I am trying to understand how to take the limit of the following Markov chain taking values in $\mathbb{R}^d$, i.e. how to formulate the continuous time version:
$x_i = \sqrt{1-\beta_i}x_{i-1} + \sqrt{\beta_i} z_{i-1},\quad i=1,\dots,N,$
where $z_{i-1} \sim \mathcal{N}(0,I)$, and $\beta_i$ are positive parameters.
I know the solution already, its:
$dx = -\frac12 \beta(t) x dt + \sqrt{\beta(t)} dW$
However, I am struggling to derive this limiting behavior for $N \to \infty$.
Here is my naive approach:
Let $\Delta t = \frac{1}{N}$. We simply reindex the chain through $t\in \{0,1,\dots, \frac{N-1}{N} \}$ and consider
$x_{t+\Delta t} = \sqrt{1-\beta_{t+\Delta t}}x_{t} + \sqrt{\beta_{t+\Delta t}} z_{t} $
However, this will not lead to the desired result for $\Delta t \to 0$. In the original derivation, which can be found in Appendix B of the paper Score-Based Generative Modeling through Stochastic Differential Equations, they do the following:
They rewrite the chain as
$x_i = \sqrt{1-\frac{N\beta_i}{N} } x_{i-1} + \sqrt{\frac{N\beta_i}{N}} z_{i-1},\quad i=1,\dots,N,$,
define $\beta(t) := \lim_{N\to \infty} N \beta_i$, and thus for $t\in \{0,1,\dots, \frac{N-1}{N} \}$,
$x_{t+\Delta t} = \sqrt{1-\beta_{t+\Delta t} \Delta t}x_{t} + \sqrt{\beta_{t+\Delta t}\Delta t} z_{t} $
which indeed leads to the desired result for $\Delta t \to 0$ after taylor expanding $\sqrt{1-x}$ for $x\to 0$.
I find the definition of $\beta(t)$ unintuitive. However, this is crucial for the appearance of $\Delta t$ and therefore the right limiting behavior. Could someone walk me through the reasoning?
In another example, they describe the limiting behaviour of
$x_i = x_{i-1} + \sqrt{\sigma_i^2-\sigma_{i-1}^2} z_{i-1}$
as follows:
$x_{t+\Delta t} = x_t + \sqrt{\sigma_{t+\Delta t}^2-\sigma_{t}^2}z_{t}$
hence, they simply replace every index with $t$. Why is this possible here, but not in the first case where this naive approach fails?
Any help is very much appreciated, thank you.
First consider that $ΔW(t_i)=(W(t_i)-W(t_{i-1}))=\sqrt{Δt}z_{i-1}$, which suggests that $\beta_i=\beta(t_i)Δt$. After that $$ \sqrt{1-\beta_i}=\sqrt{1-\beta(t_i)Δt}=1-\frac12\beta(t_i)Δt+O(Δt^2). $$ There is always some arbitrariness when converting a discrete process into some hopefully related continuous process. To hit the discrete process more exactly with the continuous process some correction terms involving $Δt$ might be necessary.
For a weakly related ODE example see Question about Euler's Method and the SIR epidemic model using a spreadsheet, in some way reversing the difference from the Euler method to the midpoint method to get an equation with an exact solution that follows closely the Euler points.