Need help for Ito Isometry proof

In summary, the conversation discusses the concept of Ito Isometry and its proof, specifically regarding the expectation of the square of the difference of two positions being equal to their time difference. The conversation also mentions Brownian motion and its normal distribution.
  • #1
Sa7oru
1
0
Hi,

There is one step in Ito Isometry proof I don't understand.

Bt(w) represents the position of w at time t, then E[(Btj+1(w)-Btj(w))2] = tj+1 - tj.

Why is the expectation of the square of the difference of 2 positions equal to their time difference?

Any hint, please.

Thank you.
 
Physics news on Phys.org
  • #2
Could you clarify what Bt(w) is & whether it's normally distributed?
 
  • #3
I'm guessing B_t(w) is Brownian motion, and yes it's normally distributed because its increments, B_t(w) - B_s(w) are normally distributed (~ N(0, t-s)) if s < t. Indeed, this is the precise property that gives the result the OP wants to understand.
 

1. What is Ito Isometry?

Ito Isometry is a mathematical concept used in probability theory and stochastic calculus. It states that the expected value of a function of a stochastic process at a certain time point is equal to the expected value of the same function of the process at a different time point, but with the time interval between the two points squared.

2. Why is Ito Isometry important?

Ito Isometry is important because it allows us to calculate expected values of functions of stochastic processes at different time points, which is useful in many applications such as finance, physics, and engineering.

3. What is the proof for Ito Isometry?

The proof for Ito Isometry involves using the properties of Ito integrals and stochastic processes, as well as the definition of expected value. It can be quite complex and involves advanced mathematical concepts, so it is best understood by those with a strong background in probability and stochastic calculus.

4. What are some examples of Ito Isometry in real life?

Ito Isometry can be applied in various real-life scenarios, such as modeling stock prices, predicting the movement of particles in physics, and analyzing the behavior of options in finance. It is also commonly used in time series analysis and forecasting.

5. Are there any limitations to Ito Isometry?

While Ito Isometry is a powerful tool in probability theory, it does have some limitations. It assumes that the stochastic process is continuous and has finite variation, which may not always be the case in real-world scenarios. It is important to carefully consider the assumptions and limitations when applying Ito Isometry in practical situations.

Similar threads

MHB Necessary and Sufficient Meaning of Isometries by D. J. H. Garling
    • Topology and Analysis
Replies
5
Views
2K
Zero K proof that a chess position contains a checkmate
    • Computing and Technology
Replies
2
Views
287
I Solving and manipulating the damped oscillator differential equation
    • Differential Equations
Replies
6
Views
2K
A Numerical Solution to Random Linear Non-Homogeneous ODE
    • Differential Equations
Replies
1
Views
990
Can a Function Have Two Different Tangent Lines at the Same Point?
    • Calculus and Beyond Homework Help
Replies
2
Views
754
A Proof of Classical Fluctuation-Dissipation Theorem
    • Classical Physics
Replies
1
Views
732
Direct Proof that every zero of p(T) is an eigenvalue of T
    • Calculus and Beyond Homework Help
Replies
24
Views
811
A Langevin equation - derivative of random force?
    • Classical Physics
Replies
8
Views
1K
B (Proof) Two right triangles are congruent.
    • General Math
Replies
2
Views
903
I On error estimates of approximate solutions
    • Differential Equations
Replies
1
Views
673

Hot Threads

Recent Insights

Back
Top