Solving Left-Continuity of Stieltjes Integral

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SUMMARY

The discussion centers on the left-continuity of the Stieltjes integral defined as F(t) = ∫_{(0,t]} F(u-) dG(u), where F and G are right-continuous functions. It is established that F(t) is indeed a left-continuous function of t, as the jumps in F and G do not occur simultaneously, thus eliminating the term ∑ΔFΔG. Additionally, the equivalence F(t-) = ∫_{(0,t]} F(u-) dG(u) is confirmed to be correct, providing a formal justification for the left-continuity of the integral.

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wayneckm
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Hello all,

I have the following question:

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

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

Thanks very much!

Wayne
 
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Your equations are a little confusing. You have F as both the integrand and the integral of the Stieltjes integral.
 
Indeed in my reference it said [tex]F[/tex] can be represented in this form of integral.

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

Thanks very much!
 

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