So I was reading and found that the following was given as an example of a complex valued local martingale: $M_t = e^{\int_0^t f(\omega,s)dB_s - \frac 12\int_0^tf(\omega,s)^2ds}$ with $f(\omega,s) = u(\omega,s) + iv(\omega,s)$ where $u,v \in \mathcal L^2_{LOC}[0,T]$.
However when I try to confirm this I get stuck since when you substitute in the definition of f, I get $e^{\int_0^t u(s)dB_s -\frac 12\int_0^tu(s)^2ds + \frac 12\int_0^tv(s)^2ds}e^{i[\int_0^tv(s)dB_s + \int_0^tu(s)v(s)ds]}$. But the definition of a complex valued local martingale is $Z_t = U_t + iV_t$ where $U_t$ and $V_t$ are local martingales and I can't think of a way to turn this into a sum besides taking the logarthm of both sides to get $logM_t = [\int_0^t u(s)dB_s -\frac 12\int_0^tu(s)^2ds + \frac 12\int_0^tv(s)^2ds] + i[\int_0^tv(s)dB_s + \int_0^tu(s)v(s)ds]$, but I don't know if this helps since at best I'd show that $logM_t$ is a complex valued local martingale.
Any help is appreciated!
If we apply Itô's formula to the function $f(x,y) := e^{x-y}$ and the (complex-valued $2$-dimensional) Itô process
$$\begin{pmatrix} X_t \\ Y_t \end{pmatrix} := \begin{pmatrix} \int_0^t f(s) \, dB_s \\ \frac{1}{2} \int_0^t f(s)^2 \, ds \end{pmatrix}$$
we find that
$$\begin{align*} e^{X_t-Y_t}-1 &= \int_0^t e^{X_s-Y_s} \, dX_s - \int_0^t e^{X_s-Y_s} \, dY_s + \frac{1}{2} \int_0^t e^{X_s-Y_s} \, d\langle X \rangle_s + \frac{1}{2} \int_0^t e^{X_s-Y_s} \, d\langle Y \rangle_s \tag{1}\end{align*}$$
where
$$\begin{align*} \langle X \rangle_t = \int_0^t f(s)^2 \, ds \qquad \quad \langle Y \rangle_t = 0 \tag{2} \end{align*}$$
denotes the quadratic variation of the process $X$ and $Y$, respectively. By definition, $M_t = e^{X_t-Y_t}$. Therefore, we see from $(1)$ and $(2)$:
$$M_t-1 = \int_0^t M_s f(s) \, dB_s.$$
Since the right-hand side is a stochastic integral, it is in particular a local martingale and therefore the claim follows.
Remark If $(M_t)_{t \geq 0}$ is a continuous martingale, then $(e^{M_t-\frac{1}{2}\langle M \rangle_t})_{t \geq 0}$ is a local martingale. It is called stochastic exponential.