Stochastic Differential Equations

In summary, a reasonable stochastic process that satisfies the three conditions given by the textbook is the Brownian motion.
  • #1
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If we have a DE of the following form:

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

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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  • #2
What is N? Your parentheses, starting between b and t are not closed. And what is the dot between [itex]\sigma(t. X_{t})[/itex] and noise supposed to denote?
 
  • #3
Dickfore said:
What is N? Your parentheses, starting between b and t are not closed. And what is the dot between [itex]\sigma(t. X_{t})[/itex] 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?
 
  • #4
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.
 

What are Stochastic Differential Equations?

Stochastic Differential Equations (SDEs) are mathematical equations used to model the evolution of a system that is subject to random fluctuations. They combine concepts from ordinary differential equations (ODEs) and probability theory, allowing for the incorporation of randomness into the dynamics of a system.

What are the applications of Stochastic Differential Equations?

SDEs are used in various fields such as physics, biology, economics, and finance to model complex systems that are influenced by random factors. They are also used in engineering and control theory to design systems that can adapt to uncertain environments.

What is the difference between a Stochastic Differential Equation and an Ordinary Differential Equation?

The main difference between SDEs and ODEs is the presence of a stochastic term in SDEs, which represents the random fluctuations in the system. This stochastic term makes the solution of SDEs a random process, whereas the solution of ODEs is a deterministic function.

How are Stochastic Differential Equations solved?

There is no general method for solving SDEs analytically. However, numerical methods such as Euler-Maruyama method and Milstein method can be used to approximate the solution of an SDE. Monte Carlo simulations can also be used to obtain numerical solutions.

What are the limitations of Stochastic Differential Equations?

SDEs assume that the random fluctuations in a system follow a specific probability distribution and are independent from each other. This may not always be the case in real-world systems, leading to limitations in the accuracy of SDE models. Additionally, SDEs can be computationally expensive to solve and may require simplifications and assumptions to make the problem tractable.

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