Non-stationary OU-processes

The most general, time dependent linear ODE

$$\frac{\mathrm{d}u_t}{\mathrm{d}t} = A_t\,u_t + f_t$$ (86)

where $A_t\in \mathbb{R}^{n\times n}$ and $f_t\in \mathbb{R}^n$. The ansatz $u_t=\int_0^t\alpha_{ts}f_s\,\mathrm{d}s+\exp\left(\int_0^t A_s\mathrm{d} s\right) u_0$ leads to the (formal) solution

$$u_t = \int_0^te^{\int_s^t A_\tau\mathrm{d}\tau}f_s\,\mathrm{d}s+e^{\int_0^t A_s\,\mathrm{d} s}u_0\quad .$$ (87)

With this result it is easy to guess a solution for the non-stationary OU-process

$$\mathrm{d}u_t = (A_t u_t+a_t)\,\mathrm{d}t+\Sigma\,\mathrm{d}W$$ (88)

as

$$u_t = \int_0^t e^{\int_s^t A_\tau\mathrm{d}\tau}(a_s\,\math... ...gma\,\mathrm{d}W_s) + e^{\int_0^t A_s\mathrm{d} s}u_0\quad .$$ (89)