Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-24T00:06:04.513Z Has data issue: false hasContentIssue false

Second Order Itô Processes

Published online by Cambridge University Press:  22 January 2016

Jerome A. Goldstein*
Affiliation:
The Institute for Advanced Study, Princeton, New Jersey
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A first order stochastic differential equation is any equation which can be expressed symbolically in the form

(1. 1)

m and σ are called the drift and diffusion coefficients and z( · ) is usually a Brownian motion process.

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1969

References

[1] Berman, S.M., “Oscillation of sample functions in diffusion processes”, Z. Warsch. verw. Geb. 1 (1963), 247250.Google Scholar
[2] Borchers, D.R., “Second order stochastic differential equations and related Itô processes”, Ph. D. Thesis, Carnegie Institute of Technology, 1964.Google Scholar
[3] Doob, J.L., “Martingales and one-dimensional diffusion”, Trans. Amer. Math. Soc. 78 (1955), 168208.Google Scholar
[4] Doob, J.L., Stochastic Processes, Wiley, New York, 1953.Google Scholar
[5] Dynkin, E.B., Markov Processes, Vol. 1, Academic Press, New York, 1965.Google Scholar
[6] Edwards, D.A. and Moyal, J.E., “Stochastic differential equations”, Proc. Cambridge Phil. Soc. 51 (1955), 663677.Google Scholar
[7] Fisk, D.L., “Quasi-martingales”, Trans. Amer. Math. Soc. 120 (1965), 369389.Google Scholar
[8] Fisk, D.L., Quasi-martingales and Stochastic Integrals, Department of Mathematics Research Monograph, Kent State University, Kent, Ohio, 1964.Google Scholar
[9] Fisk, D.L., “Sample quadratic variation of sample continuous, second order martingales”, Z. Warsch. verw. Geb. 6 (1966), 273278.Google Scholar
[10] Friedman, A., Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, New Jersey, 1964.Google Scholar
[11] Goldstein, J.A., “An existence theorem for linear stochastic differential equations”, J. Diff. Eqns. 3 (1967), 7887.Google Scholar
[12] Gray, A. H. and Caughey, T.K., “A controversy in problems involving random parametric excitation”, J. Math, and Phys. 44 (1965), 288296.Google Scholar
[13] Itô, K., “On a formula concerning stochastic differentials”, Nagoya Math. J. 3 (1951), 5565.Google Scholar
[14] Itô, K., “On stochastic differential equations”, Mem. Amer. Math. Soc, No. 4, 1951.Google Scholar
[15] Levy, P., Processes Stochastiques et Mouvement Brownien, Gauthier-Villars, Paris, 1948.Google Scholar
[16] Séminaire de Probabilities I, Université de Strasbourg, Lecture Notes in Mathematics, Vol. 39, Springer, Berlin, 1967.Google Scholar
[17] Skorokhod, A.V., “Existence and uniqueness of solutions to stochastic diffusion equations”, Sibirsk. Mat. Zh. 2 (1961), 129137 (in Russian). Translated in Selected Translations m Mathematical Statistics and Probability, Vol. 5, Providence, Rhode Island (1965), 191200.Google Scholar
[18] Skorokhod, A.V., Studies in the Theory of Random Processes, Addison-Wesley, Reading, Mass., 1965.Google Scholar
[19] Stratonovich, R.L., “A new representation for stochastic integrals and equations”, Vestnik Moskov Univ. Ser. I Mat. Meh., 1 (1964), 312 (in Russian). Translated in J. SIAM Control 4 (1966), 362371.Google Scholar
[20] Wang, J.-G., “Properties and limit theorems of sample functions of the Itô stochastic integrals”, Chinese Math.-Acta 5 (1964), 556570.Google Scholar
[21] Wong, E. and Zakai, M., “The oscillation of stochastic integrals”, Z. Warsch. verw. Geb. 4 (1965), 103112.Google Scholar