Download PROBLEMS WAS (ISA) – 2 1. Prove that in the space X

PROBLEMS WAS (ISA) – 2
1. Prove that in the space X = C = C([0, 1], R) with metric ||x − y|| = supt∈[0,1] |x(t) − y(t)| we
have σ(C) = B(C), where C denotes the set of cylinders in C.
Hint: C is separable.
2. Show that in the space X = RR+ the sets {x ∈ X : x is non − decreasing}, {x ∈ X : sup |x(t)| ≤
1} do not belong to σ(C).
Hint: Prove first that if A ∈ σ(C), then there exists a set Z ⊂ R+ at most countable, such that
from the fact that x ∈ A and ∀t∈Z x(t) = y(t) it follows that y ∈ A.
3. (Wt )t∈R+ is a Wiener process.
a) Define W 0 = 0, W t = tW1/t for t > 0. Prove that W is a Wiener process (time inversion).
b) Prove that limt→∞ Wt t = 0 a.s.
Hint for a): The event {limt&0 W t = 0} has the form {W ∈ Γ}, where Γ ⊂ σ(C) in the space
of continuous functions on (0, ∞).
4. T is an arbitrary non-empty set, and for each t ∈ T , µt is a probability measure on R.
Prove that there exists a probability space and a random function (Xt)t∈T such that Xt has
distribution µt , t ∈ T , and the random variables {Xt}t∈T are independent.
5. a)Show that the paths of the Wiener process W satisfy locally (i.e., on each interval [0, a]) the
Hölder condition with any exponent α ∈ (0, 1/2).
b)Prove that the paths of W do not satisfy the Hölder condition with exponent α = 1/2; more
precisely,
P (∃[a,b]⊂R+ W. satisfies the Hölder cond. with exponent 1/2 on [a, b]) = 0.
Hint for b): For a fixed interval [a, b] define tn,k = a + (b − a)k/n, k = 0, 1, . . ., n, and consider
Wtn,k+1 − Wtn,k .