Computation of $\int_0^t \frac 1 {M(s)^2} d\langle M\rangle(s)$

Question

Let $M$ be a positive continuous local martingale, and define $N$ to be the Ito integral of $\frac 1 M$ with respect to $M$. I am to compute the semi-martingale decomposition of $\log(M)$ in terms of $N$ and "related quantities for $N$".

I can write down Ito's formula:

$$\log(M(t)) = \int_0^t\frac1M(s)dM(s)-\frac12\int_0^t \frac 1 {M(s)^2} d\langle M\rangle(s)$$

And of course the first term is $N(t)$, but I can't think of anything intelligent to say about the second term. How can I compute or in any way rewrite an integral with respect to $\langle M\rangle$?

Answer 1

Hint: If $Z_t := \int_0^t f(s) \, dM(s)$, then

$$\langle Z\rangle_t = \int_0^t f(s)^2 \, d\langle M \rangle_s$$