Stochastic Differential Equations

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Discussion Overview

The discussion revolves around the properties and significance of stochastic differential equations (SDEs), particularly focusing on the noise term represented by stochastic processes, with an emphasis on Brownian motion. Participants explore the conditions that a stochastic process must satisfy and the implications of these conditions in the context of modeling physical phenomena.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant presents a stochastic differential equation and questions why the noise term must be represented by a process with continuous paths, specifically Brownian motion.
  • Another participant points out potential errors in the notation of the equation, seeking clarification on the meaning of certain symbols.
  • A later reply discusses the conditions that a stochastic process must satisfy, including independence, stationarity, and expectation, and questions why these conditions are necessary for the differential equation.
  • There is a mention of the limitations of other stochastic processes in satisfying the required conditions, with a specific reference to Brownian motion as the only process with continuous paths.
  • One participant expresses a desire for further reading on Brownian motion and its application in stochastic processes, indicating a need for more foundational understanding in the context of their research.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the necessity of the conditions for the stochastic process and the implications of using Brownian motion. There is no consensus on the significance of these conditions or the characterization of "reasonable" stochastic processes.

Contextual Notes

Limitations include the potential ambiguity in the definitions of "reasonable" stochastic processes and the implications of continuous paths, as well as the reliance on specific references that may not be fully explained within the discussion.

Who May Find This Useful

Readers interested in stochastic processes, differential equations, and their applications in physics and mathematics may find this discussion relevant.

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If we have a DE of the following form:

\frac{dX}{dt}=b(t,X_t)+\sigma(t,X_t).W_t

and look for a stochastic process to represent the (second) noise term. Now my textbook tells me that the only process with 'continuous paths' is Brownian motion.

The noise term denotes random, indeterministic behaviour in the physical situation the DE is modelling.

Can someone please explain why is this is the case, and why it is significant?
 
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What is N? Your parentheses, starting between b and t are not closed. And what is the dot between \sigma(t. X_{t}) and noise supposed to denote?
 
Dickfore said:
What is N? Your parentheses, starting between b and t are not closed. And what is the dot between \sigma(t. X_{t}) and noise supposed to denote?

I do apologise for my sloppiness. I've edited the post.

Now Wt has to satisfy the following conditions:

(i) t1 does not = t2 => Wt1 and Wt2 are independent.
(ii) {Wt} is stationary, i.e. the (joint) distribution of {Wt1+t,...,Wtk+t} does not depend on t.
(iii) E[Wt] = 0 for all t.

Now, my textbook says that "it turns out there does not exist any "reasonable" stochastic process satisfying (i) and (ii): Such a Wt cannot have continuous paths... The only such process with continuous paths is the Brownian motion Bt." And it simply gives an obscure reference "(See Knight (1981))"...

My question is firstly, why does the DE have to satisfy those three conditions? What is a reasonable stochastic process in this context, and why can't it have a continuous path? And why does Brownian motion in particular have a continuous path?
 
I have read that Brownian movement is used in certain stochastic processes, but I need further reading to be actually able to put some math into it. My research involves a stochastic process as well, but is an area of physics that so far I haven't been able to fully digest properly. Anyone who knows more should share for the enlightenment of all who are on the same situation, I say.
 

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