Short Rate Dynamics

The short rate under the Merton model has the following dynamics:

$$ d r_{t}= \theta dt + \sigma d w_{t}$$

Integrating from s to t, we get

$$ \int_{s}^{t}{d r_{u}}= \theta \int_{s}^{t}{ du} + \int_{s}^{t}{\sigma d w_{u}}$$ $$ r_{t}=r_{s} + \theta \left( t-s\right)+ \sigma\int_{s}^{t}{d w_{u}}$$

Which is Gaussian with mean and variance given by,

$$ E \left[ r_{t} \mid r_{s} \right]=r_{s} + \theta \left( t-s\right) $$ $$ V \left[ r_{t} \mid r_{s} \right]= V \left[ \sigma\int_{s}^{t}{d w_{u}} \right] = {\sigma}^{2} \left( t-s \right) $$