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Suppose we have the random walk with drift model

$$x_t = \delta+x_{t-1}+w_t$$ for $t = 1, 2, ...$ with initial condition $x_0=0$ and where $w_t$ is white noise. The constant $\delta$ is called the drift, when $\delta$=0, the equation above is simply a random walk. $t$ is the time. We may rewrite this as the cumulative sum of white noise variates:

$$x_t = \delta t+\sum_{j=1}^tw_j$$

What I have tried to show this:

$$\delta t+\sum_{j=1}^tw_j=\delta+x_{t-1}+w_t \\ x_{t-1}=\delta(t-1)+\sum_{j=1}^tw_j-w_t$$

If I remember summation tricks properly, then

$$\sum_{j=1}^t w_j-w_t = \sum_{j=1}^{t+1}w_j$$

So we have

$$x_{t-1} = \delta(t-1)+\sum_{j=1}^{t+1}w_j$$

Dollar Tune-bill
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  • What are your hypothesis and what are your aim here ? – EDX Jul 16 '22 at 12:19
  • @EDX My objective was to show the cumulative sum follows from the first equation. By the looks of it, I am fairly uncertain with the summation switch, as I have doubts it might be $t-1$ on the top instead. So clarification on this is really helpful! – Dollar Tune-bill Jul 16 '22 at 12:26
  • What is $\delta$ ? Please if you can set all the name and characteristics of your parameters I could help you better – EDX Jul 16 '22 at 12:27
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    @EDX I have updated it in the post – Dollar Tune-bill Jul 16 '22 at 12:29
  • If $\delta$ is a constant independant on $t$ we can write $x_t-x_{t-1}=\delta + w_t$ then telescoping the left $x_n -x_0 =(n+1)\delta + \sum_{i=0}^{n+1} w_i$ – EDX Jul 16 '22 at 12:31

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