Integral representation $B_T^3$

I have to find a $F_t$ such that $B_T^3=E[B_T^3]+\int_0^T F_t dB_t$. I have shown by ito formula that $B_T^3=\int_0^T 3 B_s^2 dB_s+\int_0^T 3 B_s ds$. Could you please help me?

Solution 1:

First of all, note that $\mathbb{E}(B_T^3)=0$, i.e. we have to find $F_t$ such that

$$B_T^3 = \int_0^T F_s \, dB_s. \tag{1}$$

It follows from Itô's formula that

$$B_t^3 = 3 \int_0^t B_s^2 \, dB_s + 3 \int_0^t B_s \, ds \tag{2}$$

(please compare this with what you wrote in your question; there are so many typos in there that I'm really not sure whether you know what you are doing). The first term on the right-hand side looks already pretty good, but we have to rewrite the second one. By the integration by parts formula we have

$$t B_t = \int_0^t B_s \, ds + \int_0^t s \, dB_s,$$

i.e.

$$\int_0^t B_s \, ds = t B_t - \int_0^t s \, dB_s = t \int_0^t \, dB_s - \int_0^t s \, dB_s. \tag{3}$$

Plugging this into $(2)$ and evaluating at $t=T$, we get

$$B_T^3 = 3 \int_0^T B_s^2 \, dB_s + 3 \left( T \int_0^T \, dB_s - \int_0^T s \, dB_s \right) = 3\int_0^T (B_s^2+T-s) \, dB_s.$$