Solving Left-Continuity of Stieltjes Integral

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In summary, the question being asked is if F and G are both right-continuous functions and F(t) is defined as the Stieltjes integral of F(u-)dG(u), is F(t) also a left-continuous function of t? The speaker is unsure due to confusion with the equations and is seeking clarification and references.
  • #1
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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  • #2
Your equations are a little confusing. You have F as both the integrand and the integral of the Stieltjes integral.
 
  • #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!
 

FAQ: Solving Left-Continuity of Stieltjes Integral

1. What is the Stieltjes integral and why is it important in mathematics?

The Stieltjes integral, also known as the Riemann-Stieltjes integral, is a generalization of the Riemann integral that allows for integration with respect to a non-decreasing function. It is important in mathematics because it provides a way to integrate more complicated functions and has applications in fields such as probability, physics, and economics.

2. How is left-continuity defined in relation to the Stieltjes integral?

Left-continuity refers to the property of a function where the limit from the left of a point is equal to the function value at that point. In the context of the Stieltjes integral, a function is considered left-continuous if the limit of the integral from the left is equal to the integral value at a point.

3. Why is solving for left-continuity of the Stieltjes integral important?

Solving for left-continuity of the Stieltjes integral is important because it ensures the validity of the integral. If a function is not left-continuous, the integral may not exist or may produce incorrect results. Therefore, it is necessary to determine left-continuity in order to properly evaluate the integral.

4. What are some common methods for solving left-continuity of the Stieltjes integral?

There are several methods for solving left-continuity of the Stieltjes integral, including using the definition of left-continuity, the properties of monotonic functions, and the Fundamental Theorem of Calculus. These methods involve evaluating limits and determining the behavior of the function at a given point.

5. Are there any practical applications for solving left-continuity of the Stieltjes integral?

Yes, there are several practical applications for solving left-continuity of the Stieltjes integral. For example, it is used in finance to evaluate financial derivatives, such as options and futures, which involve non-linear payoffs. It is also used in probability theory to calculate expected values and in physics to determine the work done by a force acting over a distance.

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