Hey guys, I am having troubles with the following transformation. I have:
\begin{align*}
Y_t^k=\int_0^t(t-s)^{\alpha-1}h(s)ds+\int_0^t (t-s)^{\alpha-1}\int_0^s(s-u)^{-\alpha}\sigma_{n_k}(X_u^{n_k})dB_u^{n_k}ds
\end{align*}
and must show that this is equivalent to
\begin{align*}
Y_t^k=\int_0^t(t-s)^{\alpha-1}h(s)ds+ c_{\alpha}\int_0^{t}\sigma_{n_k}(X_u^{n_k})dB_u^{n_k},
\end{align*}
where \begin{align*}c_\alpha =\int_0^1 (1-r)^{\alpha-1}r^{-\alpha}dr.\end{align*} My ansatz looks as follows:
Substitute r=s/t to get
\begin{align*}
Y_t^k&=\int_0^t(t-s)^{\alpha-1}h(s)ds+\int_0^1 (t-rt)^{\alpha-1}\int_0^{rt}(rt-u)^{-\alpha}\sigma_{n_k}(X_u^{n_k})dB_u^{n_k}tdr\\
&=\int_0^t(t-s)^{\alpha-1}h(s)ds+\int_0^1 (t-rt)^{\alpha-1}\int_0^{rt}(rt-u)^{-\alpha}\sigma_{n_k}(X_u^{n_k})dB_u^{n_k}(\frac{1}{t})^{\alpha-1-\alpha}dr\\
&=\int_0^t(t-s)^{\alpha-1}h(s)ds+\int_0^1 (1-r)^{\alpha-1}\int_0^{rt}(r-\frac{u}{t})^{-\alpha}\sigma_{n_k}(X_u^{n_k})dB_u^{n_k}dr.
\end{align*}
But now I am clueless how to transform the incorrect term to the thing in $c_\alpha$ and the correct integral bound. I am very thankful for ideas!
\begin{align*}
Y_t^k=\int_0^t(t-s)^{\alpha-1}h(s)ds+\int_0^t (t-s)^{\alpha-1}\int_0^s(s-u)^{-\alpha}\sigma_{n_k}(X_u^{n_k})dB_u^{n_k}ds
\end{align*}
and must show that this is equivalent to
\begin{align*}
Y_t^k=\int_0^t(t-s)^{\alpha-1}h(s)ds+ c_{\alpha}\int_0^{t}\sigma_{n_k}(X_u^{n_k})dB_u^{n_k},
\end{align*}
where \begin{align*}c_\alpha =\int_0^1 (1-r)^{\alpha-1}r^{-\alpha}dr.\end{align*} My ansatz looks as follows:
Substitute r=s/t to get
\begin{align*}
Y_t^k&=\int_0^t(t-s)^{\alpha-1}h(s)ds+\int_0^1 (t-rt)^{\alpha-1}\int_0^{rt}(rt-u)^{-\alpha}\sigma_{n_k}(X_u^{n_k})dB_u^{n_k}tdr\\
&=\int_0^t(t-s)^{\alpha-1}h(s)ds+\int_0^1 (t-rt)^{\alpha-1}\int_0^{rt}(rt-u)^{-\alpha}\sigma_{n_k}(X_u^{n_k})dB_u^{n_k}(\frac{1}{t})^{\alpha-1-\alpha}dr\\
&=\int_0^t(t-s)^{\alpha-1}h(s)ds+\int_0^1 (1-r)^{\alpha-1}\int_0^{rt}(r-\frac{u}{t})^{-\alpha}\sigma_{n_k}(X_u^{n_k})dB_u^{n_k}dr.
\end{align*}
But now I am clueless how to transform the incorrect term to the thing in $c_\alpha$ and the correct integral bound. I am very thankful for ideas!