# American Institute of Mathematical Sciences

May  2013, 7(2): 611-647. doi: 10.3934/ipi.2013.7.611

## Constructing continuous stationary covariances as limits of the second-order stochastic difference equations

 1 University of Oulu, Sodankylä Geophysical Observatory, Tähteläntie 62, FI-99600 Sodankylä 2 University of Helsinki, Department of Mathematics and Statistics, Gustaf Hällströmin katu 2b, FI-00014 University of Helsinki, Finland 3 University of Oulu, Sodankylä Geophysical Observatory, Sodankylä

Received  November 2011 Revised  August 2012 Published  May 2013

In Bayesian statistical inverse problems the a priori probability distributions are often given as stochastic difference equations. We derive a certain class of stochastic partial difference equations by starting from second-order stochastic partial differential equations in one and two dimensions. We discuss discretisation schemes on uniform lattices of these stationary continuous-time stochastic processes and convergence of the discrete-time processes to the continuous-time processes. A special emphasis is given to an analytical calculation of the covariance kernels of the processes. We find a representation for the covariance kernels in a simple parametric form with controllable parameters: correlation length and variance. In the discrete-time processes the discretisation step is also given as a parameter. Therefore, the discrete-time covariances can be considered as discretisation-invariant. In the two-dimensional cases we find rotation-invariant and anisotropic representations of the difference equations and the corresponding continuous-time covariance kernels.
Citation: Lassi Roininen, Petteri Piiroinen, Markku Lehtinen. Constructing continuous stationary covariances as limits of the second-order stochastic difference equations. Inverse Problems and Imaging, 2013, 7 (2) : 611-647. doi: 10.3934/ipi.2013.7.611
##### References:
 [1] V. I. Bogachev, "Measure Theory Vol I, II," Springer-Verlag, Berlin, 2007 doi: 10.1007/978-3-540-34514-5. [2] V. I. Bogachev and A. V. Kolesnikov, Open mappings of probability measures and Skorokhod's representation theorem, Theory Probab. Appl., 46 (2002), 20-38. doi: 10.1137/S0040585X97978701. [3] R. L. Burden, J. D. Faires and A. C. Reynolds, "Numerical Analysis," Prindle, Weber & Schmidt, 1978. [4] M. D. Donsker, An invariance principle for certain probability limit theorems, Mem. Amer. Math. Soc., 1951 (1951), 12 pp. [5] J. L. Doob, "Stochastic Processes," John Wiley & Sons, New York, 1953. [6] M. Fukushima, Y. Ōshima and M. Takeda, "Dirichlet Forms and Symmetric Markov Processes," de Gruyter Studies in Mathematics, 19, Walter de Gruyter & Co., Berlin, 1994. doi: 10.1515/9783110889741. [7] I. M. Gel'fand and N. Ya. Vilenkin, "Generalized Functions. Vol. 4. Applications Of Harmonic Analysis," Academic Press, New York-London, 1964. [8] I. S. Gradshteyn and I. M. Ryzhik, "Table of Integrals, Series and Products," $7^{th}$ edition, Academic Press, 2007. [9] T. Helin, On infinite-dimensional hierarchical probability models in statistical inverse problems, Inverse Problems and Imaging, 4 (2009), 567-597. doi: 10.3934/ipi.2009.3.567. [10] T. Hida, H-H. Kuo, J. Potthoff and L. Streit, "White Noise. An Infinite-Dimensional Calculus," Kluwer Academic Publishers Group, Dordrecht, 1993. [11] K. Itō, On stochastic differential equations, Mem. Amer. Math. Soc., 1951 (1951), 51 pp. [12] K. Itō, Stochastic integral, Proc. Imp. Acad. Tokyo, 20 (1944), 519-524. doi: 10.3792/pia/1195572786. [13] K. E. Iverson, "A Programming Language," New York: Wiley, p. 11, 1962. [14] J. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems," Springer, 2005. [15] D. Knuth, Two notes on notation, American Mathematical Monthly, 99 (1992), 403-422. doi: 10.2307/2325085. [16] H-H. Kuo, "White Noise Distribution Theory," CRC Press, Boca Raton, FL, 1996. [17] S. Lasanen, "Discretizations of Generalized Random Variables with Applications to Inverse Problems," Ph.D thesis, University of Oulu, 2002. [18] S. Lasanen, Non-Gaussian statistical inverse problems. Part I: Posterior distributions, Inverse Problems and Imaging, 6 (2012), 215-266. doi: 10.3934/ipi.2012.6.215. [19] S. Lasanen, Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns, Inverse Problems and Imaging, 6 (2012), 267-287. doi: 10.3934/ipi.2012.6.267. [20] M. Lassas, E. Saksman and S. Siltanen, Discretization invariant Bayesian inversion and Besov space priors, Inverse Problems and Imaging, 3 (2009), 87-122. doi: 10.3934/ipi.2009.3.87. [21] M. Lassas and S. Siltanen, Can one use total variation prior for edge-preserving Bayesian inversion?, Inverse Problems, 20 (2004), 1537-1563. doi: 10.1088/0266-5611/20/5/013. [22] M. S. Lehtinen, L. Päivärinta and E. Somersalo, Linear inverse problems for generalised random variables, Inverse Problems, 5 (1989), 599-612. doi: 10.1088/0266-5611/5/4/011. [23] F. Lindgren, H. Rue and J. Lindström, An explicit link between Gaussian Markov random fields: The stochastic partial differential equation approach, Journal of the Royal Statistical Society: Series B, 73 (2011), 423-498. doi: 10.1111/j.1467-9868.2011.00777.x. [24] M. Orispää and M. Lehtinen, Fortran linear inverse problem solver, Inverse Problems and Imaging, 4 (2010), 482-503. doi: 10.3934/ipi.2010.4.485. [25] P. Piiroinen, Statistical measurements, experiments and applications, Ann. Acad. Sci. Fenn. Math. Diss., (2005), 89 pp. [26] L. Roininen, M. Lehtinen, S. Lasanen, M. Orispää and M. Markkanen, Correlation priors, Inverse Problems and Imaging, 5 (2011), 167-184. doi: 10.3934/ipi.2011.5.167. [27] H. Rue and L. Held, "Gaussian Markov Random Fields: Theory and Applications," Chapman & Hall/CRC, 2005. doi: 10.1201/9780203492024. [28] D. W. Stroock and S. R. S. Varadhan, Diffusion processes with continuous coefficients. I, Comm. Pure Appl. Math., 22 (1969), 345-400. doi: 10.1002/cpa.3160220304. [29] D. W. Stroock and S. R. S. Varadhan, Diffusion processes with continuous coefficients. II, Comm. Pure Appl. Math., 22 (1969), 479-530. doi: 10.1002/cpa.3160220404. [30] D. W. Stroock and S. R. S. Varadhan, Diffusion processes with boundary conditions, Comm. Pure Appl. Math., 24 (1971), 147-225. doi: 10.1002/cpa.3160240206. [31] C. H. Su and D. Lucor, Covariance kernel representations of multidimensional second-order stochastic processes, J. Comp. Phys., 217 (2006), 82-99. doi: 10.1016/j.jcp.2006.02.006.

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##### References:
 [1] V. I. Bogachev, "Measure Theory Vol I, II," Springer-Verlag, Berlin, 2007 doi: 10.1007/978-3-540-34514-5. [2] V. I. Bogachev and A. V. Kolesnikov, Open mappings of probability measures and Skorokhod's representation theorem, Theory Probab. Appl., 46 (2002), 20-38. doi: 10.1137/S0040585X97978701. [3] R. L. Burden, J. D. Faires and A. C. Reynolds, "Numerical Analysis," Prindle, Weber & Schmidt, 1978. [4] M. D. Donsker, An invariance principle for certain probability limit theorems, Mem. Amer. Math. Soc., 1951 (1951), 12 pp. [5] J. L. Doob, "Stochastic Processes," John Wiley & Sons, New York, 1953. [6] M. Fukushima, Y. Ōshima and M. Takeda, "Dirichlet Forms and Symmetric Markov Processes," de Gruyter Studies in Mathematics, 19, Walter de Gruyter & Co., Berlin, 1994. doi: 10.1515/9783110889741. [7] I. M. Gel'fand and N. Ya. Vilenkin, "Generalized Functions. Vol. 4. Applications Of Harmonic Analysis," Academic Press, New York-London, 1964. [8] I. S. Gradshteyn and I. M. Ryzhik, "Table of Integrals, Series and Products," $7^{th}$ edition, Academic Press, 2007. [9] T. Helin, On infinite-dimensional hierarchical probability models in statistical inverse problems, Inverse Problems and Imaging, 4 (2009), 567-597. doi: 10.3934/ipi.2009.3.567. [10] T. Hida, H-H. Kuo, J. Potthoff and L. Streit, "White Noise. An Infinite-Dimensional Calculus," Kluwer Academic Publishers Group, Dordrecht, 1993. [11] K. Itō, On stochastic differential equations, Mem. Amer. Math. Soc., 1951 (1951), 51 pp. [12] K. Itō, Stochastic integral, Proc. Imp. Acad. Tokyo, 20 (1944), 519-524. doi: 10.3792/pia/1195572786. [13] K. E. Iverson, "A Programming Language," New York: Wiley, p. 11, 1962. [14] J. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems," Springer, 2005. [15] D. Knuth, Two notes on notation, American Mathematical Monthly, 99 (1992), 403-422. doi: 10.2307/2325085. [16] H-H. Kuo, "White Noise Distribution Theory," CRC Press, Boca Raton, FL, 1996. [17] S. Lasanen, "Discretizations of Generalized Random Variables with Applications to Inverse Problems," Ph.D thesis, University of Oulu, 2002. [18] S. Lasanen, Non-Gaussian statistical inverse problems. Part I: Posterior distributions, Inverse Problems and Imaging, 6 (2012), 215-266. doi: 10.3934/ipi.2012.6.215. [19] S. Lasanen, Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns, Inverse Problems and Imaging, 6 (2012), 267-287. doi: 10.3934/ipi.2012.6.267. [20] M. Lassas, E. Saksman and S. Siltanen, Discretization invariant Bayesian inversion and Besov space priors, Inverse Problems and Imaging, 3 (2009), 87-122. doi: 10.3934/ipi.2009.3.87. [21] M. Lassas and S. Siltanen, Can one use total variation prior for edge-preserving Bayesian inversion?, Inverse Problems, 20 (2004), 1537-1563. doi: 10.1088/0266-5611/20/5/013. [22] M. S. Lehtinen, L. Päivärinta and E. Somersalo, Linear inverse problems for generalised random variables, Inverse Problems, 5 (1989), 599-612. doi: 10.1088/0266-5611/5/4/011. [23] F. Lindgren, H. Rue and J. Lindström, An explicit link between Gaussian Markov random fields: The stochastic partial differential equation approach, Journal of the Royal Statistical Society: Series B, 73 (2011), 423-498. doi: 10.1111/j.1467-9868.2011.00777.x. [24] M. Orispää and M. Lehtinen, Fortran linear inverse problem solver, Inverse Problems and Imaging, 4 (2010), 482-503. doi: 10.3934/ipi.2010.4.485. [25] P. Piiroinen, Statistical measurements, experiments and applications, Ann. Acad. Sci. Fenn. Math. Diss., (2005), 89 pp. [26] L. Roininen, M. Lehtinen, S. Lasanen, M. Orispää and M. Markkanen, Correlation priors, Inverse Problems and Imaging, 5 (2011), 167-184. doi: 10.3934/ipi.2011.5.167. [27] H. Rue and L. Held, "Gaussian Markov Random Fields: Theory and Applications," Chapman & Hall/CRC, 2005. doi: 10.1201/9780203492024. [28] D. W. Stroock and S. R. S. Varadhan, Diffusion processes with continuous coefficients. I, Comm. Pure Appl. Math., 22 (1969), 345-400. doi: 10.1002/cpa.3160220304. [29] D. W. Stroock and S. R. S. Varadhan, Diffusion processes with continuous coefficients. II, Comm. Pure Appl. Math., 22 (1969), 479-530. doi: 10.1002/cpa.3160220404. [30] D. W. Stroock and S. R. S. Varadhan, Diffusion processes with boundary conditions, Comm. Pure Appl. Math., 24 (1971), 147-225. doi: 10.1002/cpa.3160240206. [31] C. H. Su and D. Lucor, Covariance kernel representations of multidimensional second-order stochastic processes, J. Comp. Phys., 217 (2006), 82-99. doi: 10.1016/j.jcp.2006.02.006.
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