Solving Left-Continuity of Stieltjes Integral

[wayneckm]

Hello all,

I have the following question:

If F,G are right-continuouse functions, and if I define F(t) = \int_{(0,t]}F(u-)dG(u), then is F(t) here a left-continuous function of t since both F,G cannot "jump together", so we eliminate the term \sum \Delta F \Delta G?

Or is it correct to say F(t-) = \int_{(0,t]}F(u-)dG(u)? If all are wrong, would anyone kindly explain and provide some references?

Thanks very much!

Wayne

 

[mathman]

Your equations are a little confusing. You have F as both the integrand and the integral of the Stieltjes integral.

 

[wayneckm]

Indeed in my reference it said F can be represented in this form of integral.

Or I should ask if both F,G are right-continuous, is the integral \int_{(0,t]}F(u-)dG(u) left-continuous? And how to justify formally?

Thanks very much!