Applying Itô Lemma to Stochastic Differential Equation

Question

Given the stochastic differential equation:

$dZ_t = -Z_t \theta_t dB_t$

for an adapted process $\theta_t$ and Brownian Motion $B_t$, how exactly do I apply Itô's Lemma to obtain:

$Z_t = exp(- \int_{0}^{t}\theta_u dB_u - \frac{1}{2}\int_{0}^{t}\theta_u^2 du)$ ?

Let $Z_0 = 1$.

Answer 1

As usual, you consider $Y_t=\ln Z_t$ to get $$ dY_t=\frac{dZ_t}{Z_t}-\frac12\frac{d\langle Z\rangle_t}{Z_t^2}=-θ_tdB_t-\frac12θ_t^2dt $$ which you now can integrate.