Ito integral exponent of Brownian Motion

Question

Let $F=e^{B_t}$, find such a process $f_s$ that $F=E[F]+\int_0^tf_sdB_s$.

I have started with $$e^{B_t}=E[e^{B_t}]+\int_0^tf_sdB_s$$ We know that $E[e^{B_t}]=e^{t/2}$ and it gives us $$e^{B_t}=e^{t/2}+\int_0^tf_sdB_s$$ I should probably use Ito Formula at this moment but I don't know how to apply this. Do you have any ideas?

Answer 1

Use ito's lemma on a function $f(x,t) = e^{x-\frac{t}{2}}$ where $x(t) = B_t$