (2014), 744-764. doi: 10.1080/10236198.2013.815748. [19] A. Rodkina and N. Dokuchaev, On asymptotic optimality of Merton's myopic portfolio strategies for discrete time market, preprint, arXiv:1403.4329, 2014. [20] C. Tudor and S. Torres, Donsker theorem for the Rosenblatt process and a binary market model, Stoch. Anal. Appl., 27 (2009), 555-573. arXiv:math/0703085. doi: 10.1080/07362990902844371. [21] A. van der Vaart and H. van Zanten, Donsker theorems for diffusions: Necessary and sufficient conditions, Annals of Probability, 33 (2005), 1422-1451. doi: 10.1214/009117905000000152.

show all references

##### References:
 [1] V. Abramov, F. Klebaner and R. Liptser, The Euler-Maruyama approximations for the CEV model, Discrete and Continuous Dynamical Systems. Series B, 16 (2011), 1-14. doi: 10.3934/dcdsb.2011.16.1. [2] E. Akyildirim, Y. Dolinsky and H. M. Soner, Approximating stochastic volatility by recombinant trees, preprint, arXiv:1205.3555, 2012. [3] K. I. Amin, On the computation of continuous time option prices using discrete approximations, Journal of Financial and Quantitative Analysis, 26 (1991), 477-495. doi: 10.2307/2331407. [4] P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968. [5] S. Borovkova, R. Burton and H. Dehling, Limit theorems for functionals of mixing processes with application to U-statistics and dimension estimation, Trans. Amer. Math. Soc., 353 (2001), 4261-4318. doi: 10.1090/S0002-9947-01-02819-7. [6] J. Dedecker and C. Prieur, New dependence coefficients. Examples and applications to statistics, Probab. Theory and Relat. Fields, 132 (2005), 203-236. doi: 10.1007/s00440-004-0394-3. [7] N. G. Dokuchaev, Mathematical Finance: Core Theory, Problems, and Statistical Algorithms, Routledge, London and New York, 2007. doi: 10.4324/9780203964729. [8] N. Dokuchaev, Discrete time market with serial correlations and optimal myopic strategies, European Journal of Operational Research, 177 (2007), 1090-1104. doi: 10.1016/j.ejor.2006.01.004. [9] N. Dokuchaev, On statistical indistinguishability of the complete and incomplete markets, preprint, arXiv:1209.4695, 2012. doi: 10.2139/ssrn.2149951. [10] M. D. Donsker, Justification and extension of Doob's heuristic approach to the Kolmogorov-Smirnov theorems, Annals of Mathematical Statistics, 23 (1952), 277-281. doi: 10.1214/aoms/1177729445. [11] D. Heath, R. Jarrow and A. Morton, Bond pricing and the term structure of interest rates: A discrete time approximation, Journal of Financial and Quantitative Analysis, 25 (1990), 419-440. doi: 10.2307/2331009. [12] D. J. Higham, X. Mao and A. M. Stuart, Strong convergence of numerical methods for nonlinear stochastic differential equations, SIAM J. Num. Anal., 40 (2002), 1041-1063. doi: 10.1137/S0036142901389530. [13] I. A. Ibragimov, Some limit theorems for stationary processes, Theory of probability and its applications, 7 (1962), 361-392. [14] I. A. Ibragimov, Properties of sample functions of stochastic processes and embedding theorems, Theory of probability and its applications, 18 (1973), 442-453. doi: 10.1137/1118059. [15] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992. doi: 10.1007/978-3-662-12616-5. [16] D. B. Nelson and K. Ramaswamy, Simple binomial processes as diffusion approximations in financial models, Review of Financial Studies, 3 (1990), 393-430. doi: 10.1093/rfs/3.3.393. [17] R. Nickl, M. Reiş, J. Söhl and M. Trabs, High-frequency Donsker theorems for Lévy measures, preprint, arXiv:1310.2523, 2013. [18] A. Rodkina and N. Dokuchaev, Instability and stability of solutions of systems of nonlinear stochastic difference equations with diagonal noise, Journal of Difference Equations and Applications, 20 (2014), 744-764. doi: 10.1080/10236198.2013.815748. [19] A. Rodkina and N. Dokuchaev, On asymptotic optimality of Merton's myopic portfolio strategies for discrete time market, preprint, arXiv:1403.4329, 2014. [20] C. Tudor and S. Torres, Donsker theorem for the Rosenblatt process and a binary market model, Stoch. Anal. Appl., 27 (2009), 555-573. arXiv:math/0703085. doi: 10.1080/07362990902844371. [21] A. van der Vaart and H. van Zanten, Donsker theorems for diffusions: Necessary and sufficient conditions, Annals of Probability, 33 (2005), 1422-1451. doi: 10.1214/009117905000000152.
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