B.M. using Levys Representation Theorem

Question

I have the following Question. Let $W(t)= \int_0^t f(t) \, dX(t)$ be a stochastic process and let $X(t)=\ln(t+1)$. Then how i have to choose $f(t)$ such that $W(t)$ becomes a B.M. My attempt goes like this: $$ dW(t)=f(t)\,dX(t)$$ hence we get $$d\langle W\rangle_t=dW(t)\cdot dW(t)=f(t)^2\cdot dX(t)\cdot dX(t) = f(t)^2\cdot\frac{1}{(t+1)^2} \, dt = dt$$ Hence $$f(t)=t+1$$ Am i right?

Answer 1

If you haven't done a typo, you are wrong. As you wrote it, your $X$ is not random an of bounded variation. As your $f$ is also not random, $W$ could'nt be random and thus no Brownian Motion. Also, the quadratic variation of $X$ would be zero.