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!