# 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.

• 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