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I'm looking for the solution of the following SDE:

$p_{t+1} = a \cdot p_t + b + c \cdot w_t$,

where $a, b$ and $c$ are constant scalars, $w_t \sim N(0, \sigma^2)$ is a white noise and $p_t$ is discrete-time 1D stochastic process.

The solution is probably something very basic, but unfortunately I'm new to SDEs and Ito calculation. Can anyone suggest a solution?

Dimka Kopitkov
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  • Suppose that $p_0$ is constant. Then $p_t$ will be normally distributed with mean and variance solving:

    $\mu_{t+1} = a \mu_t + b$ $\sigma_{t+1}^2 = a^2 \sigma_t^2 + c^2 \sigma^2$. These are first order recurrence relation which you can solve.

     – fGDu94 Jan 04 '20 at 18:29
  • This is what I was looking for. Thanks a lot! – Dimka Kopitkov Jan 05 '20 at 19:32

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