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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 & 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.  doi: 10.3934/dcdsb.2011.16.1.  Google Scholar

[2]

E. Akyildirim, Y. Dolinsky and H. M. Soner, Approximating stochastic volatility by recombinant trees,, preprint, (2012).   Google Scholar

[3]

K. I. Amin, On the computation of continuous time option prices using discrete approximations,, Journal of Financial and Quantitative Analysis, 26 (1991), 477.  doi: 10.2307/2331407.  Google Scholar

[4]

P. Billingsley, Convergence of Probability Measures,, Wiley, (1968).   Google Scholar

[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.  doi: 10.1090/S0002-9947-01-02819-7.  Google Scholar

[6]

J. Dedecker and C. Prieur, New dependence coefficients. Examples and applications to statistics,, Probab. Theory and Relat. Fields, 132 (2005), 203.  doi: 10.1007/s00440-004-0394-3.  Google Scholar

[7]

N. G. Dokuchaev, Mathematical Finance: Core Theory, Problems, and Statistical Algorithms,, Routledge, (2007).  doi: 10.4324/9780203964729.  Google Scholar

[8]

N. Dokuchaev, Discrete time market with serial correlations and optimal myopic strategies,, European Journal of Operational Research, 177 (2007), 1090.  doi: 10.1016/j.ejor.2006.01.004.  Google Scholar

[9]

N. Dokuchaev, On statistical indistinguishability of the complete and incomplete markets,, preprint, (2012).  doi: 10.2139/ssrn.2149951.  Google Scholar

[10]

M. D. Donsker, Justification and extension of Doob's heuristic approach to the Kolmogorov-Smirnov theorems,, Annals of Mathematical Statistics, 23 (1952), 277.  doi: 10.1214/aoms/1177729445.  Google Scholar

[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.  doi: 10.2307/2331009.  Google Scholar

[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.  doi: 10.1137/S0036142901389530.  Google Scholar

[13]

I. A. Ibragimov, Some limit theorems for stationary processes,, Theory of probability and its applications, 7 (1962), 361.   Google Scholar

[14]

I. A. Ibragimov, Properties of sample functions of stochastic processes and embedding theorems,, Theory of probability and its applications, 18 (1973), 442.  doi: 10.1137/1118059.  Google Scholar

[15]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations,, Springer, (1992).  doi: 10.1007/978-3-662-12616-5.  Google Scholar

[16]

D. B. Nelson and K. Ramaswamy, Simple binomial processes as diffusion approximations in financial models,, Review of Financial Studies, 3 (1990), 393.  doi: 10.1093/rfs/3.3.393.  Google Scholar

[17]

R. Nickl, M. Reiş, J. Söhl and M. Trabs, High-frequency Donsker theorems for Lévy measures,, preprint, (2013).   Google Scholar

[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.  doi: 10.1080/10236198.2013.815748.  Google Scholar

[19]

A. Rodkina and N. Dokuchaev, On asymptotic optimality of Merton's myopic portfolio strategies for discrete time market,, preprint, (2014).   Google Scholar

[20]

C. Tudor and S. Torres, Donsker theorem for the Rosenblatt process and a binary market model,, Stoch. Anal. Appl., 27 (2009), 555.  doi: 10.1080/07362990902844371.  Google Scholar

[21]

A. van der Vaart and H. van Zanten, Donsker theorems for diffusions: Necessary and sufficient conditions,, Annals of Probability, 33 (2005), 1422.  doi: 10.1214/009117905000000152.  Google Scholar

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.  doi: 10.3934/dcdsb.2011.16.1.  Google Scholar

[2]

E. Akyildirim, Y. Dolinsky and H. M. Soner, Approximating stochastic volatility by recombinant trees,, preprint, (2012).   Google Scholar

[3]

K. I. Amin, On the computation of continuous time option prices using discrete approximations,, Journal of Financial and Quantitative Analysis, 26 (1991), 477.  doi: 10.2307/2331407.  Google Scholar

[4]

P. Billingsley, Convergence of Probability Measures,, Wiley, (1968).   Google Scholar

[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.  doi: 10.1090/S0002-9947-01-02819-7.  Google Scholar

[6]

J. Dedecker and C. Prieur, New dependence coefficients. Examples and applications to statistics,, Probab. Theory and Relat. Fields, 132 (2005), 203.  doi: 10.1007/s00440-004-0394-3.  Google Scholar

[7]

N. G. Dokuchaev, Mathematical Finance: Core Theory, Problems, and Statistical Algorithms,, Routledge, (2007).  doi: 10.4324/9780203964729.  Google Scholar

[8]

N. Dokuchaev, Discrete time market with serial correlations and optimal myopic strategies,, European Journal of Operational Research, 177 (2007), 1090.  doi: 10.1016/j.ejor.2006.01.004.  Google Scholar

[9]

N. Dokuchaev, On statistical indistinguishability of the complete and incomplete markets,, preprint, (2012).  doi: 10.2139/ssrn.2149951.  Google Scholar

[10]

M. D. Donsker, Justification and extension of Doob's heuristic approach to the Kolmogorov-Smirnov theorems,, Annals of Mathematical Statistics, 23 (1952), 277.  doi: 10.1214/aoms/1177729445.  Google Scholar

[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.  doi: 10.2307/2331009.  Google Scholar

[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.  doi: 10.1137/S0036142901389530.  Google Scholar

[13]

I. A. Ibragimov, Some limit theorems for stationary processes,, Theory of probability and its applications, 7 (1962), 361.   Google Scholar

[14]

I. A. Ibragimov, Properties of sample functions of stochastic processes and embedding theorems,, Theory of probability and its applications, 18 (1973), 442.  doi: 10.1137/1118059.  Google Scholar

[15]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations,, Springer, (1992).  doi: 10.1007/978-3-662-12616-5.  Google Scholar

[16]

D. B. Nelson and K. Ramaswamy, Simple binomial processes as diffusion approximations in financial models,, Review of Financial Studies, 3 (1990), 393.  doi: 10.1093/rfs/3.3.393.  Google Scholar

[17]

R. Nickl, M. Reiş, J. Söhl and M. Trabs, High-frequency Donsker theorems for Lévy measures,, preprint, (2013).   Google Scholar

[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.  doi: 10.1080/10236198.2013.815748.  Google Scholar

[19]

A. Rodkina and N. Dokuchaev, On asymptotic optimality of Merton's myopic portfolio strategies for discrete time market,, preprint, (2014).   Google Scholar

[20]

C. Tudor and S. Torres, Donsker theorem for the Rosenblatt process and a binary market model,, Stoch. Anal. Appl., 27 (2009), 555.  doi: 10.1080/07362990902844371.  Google Scholar

[21]

A. van der Vaart and H. van Zanten, Donsker theorems for diffusions: Necessary and sufficient conditions,, Annals of Probability, 33 (2005), 1422.  doi: 10.1214/009117905000000152.  Google Scholar

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