How can I get the integral process with its quadratic variation?

Question

If I know that the quadratic variation of a stochastic integral process is $t^3$, how can I find this process.
I tried to do it with the Itô's formula,
$[X]_{t}=\int_{0}^{t} \sigma_{s}^{2} d s$ then $\sigma^2=\frac{s^2}{3}$
but I didn't know how to continue, Any advice or help

Answer 1

The quadratic variation process of the process $\int_0^t f(s)dW_s, 0 \geq t$ is $\int_0^t f^2(s)ds, 0 \geq t,$ so you need a function $f$ such that $\int_0^t f^2(s)ds=t^3.$ This is $f(s)=\sqrt{3}s.$