What's the variance of the following stochastic integral and is it weakly stationary?

Question

The stochastic integral is defined as $$u_t = \int_{t-1}^t e^{-\kappa(t-s)}\int_0^s e^{-c(s-r)} \, dW(r) \, ds.$$ where $W(t)$ is a standard Brownian motion, $\kappa$ and $c$ are both positive.

I know this integral can be viewed as $$u_t = \int_{t-1}^t e^{-\kappa(t-s)}J_c(s) \, ds,$$ where $J_c(s) = \int_0^s e^{-c(s-r)} \, dW(r)$, is an Ornstein-Uhlenbeck process. But how do we handle this double integral and use Ito's isometry to get the variance of it?

Further, does this integral admit a Wold representation, that is, $$u_t = \sum_{j = 0}^\infty F_j \varepsilon_{t-j},$$ where $\varepsilon_t \sim \mathrm{iid}(0, \sigma^2)$.

Here is my calculations, which may be wrong, I am not so sure...

$$\begin{align} Var(u_t) &= E(u_t^2) = E\left(\int_{t-1}^{t} \int_{0}^{s}e^{-\kappa(t-s)-c(s-r)}dW(r) ds\right)^2 \\ &= E\left(\int_{0}^{t} \int_{r}^{t}e^{-\kappa(t-s)-c(s-r)}ds dW(r) \right)^2 \\ &= \int_{0}^{t} \left(\int_{r}^{t}e^{-\kappa(t-s)-c(s-r)}ds\right)^2 dr \\ &= \int_{0}^{t} e^{-2\kappa t + 2cr} \left(\int_{r}^{t}e^{(\kappa-c)s}ds\right)^2 dr \\ &= \dfrac{1}{(\kappa-c)^2}\int_{0}^{t} e^{-2\kappa t + 2cr} \left(e^{(\kappa-c)t}- e^{(\kappa-c)r} \right)^2 dr \\ &= \dfrac{1}{(\kappa-c)^2}\left(\int_{0}^{t} e^{-2c(t-r)} dr + \int_{0}^{t} e^{-2\kappa(t-r)} dr - 2\int_{0}^{t} e^{-(\kappa+c)(t-r)} dr\right)\\ &= \dfrac{1}{(\kappa-c)^2}\left(\dfrac{1-e^{-2ct}}{2c} + \dfrac{1-e^{-2\kappa t}}{2\kappa} - 2\dfrac{1-e^{-(\kappa+c)t}}{\kappa+c}\right) \end{align}$$ which indicates $u_t$ exhibits heteroskedasiticity. But still, I don't know if it is weakly stationary.

Answer 1

It seems that you consider $u_t$ as being a discrete time (DT) process for $t=1$, $2$, $\dots$ yet the integral makes sense for any real positive $t$, then defining a continuous time (CT) process $u(t)$ for $t \geq 0$. Roughly speaking, $u(t)$ results from applying two linear filters to a CT Gaussian white noise, the two impulse responses $f(t)$ and $g(t)$ being $$ f(t) := e^{-ct} \,1_{[0,\,\infty)}(t), \quad g(t) := e^{-Kt} \,1_{[0,\,1]}(t). $$ The composition of the two linear filters is equivalent to a linear filter with its impulse response $h(t)$ given by the convolution of $f(t)$ and $g(t)$ $$ h(t) = \int f(s) g(t-s) \, \text{d}s. $$ However, some complications are due to the initial conditions and the related transient effects.

The Ornstein-Uhlenbeck (OU) process $J(s)=\int_0^s f(s-r) \, \text{d}W(r)$ is not weakly stationary due to its initial condition $J(0) = 0$, but it has a stationary distribution and can be considered as stationary for large $t$. Indeed, a stationary OU, say $J^\star(s)$, would be obtained by starting from the stationary distribution at $s=0$ or equivalently by starting from any distribution in the remote past. This could be written as $$ J^\star(s) := \int_{-\infty}^s e^{-c(s-r)} \, \text{d}W(r) = \int_{0}^\infty e^{-cr'} \, \text{d}W(s-r'), $$ and $J(s) - J^\star(s) = e^{-cs} X$ for some Gaussian r.v. $X$, so $J(s)$ is nearly equal to $J^\star(s)$ unless $s$ is small. In other words, $J(s)$ and $J^\star(s)$ are equal up to an initial warm-up period.

Your derivation of the variance of $u(t)$ is correct and, as you noted, the variance is not constant but tends to a constant with exponential speed. Thus none of the DT and CT processes $u_t$ and $u(t)$ can be weakly stationary, hence none has a (DT or CT) Wold representation. However, except for a warm-up period, $u(t)$ is close to the stationary processes $u^\star(t)$ with a CT Wold representation $$ u^\star(t) = \int_{0}^\infty h(s) \, \text{d}W(t-s), $$ so $u^\star(t)$ results from applying the linear filter with impulse response $h(t)$ to a white noise process. The variance of $u^\star(t)$ is given by the isometry $$ \text{Var}[u^\star(t)] = \int_0^\infty h(t)^2 \,\text{d}t. $$ After a warm-up period, the DT process $u_t$ is equal to the DT process $u^\star_t$ which is stationary and hence has a DT Wold representation. Finding this DT Wold representation does not seem to be straightforward, but the CT representation can be used instead.

While the DT Wold representation is very popular, the CT version is much underrated. See the book of D.R. Cox and H.D. Miller or the web document by John H. Cochrane Continuous Time Linear Models for linear filtering white noise.