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Stieltjes integral

  1. Jul 28, 2010 #1
    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!

  2. jcsd
  3. Jul 28, 2010 #2


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    Science Advisor

    Your equations are a little confusing. You have F as both the integrand and the integral of the Stieltjes integral.
  4. Jul 28, 2010 #3
    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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