Brownian motion integration/calculus

In summary, the integral is adding up the instantaneous changes in the function over the interval [t_1,t_2], and the summation of these infinitesimal changes is equal to the function at t_2-t_1.
  • #1
operationsres
103
0
We all know that [itex]\int_0^t dB(s) = B(t)[/itex], where [itex]B(t)[/itex] is a standard Brownian Motion. However, is this identity true?

[itex]\int_{t_1}^{t_2} dB(s) = B(t_2) - B(t_1)[/itex]
 
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  • #2
I only have a passing familiarity with Brownian motion. I assume that B(t) is a Lipschitz function. (Any real particle must have a Lipschitz path or else it is traveling faster than the speed of light).

Lipschitz functions are absolutely continuous, therefore the fundamental theorem of calculus holds as you have written it. So yes, it is true.

EDIT: Oops. I had doubts and looked it up on Wikipedia. it seems that B(t) is almost surely continuous everywhere but nowhere differentiable. Apparently I was way off base.
 
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  • #3
So your answer is "not sure"? :p
 
  • #4
operationsres said:
We all know that [itex]\int_0^t dB(s) = B(t)[/itex], where [itex]B(t)[/itex] is a standard Brownian Motion. However, is this identity true?

[itex]\int_{t_1}^{t_2} dB(s) = B(t_2) - B(t_1)[/itex]

The short answer is YES, at least if t1 and t2 are non-random, or even if they are random but "unrelated" to {B(t)}.

The longer answer (or, rather, a series of questions) is: do you know what is meant by
[tex]\int_{t_1}^{t_2} dB(s)?[/tex] (Here, I am asking for an "intuitive" understanding of the concepts, not a 100% iron-clad presentation with all the proofs.) If you do not understand the meaning of the stochastic integral, then you have a lot of preliminary work to do, reading and absorbing the basics. If you do understand what it means, then just write it all out in detail and see what you get.

RGV
 
  • #5
Ray Vickson said:
The longer answer (or, rather, a series of questions) is: do you know what is meant by
[tex]\int_{t_1}^{t_2} dB(s)?[/tex] (Here, I am asking for an "intuitive" understanding of the concepts, not a 100% iron-clad presentation with all the proofs.) If you do not understand the meaning of the stochastic integral, then you have a lot of preliminary work to do, reading and absorbing the basics. If you do understand what it means, then just write it all out in detail and see what you get.

RGV

I don't really understand what's meant when we integrate with respect to a Wiener process. I was never taught this.I know all of the properties of [itex]B(t)[/itex] though.

My qualitative, uneducated guess as to what that integral is doing is adding up all the instantaneous changes in [itex]B(t)[/itex] over the interval [itex][t_1,t_2][/itex], and the summation of these infinitesimal changes is equal to [itex]B(t_2)-B(t_1) \overset{d}{=} B(t_2-t_1) \sim \mathscr{N}(0,t_2-t_1)[/itex]. I can't really visualize this because of the fractal like property of [itex]B(t)[/itex]. I guess I would have to really break down how [itex]B(t)[/itex] is constructed to understand.

This is less confusing than something like [itex]\displaystyle \ \ \int_0^t r(s) dB(s)[/itex], where r(s) is stochastic OR deterministic. What the heck does that integral mean ... intuitively?
 

1. What is Brownian motion integration/calculus?

Brownian motion integration/calculus is a mathematical tool used to describe the random movement of particles in a fluid or gas. It is also known as stochastic calculus and is used in various fields such as physics, biology, finance, and engineering.

2. How is Brownian motion integration/calculus used in science?

Brownian motion integration/calculus is used to model the behavior of particles in a fluid or gas that are subject to random forces. It is used to analyze and predict the movement of particles, such as molecules in a liquid, and can provide insights into various physical and chemical processes.

3. What is the difference between Brownian motion integration and regular calculus?

Regular calculus deals with deterministic functions and equations, while Brownian motion integration/calculus deals with stochastic or random processes. In regular calculus, the behavior of a system can be predicted with certainty, whereas in Brownian motion integration/calculus, there is an element of randomness and uncertainty.

4. What are the applications of Brownian motion integration/calculus?

Brownian motion integration/calculus has a wide range of applications in science and engineering. It is used to study diffusion processes, chemical reactions, and the movement of particles in fluids. It is also used in finance to model stock prices and in biology to study the movement of cells and microorganisms.

5. Is Brownian motion integration/calculus difficult to understand?

Brownian motion integration/calculus can be challenging to understand, as it involves complex mathematical concepts such as stochastic processes and probability theory. However, with proper study and practice, it can be comprehended and applied successfully in various scientific and engineering fields.

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