# Simulating AR(1) Process with Initial and Final Condition

Suppose I have the following stationary $$AR(1)$$ process:

$$y_{t}=\alpha+\rho y_{t-1} + u_{t}$$

where $$u_{t} \sim \mathbb{N}(0,\sigma^{2})$$, with $$\sigma^{2}$$ known. Suppose I have an initial condition $$y_{0}$$ and terminal condition $$y_{T}$$ and I would like to simulate my process for the periods in the interim, i.e $$t = 2,\dots,T-1$$. Can someone tell me what is the right distribution from which I should draw the $$u_{t}$$ if I want to impose both the initial and the terminal condition?

• Is that so? I am not sure about that. If I start from $y_{1}$ and simulate forward drawing $u_{t} \sim \mathcal{N}(0,\sigma^{2})$, $y_{T} \neq y_{T}^{SIM}$ which is a condition I want to impose. Jan 29 '20 at 15:55
• I edited the question a little bit. The crucial thing is that I want to condition both on the initial and the terminal conditions. Jan 29 '20 at 15:58
• Have you considered drawing from the (multivariate normal) conditional distribution of $Y_2,\ldots,Y_{T-1}|{Y_1,Y_T}$? Jan 30 '20 at 11:34

However, you could look for discrete versions of the Brownian Bridge process, a continuous-time process that can be parametrized so as to simultaneously satisfy an initial and a final condition. A possible result would be an auto-regressive process like the one you describe, but with time-varying coefficients: in particular both the auto-regressive coefficient $$\rho$$ and the variance $$\sigma$$ need to be zero at time $$T$$.