The Existence and Uniqueness of Random Solution to Itô Stochastic Integral Equation

Affiliation(s)

School of Mathematics and Information Science, Northwest Normal University, Lanzhou, Gansu 730070, People’s Republic of China.

School of Mathematics and Information Science, Northwest Normal University, Lanzhou, Gansu 730070, People’s Republic of China.

ABSTRACT

The objective of this paper is to attempt to apply the theoretical techniques of probabilistic functional analysis to answer the question of existence and Uniqueness of a Random Solution to It? Stochastic Integral Equation. Another type of stochastic integral equation which has been of considerable importance to applied mathematicians and engineers is that involving the It? or It?-Doob form of stochastic integrals.

The objective of this paper is to attempt to apply the theoretical techniques of probabilistic functional analysis to answer the question of existence and Uniqueness of a Random Solution to It? Stochastic Integral Equation. Another type of stochastic integral equation which has been of considerable importance to applied mathematicians and engineers is that involving the It? or It?-Doob form of stochastic integrals.

Cite this paper

H. Alafif and C. Wang, "The Existence and Uniqueness of Random Solution to Itô Stochastic Integral Equation,"*Applied Mathematics*, Vol. 3 No. 7, 2012, pp. 800-804. doi: 10.4236/am.2012.37119.

H. Alafif and C. Wang, "The Existence and Uniqueness of Random Solution to Itô Stochastic Integral Equation,"

References

[1] K. Ito, “Stochastic Integral,” Proceedings of the Imperial Academy, Vol. 20, No. 8, 1944, pp. 519-524. doi:10.3792/pia/1195572786

[2] J. L. Doob, “Stochastic Processes,” Wiley, New York, 1953, pp. 426-432.

[3] Y. Dynkin, “Markov Processes,” Academic Press, New York, 1964, pp. 9-13.

[4] A. Jazwinski, “Stochastic Processes and Filtering Theory. Mathematics in Science and Engineering,” Vol. 64, Academic Press, New York, 1970, pp. 97-105

[5] K. Ito, “On a Stochastic Integral Equation,” Proceedings of the Japan Academy, Vol. 22, No. 2, 1946, pp. 32-35. doi:10.3792/pja/1195572371

[6] H. P. Mckean, “Stochastic Integrals,” Academic Press, New York, 1969, pp. 21-25.

[7] T. L. Satty, “Modern Nonlinear Equations,” McGrowHill, New York, 1967, pp. 216-226.

[8] L. Gikhmann and A. V. Skorokhod, “Introduction to the Theory of Random Process-Saunders,” Philadehphia, Pennsylvania, 1969, pp. 378-391.

[9] R. L. Stratonovich, “A New Representation for Stochastic Integrals and Equations,” Journal of SLAM Control, Vol. 4, 1966, pp. 362-371.

[10] E. Wong and M. Zakai, “On the Relation between Ordinary and Stochastic Differential Equations,” International Journal of Engineering Science, Vol. 3, No. 2, 1965, pp. 213-229. doi:10.1016/0020-7225(65)90045-5

[11] I. P. Natanson, “Theory of Functions of a Real Variable,” Vol. II, Ungar, New York, 2010.

[12] A. T. Bharucha-Reid, “On the Theory of Random Equations,” Proceedings of Symposia in Applied Mathematics, Vol. 16, 1964, pp. 40-69.

[13] G. Adomain, “Random Operator Equations in Mathematical Physics,” Journal of Mathematical Physics, Vol. 11, No. 3, 1970, pp. 1069-1074. doi:10.1063/1.1665198

[14] G. Adomain, “Linear Random Operator Equations in Mathematical Physics III,” Journal of Mathematical Physics, Vol. 12, No. 9, 1971, pp. 1944-1948. doi:10.1063/1.1665827

[15] G. Adomain, “Theory of Random Systems,” Transactions of the fourth Prague Conference on Information Theory, Statistical Decision Functions, Random Processes, Prague, 31 August-11 September 1965, pp. 205-222.

[1] K. Ito, “Stochastic Integral,” Proceedings of the Imperial Academy, Vol. 20, No. 8, 1944, pp. 519-524. doi:10.3792/pia/1195572786

[2] J. L. Doob, “Stochastic Processes,” Wiley, New York, 1953, pp. 426-432.

[3] Y. Dynkin, “Markov Processes,” Academic Press, New York, 1964, pp. 9-13.

[4] A. Jazwinski, “Stochastic Processes and Filtering Theory. Mathematics in Science and Engineering,” Vol. 64, Academic Press, New York, 1970, pp. 97-105

[5] K. Ito, “On a Stochastic Integral Equation,” Proceedings of the Japan Academy, Vol. 22, No. 2, 1946, pp. 32-35. doi:10.3792/pja/1195572371

[6] H. P. Mckean, “Stochastic Integrals,” Academic Press, New York, 1969, pp. 21-25.

[7] T. L. Satty, “Modern Nonlinear Equations,” McGrowHill, New York, 1967, pp. 216-226.

[8] L. Gikhmann and A. V. Skorokhod, “Introduction to the Theory of Random Process-Saunders,” Philadehphia, Pennsylvania, 1969, pp. 378-391.

[9] R. L. Stratonovich, “A New Representation for Stochastic Integrals and Equations,” Journal of SLAM Control, Vol. 4, 1966, pp. 362-371.

[10] E. Wong and M. Zakai, “On the Relation between Ordinary and Stochastic Differential Equations,” International Journal of Engineering Science, Vol. 3, No. 2, 1965, pp. 213-229. doi:10.1016/0020-7225(65)90045-5

[11] I. P. Natanson, “Theory of Functions of a Real Variable,” Vol. II, Ungar, New York, 2010.

[12] A. T. Bharucha-Reid, “On the Theory of Random Equations,” Proceedings of Symposia in Applied Mathematics, Vol. 16, 1964, pp. 40-69.

[13] G. Adomain, “Random Operator Equations in Mathematical Physics,” Journal of Mathematical Physics, Vol. 11, No. 3, 1970, pp. 1069-1074. doi:10.1063/1.1665198

[14] G. Adomain, “Linear Random Operator Equations in Mathematical Physics III,” Journal of Mathematical Physics, Vol. 12, No. 9, 1971, pp. 1944-1948. doi:10.1063/1.1665827

[15] G. Adomain, “Theory of Random Systems,” Transactions of the fourth Prague Conference on Information Theory, Statistical Decision Functions, Random Processes, Prague, 31 August-11 September 1965, pp. 205-222.