Apply Isometry property to compute the variance of the stochastic integral $\int_0^T B(t)dB(t)$.
Added by Nathaniel P.
Close
Step 1
The isometry property states that for a stochastic process X(t) and a deterministic function f(t), the variance of the stochastic integral \int_0^T f(t)X(t)dW(t) is given by E[\int_0^T [f(t)]^2 dt]. Show more…
Show all steps
Your feedback will help us improve your experience
Ameer Said and 80 other Calculus 3 educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Consider the estimator of the pmf in expression (4.1.10). In equation (4.1.11), we showed that this estimator is unbiased. Find the variance of the estimator and its mgf.
Some Elementary Statistical Inferences
Sampling and Statistics
Now consider an Ornstein-Uhlenbeck process X = (Xt)t≥0, defined by the stochastic differential equation dXt = -ΙXtdt + σdZt, where Z = (Zt)t≥0 is a standard Brownian motion under the probability measure P and λ > 0. (i) Given a starting value, X0 = 0, solve the stochastic differential equation.
Adi S.
Problem 5. The operator T : H → H is an isometry if ||Tf|| = ||f|| for all f ∈ H. (a) Please prove that if T is an isometry, then (Tf, Tg) = (f, g) for all f, g ∈ H. (b) Now prove that if T is an isometry, then T*T = I. (c) Now prove that if T is surjective and an isometry (and thus unitary), then TT* = I. (d) Give an example of an isometry T that is not unitary. Hint: consider l2(N) and the map which takes (a1, a2, . . .) to (0, a1, a2, . . .). (e) Now prove that if T*T is unitary, then T is an isometry. Hint: Start with ||Tf||^2 = (f, T*Tf) and use Holder's inequality to get ||Tf|| ≤ ||f||. Next, consider ||f|| = ||T*Tf|| to get the opposite inequality.
Madhur L.
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD