G'day, today I have managed to derive $W^n_{t}$ in Integral form. However I am missing some critical rules to compute $W^1_{t}, W^2_{t}, W^3_{t}, W^{4}_{t}$.
$W^n_{t} = \frac{n(n-1)}{2} \int_0^{t} W^{n-2}_{s}ds+ n\int_0^{t}W^{n-1}_{s}dW_s$
a) n = 1 $W^1_{t} = \frac{1(1-1)}{2} \int_0^{t} W^{1-2}_{s}ds+ 1\int_0^{t}W^{1-1}_{s}dW_s = 0 \int_0^{t} W^{-1}_{s}ds+ 1\int_0^{t}W^{0}_{s}dW_s = 1 \int_0^{t}W_sdW_s =???$
b) n = 2 $W^2_{t} = \frac{2(2-1)}{2} \int_0^{t} W^{2-2}_{s}ds+ 2\int_0^{t}W^{2-1}_{s}dW_s = 1 \int_0^{t} W^{0}_{s}ds+ 2\int_0^{t}W^{1}_{s}dW_s = t + 2\int_0^{t}W^{1}_{s}dW_s = ???$
Can I assume here that $t = 1 \int_0^{t} W^{0}_{s}ds$ ? and if so why ?
This is obtained exactly in the same way as in your previous question. Notice that $W^0_t = 1$, so that in the case where $n=1$ you get $W_t = \int_0^t dW_s = W_t - W_0 = W_t$. What you got for $n=2$ is correct.