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Hi

On page 98 "Stochastic differential equations" of Øksendal, 6th edition,

the author writes that $$\int_{0}^{u}\Big(\int_{0}^{t}\frac{\partial}{\partial t}f(s,t)dR_{s}\Big)dt=\int_{0}^{u}\Big(\int_{s}^{u}\frac{\partial}{\partial t}f(s,t)dt\Big)dR_{s}$$

could some one please tell me why? thanks so much for your time !

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  • $\begingroup$ It will probably help your cause to provide some definitions and/or context -- what ideas have you tried on your own? $\endgroup$
    – Cam McLeman
    Commented May 13, 2010 at 22:18
  • $\begingroup$ Indeed! That equation can go from trivial to very deep depending on the intended meaning of the symbols... $\endgroup$
    – Mariano Suárez-Álvarez
    Commented May 13, 2010 at 22:20
  • $\begingroup$ it is too long to write down all definitions, hope some one already read the Oksendal's book and give me a hand. Thanks $\endgroup$
    – Steven
    Commented May 13, 2010 at 22:29
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    $\begingroup$ This has nothing to do with stochastic differential equations. It's an issue in multivariable calculus: Oksendal is just switching the order of integration (on the left, the inside is w.r.t. s and outside is w.r.t. t, while on the right the inside is w.r.t. t and the outside is w.r.t. s). How the bounds of integration change is a standard technique in double integrals. Please review your multivariable calculus. Look at examples and pictures in any book covering double integrals. When you know how to change the order of integration in a double integral you'll have answered your own question. $\endgroup$
    – KConrad
    Commented May 13, 2010 at 22:37
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    $\begingroup$ Steven, it is really preferred that you give context and meaning to your questions. Please think about how this looks from the viewpoint of those on the site: you show up and just throw an equation on the page and ask what it is all about. Imagine a stranger came into your office, wrote some expression on the blackboard, and then looked at you and said "What is this?" That's how these Oksendal questions appear to the rest of us. $\endgroup$
    – KConrad
    Commented May 13, 2010 at 22:41

2 Answers 2

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i am also having some problems with oksendal sometimes but here i can help u: Its basically Fubini theorem where one can change the order of integration: on the left the inner integrand is being integrated over 0<=s<=t and outer integral over 0<=t<=u. When u combine these two inequalities you get 0<=s<=t<=u. So on the right u can now have the inner integral wrt t as s<=t<=u and the outer over 0<=s<=u. hope that helps.

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As an exercise, prove a Fubini theorem for Itó and Lebesgue integrals: for any $F \in L^2([0,u]^2)$ we have $$\int_0^u \int_0^u F(s,t)\;dR_s\;dt = \int_0^u \int_0^u F(s,t)\;dt\;dR_s.$$ (Hint: start with elementary functions.) Then apply it with $F(s,t) = 1_{s \le t} \frac{\partial}{\partial t}f(s,t)$.

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