# What is augmented data when simulating stochastic differential equations using Gibbs Sampler?

I am reading this paper on Bayesian Estimation of CIR Model.

Basically, it is about estimating parameters using Bayesian inference.

It estimates this stochastic differential equation: $$dy(t)=\{ \alpha-\beta y(t)\}dt+ \sigma \sqrt{y(t)}dB(t)$$ where $B(t)$ is standard Brownian motion.

by using this approximation: $$y(t+ {\Delta}^{+})=y(t)+\{\alpha-\beta y(t) \}{\Delta}^{+}+\sigma \sqrt{y(t)} {\epsilon}_{t}$$ ${\epsilon}_{t} \tilde{\ }N(0, {\Delta}^{+})$

My question is: Let $Y=({y}_{1},...,{y}_{T})$ denote observation data and
${Y}^{*}=(y_{1}^{*},...,y_{T-1}^{*})$ be AUGMENTED data, where $y_{*}^{t}=\{ y_{t,1}^{*},..., y_{t,M}^{*} \}$

What is augmented data?

I see that $y_{1}^{*}$ has elements $y_{1,1}^{*},..., y_{1,M}^{*}$. Is this what's so called augmented data? Why do we need this?

This seems like a finance concept I am missing.

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This is not a finance concept. Augmented data is related to Bayesian inference. It's essentially a way to improve maximum likelihood estimation from incomplete data. For details see the article "The calculation of posterior distributions by data augmentation" by Martin A. Tanner and Wing Hung Wong (it's referred in the paper you are reading).

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Thanks for pointing that out, I should've digged into that paper. – Mike Apr 23 '12 at 19:22
What data augmentation is, is basically filling in missing values. Why are we filling in missing values/data augmenting? What's the point here? – Mike Apr 23 '12 at 19:24
@Mike It seems that it is used to model latent data for the Treasury bills yields, i.e. latent points are placed between each consecutive pair of observations. – Alexey Kalmykov Apr 23 '12 at 20:41
thanks, that makes a lot of sense now. Just to confirm, in this case, since it's weekly data, so I am filling in 4 other days of the week. – Mike Apr 23 '12 at 23:11
But, M=20 not 4, in the paper, page 6(page # is 276) bottom paragraph. btw, what is $\Delta$ with a cross sign on the top (that same paragraph), is that a plus sign? – Mike Apr 23 '12 at 23:49