Engelbert–Schmidt zero–one law explained

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 and the probabilist Wolfgang Schmidt (not to be confused with the number theorist Wolfgang M. Schmidt).

Engelbert–Schmidt 0–1 law

Let

l{F} be a σ-algebra and let F = ( l{F} _{t)} _{t} be an increasing family of sub-σ-algebras of l{F} . Let (W, F) be a Wiener process on the probability space (\Omega, l{F}, P) .Suppose that f is a Borel measurable function of the real line into [0,∞].Then the following three assertions are equivalent:

(i)

P ( \int ^{t} _{0} f (W _{s)ds<} inf ty forall t \ge 0 ) > 0 .

(ii)

P ( \int ^{t} _{0} f (W _{s)ds<} inf ty forall t \ge 0 ) = 1 .

(iii)

\int _{K} f (y) dy < inf ty for all compact subsets K of the real line.^[2]

Extension to stable processes

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

\alpha = 2 .

Let

X be a R -valued stable process of index \alpha\in( 1 , 2 ] on the filtered probability space (\Omega, l{F}, ( l{F} _{t),} P) .Suppose that f: R \to [ 0 , inf ty] is a Borel measurable function.Then the following three assertions are equivalent:

(i)

P ( \int ^{t} _{0} f (X _{s)ds<} inf ty forall t \ge 0 ) > 0 .

(ii)

P ( \int ^{t} _{0} f (X _{s)ds<} inf ty forall t \ge 0 ) = 1 .

(iii)

\int _{K} f (y) dy < inf ty for all compact subsets K of the real line.^[3]

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

\alpha\in( 1 , 2 ] , which is known to be jointly continuous.^[4]

See also

• zero-one law

Notes and References

1. Book: Karatzas, Ioannis. Shreve, Steven. Brownian motion and stochastic calculus. Springer. 2012. 215.
2. Book: Engelbert, H. J.. Schmidt, W.. On the behavior of certain functionals of the Wiener process and applications to stochastic differential equations. Arató, M.. Vermes, D.. Balakrishnan, A. V.. Stochastic Differential Systems. Lectures Notes in Control and Information Sciences, vol. 36. 1981. Springer. Berlin; Heidelberg. 47–55. 10.1007/BFb0006406.
3. Zanzotto, P. A.. On solutions of one-dimensional stochastic differential equations driven by stable Lévy motion. 1997. Stochastic Processes and their Applications. 68. 209–228. 10.1214/aop/1023481008.
4. Book: Bertoin, J.. Lévy Processes, Theorems V.1, V.15. 1996. Cambridge University Press.