Proving a process is Brownian Motion

In summary, the conversation discusses the process \{X(t)\}_{t\geq 0}, where X(t)=\rho B_1(t)+\sqrt{1-\rho^2}B_2(t), and determines whether it is a standard Brownian Motion. The process is shown to have a mean of 0 and a variance of t_2-t_1. The question is raised of whether X(t_3)-X(t_2) and X(t_2)-X(t_1) are independent, which can be proven by showing that A+B and C+D are independent for any measurable functions f(.,.) and g(.,.).
  • #1
IniquiTrance
190
0

Homework Statement


Is the process [itex]\{X(t)\}_{t\geq 0}[/itex], where [itex]X(t)=\rho B_1(t)+\sqrt{1-\rho^2}B_2(t)[/itex] Standard Brownian Motion?

Where [itex]\rho\in(0,1), \ B_1(t)[/itex] and [itex]B_2(t) [/itex] are independent standard brownian motions

Homework Equations


The Attempt at a Solution



Obviously [itex]X(0)=0[/itex]. Now let [itex]0\leq t_1<t_2<t_3[/itex]. Then, I can show that [itex]X(t_2)-X(t_1)\sim\mathcal{N}(0, t_2-t_1)[/itex]. My problem is showing that [itex]X(t_3)-X(t_2)[/itex] and [itex]X(t_2)-X(t_1)[/itex] are independent. I can show their covariance is 0, but that only implies independence if the process is Gaussian, which I have not shown. Any help would be much appreciated!
 
Physics news on Phys.org
  • #2
IniquiTrance said:

Homework Statement


Is the process [itex]\{X(t)\}_{t\geq 0}[/itex], where [itex]X(t)=\rho B_1(t)+\sqrt{1-\rho^2}B_2(t)[/itex] Standard Brownian Motion?

Where [itex]\rho\in(0,1), \ B_1(t)[/itex] and [itex]B_2(t) [/itex] are independent standard brownian motions


Homework Equations





The Attempt at a Solution



Obviously [itex]X(0)=0[/itex]. Now let [itex]0\leq t_1<t_2<t_3[/itex]. Then, I can show that [itex]X(t_2)-X(t_1)\sim\mathcal{N}(0, t_2-t_1)[/itex]. My problem is showing that [itex]X(t_3)-X(t_2)[/itex] and [itex]X(t_2)-X(t_1)[/itex] are independent. I can show their covariance is 0, but that only implies independence if the process is Gaussian, which I have not shown. Any help would be much appreciated!

For i = 1,2 and j = 1,2, are the ##B_i(t_3)-B_i(t_2)## independent of the ##B_j(t_2)-B_j(t_1)##?
 
  • #3
Ray Vickson said:
For i = 1,2 and j = 1,2, are the ##B_i(t_3)-B_i(t_2)## independent of the ##B_j(t_2)-B_j(t_1)##?

Hi Ray, yes they are.
 
  • #4
Not sure how to proceed.
 
  • #5
Are you implying that if [itex]A\perp B\perp C \perp D[/itex], then [itex]A+B \perp C+D[/itex], where [itex]\perp[/itex] means independent?
 
  • #6
IniquiTrance said:
Are you implying that if [itex]A\perp B\perp C \perp D[/itex], then [itex]A+B \perp C+D[/itex], where [itex]\perp[/itex] means independent?

More than that: if [itex]A\perp B\perp C \perp D[/itex], then [itex] f(A,B) \perp g(C,D)[/itex] for any (measurable) functions f(.,.) and g(.,.).
 
  • #7
Ray Vickson said:
More than that: if [itex]A\perp B\perp C \perp D[/itex], then [itex] f(A,B) \perp g(C,D)[/itex] for any (measurable) functions f(.,.) and g(.,.).

Thanks. How can I go about proving that?
 
  • #8
IniquiTrance said:
Thanks. How can I go about proving that?

It depends on what you know already. Anyway, all you need is the result for A+B and C+D, and you can use any number of standard tools, but first you need to know what those tools are. Do you know about characteristic functions? What can you say about the density (or cdf) of a set of independent random variables? How can you test whether two random variables (A+B and C+D in this case) are independent?

All of these things can be found in books and articles---many freely available on-line---so I will leave it to you to search for the answers if you do not know them already. If you do know them already, just think about how you would use them.
 

1. What is Brownian Motion?

Brownian Motion is a physical phenomenon that describes the random movement of particles in a fluid medium, caused by collisions with other molecules in the fluid.

2. How can you prove that a process is Brownian Motion?

To prove that a process is Brownian Motion, we need to observe the movement of particles in a fluid and show that it follows certain characteristics, such as random movement, small particle size, and the absence of external forces.

3. What are the characteristics of Brownian Motion?

The characteristics of Brownian Motion include random movement, small particle size, the absence of external forces, and the particles' trajectory being independent of each other.

4. Can Brownian Motion occur in different mediums?

Yes, Brownian Motion can occur in different mediums such as air, water, and other fluids. It is a universal phenomenon that can be observed in any medium with small enough particles.

5. How is Brownian Motion relevant in scientific research?

Brownian Motion is relevant in scientific research as it provides evidence for the existence of atoms and molecules, and it is used to study diffusion, viscosity, and other physical properties of fluids. It also has applications in fields such as biology, chemistry, and finance.

Similar threads

  • Calculus and Beyond Homework Help
Replies
5
Views
141
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
932
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
17
Views
905
  • Calculus and Beyond Homework Help
Replies
2
Views
723
  • Differential Equations
Replies
5
Views
608
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
Back
Top