# Applyin Itô's formula in function of quadratic variation

I am learning some basic stochastic calculus and came across the following exercise:

Consider a local martingale $$M$$ with continuous trajectories. Let $$Z_t = \exp(M_t −0.5[M]_t)$$. Show that Z satisfies the equation $$Z_t = Z_0 +\int_0^tZ_sdMs$$ Is $$Z$$ a local martingale? Compute $$Z$$ for the case where $$M_t = σB_t$$ for a standard Brownian motion $$B$$.

What confuses me, is the presence of quadratic variation in the function of a martingale. I've tried to apply standard Ito's formula:

$$f(X) = f(X_0)+\int f^\prime(X)\,dX + \frac{1}{2}\int f^{\prime\prime}(X)\,d[X]$$

But I am not sure how to treat the $$[M]_t$$ element in the function. Could someone help? Thanks!

I assume that the Ito formula that you wrote refers to a semi-martingale $$X_t$$. A semi-martingale is the sum of a local martingale and a process with bounded variation: $$X_t = M_t + A_t$$. But the process with bounded variation is irrelevant in the computation of the quadratic variation, namely $$[X]_t = [M]_t$$. So in your case, if $$X_t = \sigma B_t - \frac12 \sigma^2 t,$$ then $$[X]_t = \sigma^2 [B]_t = \sigma^2 t.$$