I have encountered a problem in a paper called Volatility is rough by Jim Gatheral et al.
A stationary fractional Ornstein–Uhlenbeck process ($X_t$) is defined as the stationary solution of the stochastic differential equation $$dX_t = \nu dW^H_t− \alpha (X_t − m)dt$$ where $m ∈ \mathbb{R}$ and $\nu$ and $\alpha$ are positive parameters (see Cheridito et al. 2003). As for usual Ornstein–Uhlenbeck processes, there is an explicit form for the solution which is given by $$X_t = \nu \int^t_{−\infty}e^{−\alpha (t−s)}dW^H_t+ m \qquad \quad (3.3) $$ Here, the stochastic integral with respect to fBm is simply a pathwise Riemann–Stieltjes integral, see again Cheridito et al. (2003).
Proposition 3.1 Let $W_H$ be a fBm and $X^\alpha$ defined by (3.3) for a given $\alpha > 0$. As \alpha tends to zero, $$E\left[\sup_{t\in[0,T]}|X^\alpha_t− X^\alpha_0− \nu W^H_t|\right]\to 0$$
Then the proof goes like this:
Starting from equation (3.3) and applying integration by parts, we get $$X^\alpha_t = \nu W^H_t −\int_t^{−\infty}\nu \alpha e^{−\alpha(t−s)}W^H_s ds + m$$
and some proofs that I know; it is just this part that I don't understand.
So my question is: How do I even perform integration by parts for the fractional Ornstein-Uhlenbeck process in this proof. I don't see anyway how to obtain the answer by integration by parts. I'd appreciate if someone can answer or link me to somewhere that has methods/answers!
You can look at the Proposition A.1 in link, they provide a more rigourous proof.
Essentially, we apply the integration by part to $(e^{\alpha t}W_t^H)$ \begin{equation} d(e^{\alpha t}W_t^H) = \alpha e^{\alpha t}W_t^Hdt +e^{\alpha t}dW_t^H \end{equation} Then integrate \begin{equation} e^{\alpha t}W_t^H - e^{\alpha b}W_b^H = \int_b^t \alpha e^{\alpha s}W_s^Hds + \int_b^te^{\alpha s}dW_s^H\end{equation} So we have : $\int_b^te^{\alpha s}dW_s^H = e^{\alpha t}W_t^H - e^{\alpha b}W_b^H - \int_b^t \alpha e^{\alpha s}W_s^Hds$. Replacing the latter in $(3.3)$ by taking $b\to-\infty$, you will find the answer.