Solution: Integration by Parts for Stieltjes Integrals

Exercise: Integration by Parts for Stieltjes Integrals

Solution

First, $\int_0^1 x\,d(x^2)=\int_0^1 2x^2\,dx=2/3$. Second, $\int_0^1 x^2\,d(x)=\int_0^1 x^2\,dx=1/3$. Their sum is $1$.

The right side is $f(1)g(1)-f(0)g(0)=1\cdot1-0\cdot0=1$.

Takeaways

• Stieltjes integration has its own integration-by-parts identity.
• For smooth functions it matches the ordinary calculus result.
• Stochastic calculus modifies this identity with a quadratic-variation term.