Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I want to show that the following expression is a local martingale: $$M_t=V(X_{t\wedge T})+\int_0^{t\wedge T}f(X_s)ds,$$ where $T$ is some stopping time (there are more conditions, but they are not important for my question).

Where $X$ is some process, and V is sufficiently nice for Ito to be applied. My problem is how to get around the problem, that the integral on the right ranges from $0$ to $t\wedge T$. My first inkling was to apply Ito to $X_{t\wedge T}$ first, but it does not seem to be working.

share|improve this question
add comment

1 Answer 1

$$\mathrm dM_t=V'(X_t)\,[t\lt T]\,\mathrm dX_t+\frac12V''(X_t)\,[t\lt T]\,\mathrm d\langle X\rangle_t+f(X_t)\,[t\lt T]\mathrm dt$$

share|improve this answer
    
I have no doubt that this is correct, but could you clarify which version of Ito you are applying? My question was, essentially, how to make it rigorous. –  Tom Artiom Fiodorov May 17 '12 at 22:38
    
I have no doubt either. I know only one version of Itô equality. –  Did Sep 19 '12 at 18:19
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.