| Title:
|
A note on measurable modifications (English) |
| Author:
|
Ondreját, Martin |
| Author:
|
Seidler, Jan |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 (print) |
| ISSN:
|
1805-949X (online) |
| Volume:
|
62 |
| Issue:
|
1 |
| Year:
|
2026 |
| Pages:
|
7-17 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We present, with purely didactic aims, a simple and essentially self-contained proof of two necessary and sufficient conditions for existence of a measurable modification of a stochastic process with values in a separable complete metric space. Existence of a measurable modification of a stochastic process continuous in probability is an immediate consequence. (English) |
| Keyword:
|
stochastic processes |
| Keyword:
|
measurable modification |
| MSC:
|
60G05 |
| DOI:
|
10.14736/kyb-2026-1-0007 |
| . |
| Date available:
|
2026-03-03T15:18:48Z |
| Last updated:
|
2026-03-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153532 |
| . |
| Reference:
|
[1] Burke, M. R., Macheras, N. D., Strauss, W.: Measurability of stochastic processes on topological probability spaces..Topology Appl. 370 (2025), 109438, 37 pp. |
| Reference:
|
[2] Cohn, D. L.: Measurable choice of limit points and the existence of separable and measurable processes..Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 22 (1972), 161-165. |
| Reference:
|
[3] Chung, K. L., Doob, J. L.: Fields, optionality and measurability..Amer. J. Math. 87 (1965), 397-424. 10.2307/2373011 |
| Reference:
|
[4] Prato, G. Da, Zabczyk, J.: Stochastic Equations in Infinite Dimensions..Cambridge University Press, Cambridge 1992. |
| Reference:
|
[5] Dellacherie, C., Meyer, P.-A.: Probabilités et potentiel, Chapitres I à IV..Hermann, Paris 1975. |
| Reference:
|
[6] Doob, J. L.: Stochastic Processes..Wiley, New York 1953. |
| Reference:
|
[7] Fremlin, D. H.: Measure Theory, Vol. 2. Second edition..Torres Fremlin, Colchester 2010. |
| Reference:
|
[8] Geiss, H., Geiss, S.: Measure, Probability and Functional Analysis..Springer, Cham 2025. |
| Reference:
|
[9] Gikhman, I. I., Skorokhod, A. V.: Introduction to the Theory of Random Processes..W. B. Saunders Co., Philadelphia 1969. |
| Reference:
|
[10] Gihman, I. I., Skorohod, A. V.: The Theory of Stochastistic Processes I..Springer-Verlag, New York 1974. |
| Reference:
|
[11] He, S.-w., Wang, J.-g., Yan, J.-a.: Semimartingale Theory and Stochastic Calculus..CRC, Boca Raton 1992. |
| Reference:
|
[12] Hoffmann-Jørgensen, J.: Existence of measurable modifications of stochastic processes..Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 25 (1972/73), 205-207. |
| Reference:
|
[13] Hoffmann-Jørgensen, J.: Stochastic Processes on Polish Spaces..Various Publ. Ser. (Aarhus) 39, Aarhus Universitet, Matematisk Institut, Aarhus 1991. |
| Reference:
|
[14] Kawada, Y.: On the measurable stochastic processes..Proc. Phys.-Math. Soc. Japan (3) 23 (1941), 512-527. |
| Reference:
|
[15] Klenke, A.: Probability Theory. Third edition..Springer, Cham 2020. |
| Reference:
|
[16] Mansfield, R., Weitkamp, S. G.: Recursive Aspects of Descriptive Set Theory..Oxford University Press, New York 1985. |
| Reference:
|
[17] Meyer, P.-A.: Probability and Potentials..Blaisdell, Waltham 1966. |
| Reference:
|
[18] Neveu, J.: Mathematical Foundations of the Calculus of Probability..Holden-Day, San Francisco 1965. |
| Reference:
|
[19] Ondreját, M., Seidler, J.: On existence of progressively measurable modifications..Electron. Commun. Probab. 18 (2013), 20, 6 pp. |
| Reference:
|
[20] Schilling, R. L., Kühn, F.: Counterexamples in Measure and Integration..Cambridge University Press, Cambridge 2021. |
| Reference:
|
[21] Skorokhod, A. V.: On measurability of stochastic processes..Theory Probab. Appl. 25 (1980), 139-141. |
| Reference:
|
[22] Srivastava, S. M.: A Course on Borel Sets..Springer, New York 1998. |
| . |
