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Consider the following mean-square differential equation: $Y^{'}_t+\alpha Y_{t} = X_{t}$ m.s. with $ Y_{0}=0$ where $X_t$ is a Gaussian random process with $\mu_{X}(t)=0$ and $R_X(\tau)=a^2\delta(\tau)$.

Find a partial differential equation for crosscorrelation function between ${Y_{t}; t\in R}$ and ${X_{t}; t\in R}$ with appropriate boundary conditions.

Is $Y_t$ wide sense stationary?

I tried squaring both sides and taking the expectation but I'm not sure that's what the question is asking for. Any help?

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There is a problem of definition here.

The only way I know to make sense of your equation is to replace it by the stochastic differential equation $\mathrm dY_t=\alpha Y_t\mathrm dt+\mathrm dW_t$ where $(W_t)_{t\geqslant0}$ is a standard Brownian motion. But then the autocorrelation of $Y_t$ and $R_t$ does not make sense since $(R_t)_{t\geqslant0}$ is not defined. So, the first thing to do is to explain what you mean by the equation $Y'_t=\alpha Y_t+R_t$, mean square.

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