need help in understanding the following integration

[tapaspanda]

I was going through a presentation of Ito's process and I read the following

dx = a(x,t)dt

x(t) = x(0) + intgration from 0 to t (a(x,s)ds

What I do not understand is from where did 's' come and what is 's'? I think I have forgotten the basics of integration.

Thank you for your help.

 

[DrPhil]

I was going through a presentation of Ito's process and I read the following

dx = a(x,t)dt

x(t) = x(0) + intgration from 0 to t (a(x,s)ds

What I do not understand is from where did 's' come and what is 's'? I think I have forgotten the basics of integration.

Thank you for your help.

"s" is just a dummy variable name, so that "t" can be used as a limit of integration.

 

[tapaspanda]

Thank you for your reply. What is the need for using dummy variable? Sorry for the dumb question.

 

[DrPhil]

Thank you for your reply. What is the need for using dummy variable? Sorry for the dumb question.

You should never use the same name for two different things. As far as the integration is concerned, "t" is a constant, the upper limit of the integration. It can not also be the variable in the integrand. The integral is a function of s, which is then evaluated at s=t and at s=0.