Let $(\Omega,\mathcal{F},(\mathcal{F}_t:t\ge{0}),P)$ be a stochastic basis, $B=(B_t:t\ge{0})$ the Brownian Motion and $\forall{n=1,\dots,{N}}$ we have $f_n\in{L}^\infty({{\Omega},\mathcal{F}_n,P})$. I need to show that: $$\prod_{n=1}^N{\exp[f_n\cdot(B_{T_{n+1}}-B_{T_n})]}\in{L}^1({{\Omega},\mathcal{F}_{T_{N+1}},P}),$$ which means: $$\mathbb{E}\{\prod_{n=1}^N{\exp[f_n\cdot(B_{T_{n+1}}-B_{T_n})]}\}<\infty.$$
I think I should use the tower property, conditioning to $\mathcal{F}_{T_{n+1}}$, combined with the "taking out what is known" property; but this require at least that $\forall{n=1,...,N}$ we have: $$\exp(f_n(B_{T_{n+1}}-B_{T_n}))\in{L}^1({{\Omega},\mathcal{F}_{T_{n+1}},P}).$$ Since I don't know the distribution of $f_n$, I can't compute the expected value. I even thought about the moment generating function of the normal distribution, but $f_n$ is not a constant. Where can I start from?