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Under the Jarrow-Yildirim model, the nominal short rate $r_n$, the real rate $r_r$ and index $I$ are modelled according to the following stochastic differential equations under the Martingale measure $\mathbb{Q}$:

\begin{eqnarray*} d r_n (t) & = & [\theta_n (t) - a_n r_n (t)] d t + \sigma_n d W_n (t)\\ d r_r (t) & = & [\theta_r (t) - \sigma_r \sigma_i \rho_{r, I} - a_r r_r (t)] d t + \sigma_r d W_r (t)\\ \frac{d I (t)}{I (t)} & = & [r_n (t) - r_r (t)] d t + \sigma_I d W_I (t) \end{eqnarray*}

where $W_n$, $W_r$ and $W_I$ are correlated Brownian motions, $\rho_{r, I}$ is the correlation between $W_r$ and $W_I$, $\theta_n$ and $\theta_r$ are drift functions used to match the term structure, $a_n$ and $a_r$, are constant mean reversion speeds and $\sigma_n$, $\sigma_r$ and $\sigma_I$ are constant volatility parameters.

What is the purpose of the $\sigma_I$ parameter? I am confused since my naiive intuition tells me that an instantaneous percentage increment of the inflation index should be fully determined by the inflation short rate: $r_n (t) - r_r (t)$. I.e.

\begin{eqnarray*} \frac{d I (t)}{I (t)} & = & [r_n (t) - r_r (t)] d t \end{eqnarray*}

Any pointers as to where my logic is failing would be appreciated.

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    $\begingroup$ Isn’t it because the nominal-real interest rate differential gives the expected inflation, not the actual realized inflation. The latter has some randomness on top. $\endgroup$ – dm63 Mar 24 at 7:37
  • $\begingroup$ Thanks dm63. I think that you are right. Many thanks $\endgroup$ – Nick Deguillaume Mar 24 at 8:40

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