The Engelbert–Schmidt zero–one law is a theorem that gives a mathematical criterion for an event associated with a continuous, non-decreasing additive functional of Brownian motion to have probability either 0 or 1, without the possibility of an intermediate value. This zero-one law is used in the study of questions of finiteness and asymptotic behavior for stochastic differential equations.[1] (A Wiener process is a mathematical formalization of Brownian motion used in the statement of the theorem.) This 0-1 law, published in 1981, is named after Hans-Jürgen Engelbert[2] and the probabilist Wolfgang Schmidt[3] (not to be confused with the number theorist Wolfgang M. Schmidt).
Engelbert–Schmidt 0–1 law
[edit]Let be a σ-algebra and let be an increasing family of sub-σ-algebras of . Let be a Wiener process on the probability space . Suppose that is a Borel measurable function of the real line into [0,∞]. Then the following three assertions are equivalent:
(i) .
(ii) .
(iii) for all compact subsets of the real line.[4]
Extension to stable processes
[edit]In 1997 Pio Andrea Zanzotto proved the following extension of the Engelbert–Schmidt zero-one law. It contains Engelbert and Schmidt's result as a special case, since the Wiener process is a real-valued stable process of index .
Let be a -valued stable process of index on the filtered probability space . Suppose that is a Borel measurable function. Then the following three assertions are equivalent:
(i) .
(ii) .
(iii) for all compact subsets of the real line.[5]
The proof of Zanzotto's result is almost identical to that of the Engelbert–Schmidt zero-one law. The key object in the proof is the local time process associated with stable processes of index , which is known to be jointly continuous.[6]
See also
[edit]References
[edit]- ^ Karatzas, Ioannis; Shreve, Steven (2012). Brownian motion and stochastic calculus. Springer. p. 215.
- ^ Hans-Jürgen Engelbert at the Mathematics Genealogy Project
- ^ Wolfgang Schmidt at the Mathematics Genealogy Project
- ^ Engelbert, H. J.; Schmidt, W. (1981). "On the behavior of certain functionals of the Wiener process and applications to stochastic differential equations". In Arató, M.; Vermes, D.; Balakrishnan, A. V. (eds.). Stochastic Differential Systems. Lectures Notes in Control and Information Sciences, vol. 36. Berlin; Heidelberg: Springer. pp. 47–55. doi:10.1007/BFb0006406.
- ^ Zanzotto, P. A. (1997). "On solutions of one-dimensional stochastic differential equations driven by stable Lévy motion" (PDF). Stochastic Processes and their Applications. 68: 209–228. doi:10.1214/aop/1023481008.
- ^ Bertoin, J. (1996). Lévy Processes, Theorems V.1, V.15. Cambridge University Press.