Help for the stochastic differential equations

In summary, the conversation is about solving the equation dX =sqrt(X) dB, where X is a process, B is a Brownian motion, and sqrt(X) is the square root of X. The poster is asking for help in finding a solution and mentions using Ito's lemma, but faces difficulty in finding a suitable substitution. They also mention the existence and uniqueness theorem for Ito-diffusions.
  • #1
ptc_scr
2
0
Hi,

Could some one help me to solve the equations ?
dX =sqrt(X) dB

where X is a process; B is a Brownian motion with B(0,w) =0;sqrt(X) is squart root of X.
 
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  • #2
ptc_scr said:
Hi,

Could some one help me to solve the equations ?
dX =sqrt(X) dB

where X is a process; B is a Brownian motion with B(0,w) =0;sqrt(X) is squart root of X.

Hey ptc_scr and welcome to the forums.

In these forums, we require the poster to show any work that they have done before we can help them. We do this so that you can actually learn for yourself what is going on so that you do the work and end up understanding it yourself.

So first I ask you to show any working, and secondly what do you know about solving SDE's with Brownian motion? Do you know about Ito's lemma and its assumptions?
 
  • #3
Hi,

I just try to assign Y=sqrt(X) and use Ito lemma to solve the problem. so
dY= 1/2 dB+ 1/(4Y) dt.

Obviously, we cannot put Y one left side. So the substitution is failed.
ANy one can show me how to find a good substitution or show me it is impossible to solve the problem ?
But for existence and uniqueness theorm for Ito-diffusion, it seems that the problem can be solve ?
because sqrt(X) <= C(1+|X|) for some certain C.

Thanks
 

FAQ: Help for the stochastic differential equations

What is a stochastic differential equation (SDE)?

A stochastic differential equation is a type of differential equation that involves both deterministic and random components. It is used to model systems that exhibit randomness, such as financial markets, population growth, and chemical reactions.

How is an SDE different from a regular differential equation?

An SDE incorporates random variables or processes, while a regular differential equation involves only deterministic functions. This means that the solution to an SDE will also have an element of randomness, whereas the solution to a regular differential equation will be deterministic.

What are some applications of SDEs?

SDEs are used in a wide range of fields, including finance, physics, biology, and engineering. They are commonly used to model and predict the behavior of complex systems that are affected by random factors.

What techniques are used to solve SDEs?

There are several techniques for solving SDEs, including numerical methods such as the Euler-Maruyama method, and analytical methods such as the Fokker-Planck equation. The choice of method depends on the specific characteristics of the SDE and the desired level of accuracy.

How can SDEs be used to simulate real-world systems?

SDEs can be used to simulate the behavior of real-world systems by incorporating random factors into the model. This allows for a more realistic representation of the system, and can help researchers better understand and predict its behavior under different conditions.

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