Help for the stochastic differential equations

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SUMMARY

The discussion centers on solving the stochastic differential equation (SDE) dX = sqrt(X) dB, where X is a stochastic process and B represents Brownian motion. The user attempts to apply Ito's lemma but encounters difficulties with substitutions. The conversation emphasizes the importance of demonstrating prior work and understanding Ito's lemma and its assumptions. The existence and uniqueness theorem for Ito-diffusions suggests that a solution may exist under certain conditions.

PREREQUISITES
  • Understanding of stochastic differential equations (SDEs)
  • Familiarity with Brownian motion and its properties
  • Knowledge of Ito's lemma and its assumptions
  • Concept of existence and uniqueness theorems in stochastic processes
NEXT STEPS
  • Study the application of Ito's lemma in solving SDEs
  • Research the existence and uniqueness theorem for Ito-diffusions
  • Explore alternative substitution methods for SDEs
  • Learn about numerical methods for simulating solutions to SDEs
USEFUL FOR

Mathematicians, financial analysts, and researchers working with stochastic processes, particularly those interested in solving stochastic differential equations and applying Ito's lemma.

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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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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?
 
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 theorem for Ito-diffusion, it seems that the problem can be solve ?
because sqrt(X) <= C(1+|X|) for some certain C.

Thanks
 

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